Average Error: 0.6 → 1.4
Time: 18.9s
Precision: 64
Internal Precision: 320
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\sqrt[3]{e^{a}} \cdot \sqrt[3]{e^{a}}}{\sqrt[3]{e^{a} + e^{b}} \cdot \sqrt[3]{e^{a} + e^{b}}} \cdot \frac{\sqrt[3]{e^{a}}}{\sqrt[3]{e^{a} + e^{b}}}\]

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
Herbie1.4
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-cube-cbrt1.4

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

  4. Applied add-cube-cbrt1.4

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

    \[\leadsto \frac{\sqrt[3]{e^{a}} \cdot \sqrt[3]{e^{a}}}{\sqrt[3]{e^{a} + e^{b}} \cdot \sqrt[3]{e^{a} + e^{b}}} \cdot \frac{\sqrt[3]{e^{a}}}{\sqrt[3]{e^{a} + e^{b}}}\]

Runtime

Time bar (total: 18.9s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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