Average Error: 0.7 → 1.1
Time: 13.4s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a} + e^{b}\right)\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{e^{a}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a} + e^{b}\right)\right)}
double f(double a, double b) {
        double r93189 = a;
        double r93190 = exp(r93189);
        double r93191 = b;
        double r93192 = exp(r93191);
        double r93193 = r93190 + r93192;
        double r93194 = r93190 / r93193;
        return r93194;
}
↓
double f(double a, double b) {
        double r93195 = a;
        double r93196 = exp(r93195);
        double r93197 = b;
        double r93198 = exp(r93197);
        double r93199 = r93196 + r93198;
        double r93200 = log1p(r93199);
        double r93201 = expm1(r93200);
        double r93202 = r93196 / r93201;
        return r93202;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie1.1
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied expm1-log1p-u1.1

    \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a} + e^{b}\right)\right)}}\]

  4. Final simplification1.1

    \[\leadsto \frac{e^{a}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a} + e^{b}\right)\right)}\]

Reproduce

herbie shell --seed 2019212 +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))))