Average Error: 0.7 → 0.7
Time: 16.8s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{e^{a} + e^{b}}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

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Target

Original0.7
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  2. Initial simplification0.7

    \[\leadsto \frac{e^{a}}{e^{a} + e^{b}}\]
  3. Using strategy rm

  4. Applied add-exp-log0.7

    \[\leadsto \frac{e^{a}}{\color{blue}{e^{\log \left(e^{a} + e^{b}\right)}}}\]

  5. Applied div-exp0.5

    \[\leadsto \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}}\]

  6. Taylor expanded around -inf 0.5

    \[\leadsto \color{blue}{e^{a - \log \left(e^{b} + e^{a}\right)}}\]

  7. Simplified0.7

    \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}}\]

  8. Final simplification0.7

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

Runtime

Time bar (total: 16.8s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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