Average Error: 0.6 → 1.0
Time: 39.0s
Precision: 64
Internal Precision: 320
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\sqrt[3]{{\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.0
Herbie1.0
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-cbrt-cube0.7

    \[\leadsto \frac{e^{a}}{\color{blue}{\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}}}\]

  4. Applied add-cbrt-cube0.7

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(e^{a} \cdot e^{a}\right) \cdot e^{a}}}}{\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}}\]

  5. Applied cbrt-undiv1.1

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(e^{a} \cdot e^{a}\right) \cdot e^{a}}{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}}}\]

  6. Simplified1.0

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}}}\]

  7. Final simplification1.0

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

Runtime

Time bar (total: 39.0s)Debug logProfile

herbie shell --seed 2018217 +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))))