Average Error: 0.5 → 0.5

Time: 15.0s

Precision: 64

Internal Precision: 128

\[\frac{e^{a}}{e^{a} + e^{b}}\]

↓

\[\frac{e^{a}}{e^{\log \left((\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*\right)}}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

Target

Original	0.5
Target	0.0
Herbie	0.5

\[\frac{1}{1 + e^{b - a}}\]

Derivation

Initial program 0.5
\[\frac{e^{a}}{e^{a} + e^{b}}\]
Initial simplification0.5
\[\leadsto \frac{e^{a}}{e^{a} + e^{b}}\]
Using strategy rm
Applied add-sqr-sqrt0.5
\[\leadsto \frac{e^{a}}{\color{blue}{\sqrt{e^{a}} \cdot \sqrt{e^{a}}} + e^{b}}\]
Applied fma-def0.5
\[\leadsto \frac{e^{a}}{\color{blue}{(\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*}}\]
Using strategy rm
Applied add-exp-log0.5
\[\leadsto \frac{e^{a}}{\color{blue}{e^{\log \left((\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*\right)}}}\]
Final simplification0.5
\[\leadsto \frac{e^{a}}{e^{\log \left((\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*\right)}}\]

Runtime

herbie shell --seed 2018348 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

  :herbie-target
  (/ 1 (+ 1 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))