Average Error: 0.6 → 0.5
Time: 19.9s
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 r5586360 = a;
        double r5586361 = exp(r5586360);
        double r5586362 = b;
        double r5586363 = exp(r5586362);
        double r5586364 = r5586361 + r5586363;
        double r5586365 = r5586361 / r5586364;
        return r5586365;
}
↓
double f(double a, double b) {
        double r5586366 = a;
        double r5586367 = exp(r5586366);
        double r5586368 = b;
        double r5586369 = exp(r5586368);
        double r5586370 = r5586367 + r5586369;
        double r5586371 = log(r5586370);
        double r5586372 = r5586366 - r5586371;
        double r5586373 = exp(r5586372);
        return r5586373;
}

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 2019158 +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))))