Average Error: 0.6 → 0.5
Time: 10.2s
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 r5464639 = a;
        double r5464640 = exp(r5464639);
        double r5464641 = b;
        double r5464642 = exp(r5464641);
        double r5464643 = r5464640 + r5464642;
        double r5464644 = r5464640 / r5464643;
        return r5464644;
}
↓
double f(double a, double b) {
        double r5464645 = a;
        double r5464646 = exp(r5464645);
        double r5464647 = b;
        double r5464648 = exp(r5464647);
        double r5464649 = r5464646 + r5464648;
        double r5464650 = log(r5464649);
        double r5464651 = r5464645 - r5464650;
        double r5464652 = exp(r5464651);
        return r5464652;
}

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

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2019142 +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))))