Average Error: 0.7 → 0.5
Time: 3.8s
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 r110908 = a;
        double r110909 = exp(r110908);
        double r110910 = b;
        double r110911 = exp(r110910);
        double r110912 = r110909 + r110911;
        double r110913 = r110909 / r110912;
        return r110913;
}
↓
double f(double a, double b) {
        double r110914 = a;
        double r110915 = exp(r110914);
        double r110916 = b;
        double r110917 = exp(r110916);
        double r110918 = r110915 + r110917;
        double r110919 = log(r110918);
        double r110920 = r110914 - r110919;
        double r110921 = exp(r110920);
        return r110921;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-exp-log0.7

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

  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 2020003 +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))))