Average Error: 0.6 → 0.6
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 r108130 = a;
        double r108131 = exp(r108130);
        double r108132 = b;
        double r108133 = exp(r108132);
        double r108134 = r108131 + r108133;
        double r108135 = r108131 / r108134;
        return r108135;
}
↓
double f(double a, double b) {
        double r108136 = a;
        double r108137 = exp(r108136);
        double r108138 = sqrt(r108137);
        double r108139 = b;
        double r108140 = exp(r108139);
        double r108141 = fma(r108138, r108138, r108140);
        double r108142 = r108137 / r108141;
        return r108142;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.6
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-sqr-sqrt0.6

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

  4. Applied fma-def0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 +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))))