Average Error: 0.6 → 0.6
Time: 3.1s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left(e^{e^{a - \log \left(e^{a} + e^{b}\right)}}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left(e^{e^{a - \log \left(e^{a} + e^{b}\right)}}\right)
double f(double a, double b) {
        double r119863 = a;
        double r119864 = exp(r119863);
        double r119865 = b;
        double r119866 = exp(r119865);
        double r119867 = r119864 + r119866;
        double r119868 = r119864 / r119867;
        return r119868;
}
↓
double f(double a, double b) {
        double r119869 = a;
        double r119870 = exp(r119869);
        double r119871 = b;
        double r119872 = exp(r119871);
        double r119873 = r119870 + r119872;
        double r119874 = log(r119873);
        double r119875 = r119869 - r119874;
        double r119876 = exp(r119875);
        double r119877 = exp(r119876);
        double r119878 = log(r119877);
        return r119878;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 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. Using strategy rm

  6. Applied add-log-exp0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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