Average Error: 0.7 → 0.6
Time: 24.8s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]
double f(double a, double b) {
        double r15554847 = a;
        double r15554848 = exp(r15554847);
        double r15554849 = b;
        double r15554850 = exp(r15554849);
        double r15554851 = r15554848 + r15554850;
        double r15554852 = r15554848 / r15554851;
        return r15554852;
}
↓
double f(double a, double b) {
        double r15554853 = a;
        double r15554854 = exp(r15554853);
        double r15554855 = b;
        double r15554856 = exp(r15554855);
        double r15554857 = r15554854 + r15554856;
        double r15554858 = log(r15554857);
        double r15554859 = r15554853 - r15554858;
        double r15554860 = exp(r15554859);
        return r15554860;
}


\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(e^{a} + e^{b}\right)}

Error

Bits error versus a

Bits error versus b

Target

Original0.7
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

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-exp0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019101 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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