Average Error: 0.6 → 0.6

Time: 7.3s

Precision: 64

Internal Precision: 320

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

↓

\[\frac{e^{a}}{(\left(\sqrt[3]{e^{b}} \cdot \sqrt[3]{e^{b}}\right) \cdot \left(\sqrt[3]{e^{b}}\right) + \left(e^{a}\right))_*}\]

Error

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

Target

Original	0.6
Target	0.0
Herbie	0.6

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

Derivation

Initial program 0.6
\[\frac{e^{a}}{e^{a} + e^{b}}\]
Initial simplification0.6
\[\leadsto \frac{e^{a}}{e^{a} + e^{b}}\]
Taylor expanded around inf 0.6
\[\leadsto \color{blue}{\frac{e^{a}}{e^{b} + e^{a}}}\]
Using strategy rm
Applied add-cube-cbrt0.6
\[\leadsto \frac{e^{a}}{\color{blue}{\left(\sqrt[3]{e^{b}} \cdot \sqrt[3]{e^{b}}\right) \cdot \sqrt[3]{e^{b}}} + e^{a}}\]
Applied fma-def0.6
\[\leadsto \frac{e^{a}}{\color{blue}{(\left(\sqrt[3]{e^{b}} \cdot \sqrt[3]{e^{b}}\right) \cdot \left(\sqrt[3]{e^{b}}\right) + \left(e^{a}\right))_*}}\]
Final simplification0.6
\[\leadsto \frac{e^{a}}{(\left(\sqrt[3]{e^{b}} \cdot \sqrt[3]{e^{b}}\right) \cdot \left(\sqrt[3]{e^{b}}\right) + \left(e^{a}\right))_*}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.6	0.6	0.6	0.0	0%

herbie shell --seed 2018286 +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))))