Average Error: 0.7 → 0.6
Time: 9.0s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{\mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, -\log \left(e^{a} + e^{b}\right)\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{\mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, -\log \left(e^{a} + e^{b}\right)\right)}
double f(double a, double b) {
        double r2247798 = a;
        double r2247799 = exp(r2247798);
        double r2247800 = b;
        double r2247801 = exp(r2247800);
        double r2247802 = r2247799 + r2247801;
        double r2247803 = r2247799 / r2247802;
        return r2247803;
}
↓
double f(double a, double b) {
        double r2247804 = a;
        double r2247805 = cbrt(r2247804);
        double r2247806 = r2247805 * r2247805;
        double r2247807 = exp(r2247804);
        double r2247808 = b;
        double r2247809 = exp(r2247808);
        double r2247810 = r2247807 + r2247809;
        double r2247811 = log(r2247810);
        double r2247812 = -r2247811;
        double r2247813 = fma(r2247806, r2247805, r2247812);
        double r2247814 = exp(r2247813);
        return r2247814;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.7
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-exp-log0.7

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

  4. Applied div-exp0.6

    \[\leadsto \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.6

    \[\leadsto e^{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}} - \log \left(e^{a} + e^{b}\right)}\]

  7. Applied fma-neg0.6

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, -\log \left(e^{a} + e^{b}\right)\right)}}\]

  8. Final simplification0.6

    \[\leadsto e^{\mathsf{fma}\left(\sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a}, -\log \left(e^{a} + e^{b}\right)\right)}\]

Reproduce

herbie shell --seed 2019156 +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))))