Average Error: 0.7 → 0.6
Time: 38.7s
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 r4128593 = a;
        double r4128594 = exp(r4128593);
        double r4128595 = b;
        double r4128596 = exp(r4128595);
        double r4128597 = r4128594 + r4128596;
        double r4128598 = r4128594 / r4128597;
        return r4128598;
}
↓
double f(double a, double b) {
        double r4128599 = a;
        double r4128600 = exp(r4128599);
        double r4128601 = b;
        double r4128602 = exp(r4128601);
        double r4128603 = r4128600 + r4128602;
        double r4128604 = log(r4128603);
        double r4128605 = r4128599 - r4128604;
        double r4128606 = exp(r4128605);
        return r4128606;
}


\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 +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))))