Average Error: 0.7 → 0.7
Time: 10.5s
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 r85815 = a;
        double r85816 = exp(r85815);
        double r85817 = b;
        double r85818 = exp(r85817);
        double r85819 = r85816 + r85818;
        double r85820 = r85816 / r85819;
        return r85820;
}
↓
double f(double a, double b) {
        double r85821 = a;
        double r85822 = exp(r85821);
        double r85823 = sqrt(r85822);
        double r85824 = b;
        double r85825 = exp(r85824);
        double r85826 = fma(r85823, r85823, r85825);
        double r85827 = r85822 / r85826;
        return r85827;
}

Error

Bits error versus a

Bits error versus b

Target

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

Derivation

  1. Initial program 0.7

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

  3. Applied add-sqr-sqrt0.7

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

  4. Applied fma-def0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))