Average Error: 0.6 → 0.5
Time: 14.3s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(e^{a} + e^{b}\right)}
double f(double a, double b) {
        double r5522011 = a;
        double r5522012 = exp(r5522011);
        double r5522013 = b;
        double r5522014 = exp(r5522013);
        double r5522015 = r5522012 + r5522014;
        double r5522016 = r5522012 / r5522015;
        return r5522016;
}
↓
double f(double a, double b) {
        double r5522017 = a;
        double r5522018 = exp(r5522017);
        double r5522019 = b;
        double r5522020 = exp(r5522019);
        double r5522021 = r5522018 + r5522020;
        double r5522022 = log(r5522021);
        double r5522023 = r5522017 - r5522022;
        double r5522024 = exp(r5522023);
        return r5522024;
}

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
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-exp-log0.6

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

  4. Applied div-exp0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019141 +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))))