Average Error: 0.5 → 0.4
Time: 9.5s
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 r1308322 = a;
        double r1308323 = exp(r1308322);
        double r1308324 = b;
        double r1308325 = exp(r1308324);
        double r1308326 = r1308323 + r1308325;
        double r1308327 = r1308323 / r1308326;
        return r1308327;
}
↓
double f(double a, double b) {
        double r1308328 = a;
        double r1308329 = exp(r1308328);
        double r1308330 = b;
        double r1308331 = exp(r1308330);
        double r1308332 = r1308329 + r1308331;
        double r1308333 = log(r1308332);
        double r1308334 = r1308328 - r1308333;
        double r1308335 = exp(r1308334);
        return r1308335;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.5
Target0.0
Herbie0.4
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.5

    \[\frac{e^{a}}{e^{a} + e^{b}}\]
  2. Using strategy rm

  3. Applied add-exp-log0.5

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

  4. Applied div-exp0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019154 +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))))