Average Error: 0.7 → 1.0
Time: 11.2s
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 r4793425 = a;
        double r4793426 = exp(r4793425);
        double r4793427 = b;
        double r4793428 = exp(r4793427);
        double r4793429 = r4793426 + r4793428;
        double r4793430 = r4793426 / r4793429;
        return r4793430;
}
↓
double f(double a, double b) {
        double r4793431 = a;
        double r4793432 = exp(r4793431);
        double r4793433 = b;
        double r4793434 = exp(r4793433);
        double r4793435 = r4793432 + r4793434;
        double r4793436 = log1p(r4793435);
        double r4793437 = expm1(r4793436);
        double r4793438 = r4793432 / r4793437;
        return r4793438;
}

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

Derivation

  1. Initial program 0.7

    \[\frac{e^{a}}{e^{a} + e^{b}}\]

  2. Taylor expanded around -inf 0.7

    \[\leadsto \frac{e^{a}}{\color{blue}{e^{b} + e^{a}}}\]
  3. Using strategy rm

  4. Applied expm1-log1p-u1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2019144 +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))))