Average Error: 0.8 → 0.7
Time: 17.1s
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 r21118026 = a;
        double r21118027 = exp(r21118026);
        double r21118028 = b;
        double r21118029 = exp(r21118028);
        double r21118030 = r21118027 + r21118029;
        double r21118031 = r21118027 / r21118030;
        return r21118031;
}
↓
double f(double a, double b) {
        double r21118032 = a;
        double r21118033 = exp(r21118032);
        double r21118034 = b;
        double r21118035 = exp(r21118034);
        double r21118036 = r21118033 + r21118035;
        double r21118037 = log(r21118036);
        double r21118038 = r21118032 - r21118037;
        double r21118039 = exp(r21118038);
        return r21118039;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.8
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.8

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

  3. Applied add-exp-log0.8

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

  4. Applied div-exp0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019124 +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))))