Average Error: 0.6 → 0.6
Time: 20.7s
Precision: 64
Internal Precision: 576
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\sqrt{e^{a}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

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Target

Original0.6
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied *-un-lft-identity0.6

    \[\leadsto \frac{e^{a}}{\color{blue}{1 \cdot \left(e^{a} + e^{b}\right)}}\]

  4. Applied add-sqr-sqrt0.6

    \[\leadsto \frac{\color{blue}{\sqrt{e^{a}} \cdot \sqrt{e^{a}}}}{1 \cdot \left(e^{a} + e^{b}\right)}\]

  5. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{\sqrt{e^{a}}}{1} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}}\]

  6. Applied simplify0.6

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

Runtime

Time bar (total: 20.7s)Debug logProfile

herbie shell --seed 2018170 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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