Average Error: 0.7 → 0.6
Time: 7.8s
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 r1293100 = a;
        double r1293101 = exp(r1293100);
        double r1293102 = b;
        double r1293103 = exp(r1293102);
        double r1293104 = r1293101 + r1293103;
        double r1293105 = r1293101 / r1293104;
        return r1293105;
}
↓
double f(double a, double b) {
        double r1293106 = a;
        double r1293107 = exp(r1293106);
        double r1293108 = b;
        double r1293109 = exp(r1293108);
        double r1293110 = r1293107 + r1293109;
        double r1293111 = log(r1293110);
        double r1293112 = r1293106 - r1293111;
        double r1293113 = exp(r1293112);
        return r1293113;
}

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

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 2019153 +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))))