Average Error: 0.8 → 0.7
Time: 11.8s
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 r3791870 = a;
        double r3791871 = exp(r3791870);
        double r3791872 = b;
        double r3791873 = exp(r3791872);
        double r3791874 = r3791871 + r3791873;
        double r3791875 = r3791871 / r3791874;
        return r3791875;
}
↓
double f(double a, double b) {
        double r3791876 = a;
        double r3791877 = exp(r3791876);
        double r3791878 = b;
        double r3791879 = exp(r3791878);
        double r3791880 = r3791877 + r3791879;
        double r3791881 = log(r3791880);
        double r3791882 = r3791876 - r3791881;
        double r3791883 = exp(r3791882);
        return r3791883;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.8
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.8

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

  3. Applied add-exp-log0.8

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

  4. Applied div-exp0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019134 +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))))