Average Error: 0.5 → 1.1

Time: 24.4s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

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Out

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Target

Original	0.5
Target	0.0
Herbie	1.1

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

Derivation

Initial program 0.5
\[\frac{e^{a}}{e^{a} + e^{b}}\]
Initial simplification0.5
\[\leadsto \frac{e^{a}}{e^{a} + e^{b}}\]
Taylor expanded around -inf 0.5
\[\leadsto \frac{e^{a}}{\color{blue}{e^{b} + e^{a}}}\]
Using strategy rm
Applied add-sqr-sqrt1.1
\[\leadsto \frac{e^{a}}{\color{blue}{\sqrt{e^{b} + e^{a}} \cdot \sqrt{e^{b} + e^{a}}}}\]
Applied add-sqr-sqrt1.1
\[\leadsto \frac{\color{blue}{\sqrt{e^{a}} \cdot \sqrt{e^{a}}}}{\sqrt{e^{b} + e^{a}} \cdot \sqrt{e^{b} + e^{a}}}\]
Applied times-frac1.1
\[\leadsto \color{blue}{\frac{\sqrt{e^{a}}}{\sqrt{e^{b} + e^{a}}} \cdot \frac{\sqrt{e^{a}}}{\sqrt{e^{b} + e^{a}}}}\]
Final simplification1.1
\[\leadsto \frac{\sqrt{e^{a}}}{\sqrt{e^{a} + e^{b}}} \cdot \frac{\sqrt{e^{a}}}{\sqrt{e^{a} + e^{b}}}\]

Runtime

herbie shell --seed 2018273 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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