Average Error: 0.7 → 0.6
Time: 13.5s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(\mathsf{fma}\left(\sqrt[3]{e^{a}} \cdot \sqrt[3]{e^{a}}, \sqrt[3]{e^{a}}, e^{b}\right)\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(\mathsf{fma}\left(\sqrt[3]{e^{a}} \cdot \sqrt[3]{e^{a}}, \sqrt[3]{e^{a}}, e^{b}\right)\right)}
double f(double a, double b) {
        double r4474086 = a;
        double r4474087 = exp(r4474086);
        double r4474088 = b;
        double r4474089 = exp(r4474088);
        double r4474090 = r4474087 + r4474089;
        double r4474091 = r4474087 / r4474090;
        return r4474091;
}
↓
double f(double a, double b) {
        double r4474092 = a;
        double r4474093 = exp(r4474092);
        double r4474094 = cbrt(r4474093);
        double r4474095 = r4474094 * r4474094;
        double r4474096 = b;
        double r4474097 = exp(r4474096);
        double r4474098 = fma(r4474095, r4474094, r4474097);
        double r4474099 = log(r4474098);
        double r4474100 = r4474092 - r4474099;
        double r4474101 = exp(r4474100);
        return r4474101;
}

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

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

  7. Applied fma-def0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019146 +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))))