[PATCH] Fix error in exp in magnitude [2e-32,2e-28]

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[PATCH] Fix error in exp in magnitude [2e-32,2e-28]

Fabian Schriever
While testing the exp function we noticed some errors at the specified
magnitude. Within this range the exp function returns the input value +1
as an output. We chose to run a test of 1m exponentially spaced values
in the ranges [-2^-27,-2^-32] and [2^-32,2^-27] which showed 7603 and
3912 results with an error of >=0.5 ULP (compared with MPFR in 128 bit)
with the highest being 0.56 ULP and 0.53 ULP.

It's easy to fix by changing the magnitude at which the input value +1
is returned from <2^-28 to <2^-32 and using the polynomial instead. This
reduces the number of results with an error of >=0.5 ULP to 485 and 479
in above tests, all of which are exactly 0.5 ULP.

As we were already checking on exp we also took a look at expf. For expf
the magnitude where the input value +1 is returned can be increased from
<2^-28 to <2^-23 without accuracy loss for a slight performance
improvement. To ensure this was the correct value we tested all values
in the ranges [-2^-17,-2^-28] and [2^-28,2^-17] (~92.3m values each).
---
 newlib/libm/math/e_exp.c  | 2 +-
 newlib/libm/math/ef_exp.c | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/newlib/libm/math/e_exp.c b/newlib/libm/math/e_exp.c
index 81ea64dfb..d23b1162b 100644
--- a/newlib/libm/math/e_exp.c
+++ b/newlib/libm/math/e_exp.c
@@ -142,7 +142,7 @@ P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
     }
     x  = hi - lo;
  }
- else if(hx < 0x3e300000)  { /* when |x|<2**-28 */
+ else if(hx < 0x3df00000)  { /* when |x|<2**-32 */
     if(huge+x>one) return one+x;/* trigger inexact */
  }
 
diff --git a/newlib/libm/math/ef_exp.c b/newlib/libm/math/ef_exp.c
index e817370ac..fb3e2ffe6 100644
--- a/newlib/libm/math/ef_exp.c
+++ b/newlib/libm/math/ef_exp.c
@@ -77,7 +77,7 @@ P5   =  4.1381369442e-08; /* 0x3331bb4c */
     }
     x  = hi - lo;
  }
- else if(hx < 0x31800000)  { /* when |x|<2**-28 */
+ else if(hx < 0x34000000)  { /* when |x|<2**-23 */
     if(huge+x>one) return one+x;/* trigger inexact */
  }
 
--
2.24.1.windows.2


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Re: [PATCH] Fix error in exp in magnitude [2e-32,2e-28]

Corinna Vinschen
On Mar  6 15:46, Fabian Schriever wrote:

> While testing the exp function we noticed some errors at the specified
> magnitude. Within this range the exp function returns the input value +1
> as an output. We chose to run a test of 1m exponentially spaced values
> in the ranges [-2^-27,-2^-32] and [2^-32,2^-27] which showed 7603 and
> 3912 results with an error of >=0.5 ULP (compared with MPFR in 128 bit)
> with the highest being 0.56 ULP and 0.53 ULP.
>
> It's easy to fix by changing the magnitude at which the input value +1
> is returned from <2^-28 to <2^-32 and using the polynomial instead. This
> reduces the number of results with an error of >=0.5 ULP to 485 and 479
> in above tests, all of which are exactly 0.5 ULP.
>
> As we were already checking on exp we also took a look at expf. For expf
> the magnitude where the input value +1 is returned can be increased from
> <2^-28 to <2^-23 without accuracy loss for a slight performance
> improvement. To ensure this was the correct value we tested all values
> in the ranges [-2^-17,-2^-28] and [2^-28,2^-17] (~92.3m values each).
> ---
>  newlib/libm/math/e_exp.c  | 2 +-
>  newlib/libm/math/ef_exp.c | 2 +-
>  2 files changed, 2 insertions(+), 2 deletions(-)
>
> diff --git a/newlib/libm/math/e_exp.c b/newlib/libm/math/e_exp.c
> index 81ea64dfb..d23b1162b 100644
> --- a/newlib/libm/math/e_exp.c
> +++ b/newlib/libm/math/e_exp.c
> @@ -142,7 +142,7 @@ P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
>      }
>      x  = hi - lo;
>   }
> - else if(hx < 0x3e300000)  { /* when |x|<2**-28 */
> + else if(hx < 0x3df00000)  { /* when |x|<2**-32 */
>      if(huge+x>one) return one+x;/* trigger inexact */
>   }
>  
> diff --git a/newlib/libm/math/ef_exp.c b/newlib/libm/math/ef_exp.c
> index e817370ac..fb3e2ffe6 100644
> --- a/newlib/libm/math/ef_exp.c
> +++ b/newlib/libm/math/ef_exp.c
> @@ -77,7 +77,7 @@ P5   =  4.1381369442e-08; /* 0x3331bb4c */
>      }
>      x  = hi - lo;
>   }
> - else if(hx < 0x31800000)  { /* when |x|<2**-28 */
> + else if(hx < 0x34000000)  { /* when |x|<2**-23 */
>      if(huge+x>one) return one+x;/* trigger inexact */
>   }
>  
> --
> 2.24.1.windows.2
>
Pushed.


Thanks,
Corinna

--
Corinna Vinschen
Cygwin Maintainer
Red Hat

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