Average Error: 0.6 → 0.5
Time: 15.4s
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 r109621 = a;
        double r109622 = exp(r109621);
        double r109623 = b;
        double r109624 = exp(r109623);
        double r109625 = r109622 + r109624;
        double r109626 = r109622 / r109625;
        return r109626;
}
↓
double f(double a, double b) {
        double r109627 = a;
        double r109628 = exp(r109627);
        double r109629 = b;
        double r109630 = exp(r109629);
        double r109631 = r109628 + r109630;
        double r109632 = log(r109631);
        double r109633 = r109627 - r109632;
        double r109634 = exp(r109633);
        return r109634;
}

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 2019323 +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))))