Average Error: 0.8 → 0.8
Time: 2.9s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}
double f(double a, double b) {
        double r125287 = a;
        double r125288 = exp(r125287);
        double r125289 = b;
        double r125290 = exp(r125289);
        double r125291 = r125288 + r125290;
        double r125292 = r125288 / r125291;
        return r125292;
}
↓
double f(double a, double b) {
        double r125293 = a;
        double r125294 = exp(r125293);
        double r125295 = sqrt(r125294);
        double r125296 = b;
        double r125297 = exp(r125296);
        double r125298 = fma(r125295, r125295, r125297);
        double r125299 = r125294 / r125298;
        return r125299;
}

Error

Bits error versus a

Bits error versus b

Target

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

Derivation

  1. Initial program 0.8

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

  3. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{e^{a}}{\color{blue}{\sqrt{e^{a}} \cdot \sqrt{e^{a}}} + e^{b}}\]

  4. Applied fma-def0.8

    \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}}\]

  5. Final simplification0.8

    \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}\]

Reproduce

herbie shell --seed 2020083 +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))))