Average Error: 0.7 → 1.0
Time: 12.6s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}} \cdot \left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}} \cdot \left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}
double f(double a, double b) {
        double r4754396 = a;
        double r4754397 = exp(r4754396);
        double r4754398 = b;
        double r4754399 = exp(r4754398);
        double r4754400 = r4754397 + r4754399;
        double r4754401 = r4754397 / r4754400;
        return r4754401;
}
↓
double f(double a, double b) {
        double r4754402 = a;
        double r4754403 = exp(r4754402);
        double r4754404 = b;
        double r4754405 = exp(r4754404);
        double r4754406 = r4754403 + r4754405;
        double r4754407 = r4754403 / r4754406;
        double r4754408 = r4754407 * r4754407;
        double r4754409 = r4754407 * r4754408;
        double r4754410 = cbrt(r4754409);
        return r4754410;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie1.0
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-cbrt-cube0.8

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

  4. Applied add-cbrt-cube0.8

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

  5. Applied cbrt-undiv1.1

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

  6. Simplified1.0

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

  7. Final simplification1.0

    \[\leadsto \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}} \cdot \left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}\]

Reproduce

herbie shell --seed 2019164 +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))))