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 r101377 = a;
        double r101378 = exp(r101377);
        double r101379 = b;
        double r101380 = exp(r101379);
        double r101381 = r101378 + r101380;
        double r101382 = r101378 / r101381;
        return r101382;
}
↓
double f(double a, double b) {
        double r101383 = a;
        double r101384 = exp(r101383);
        double r101385 = b;
        double r101386 = exp(r101385);
        double r101387 = r101384 + r101386;
        double r101388 = log(r101387);
        double r101389 = r101383 - r101388;
        double r101390 = exp(r101389);
        return r101390;
}

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 2019208 +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))))