Average Error: 0.7 → 0.5
Time: 3.8s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(e^{a} + e^{b}\right)}
double f(double a, double b) {
        double r166522 = a;
        double r166523 = exp(r166522);
        double r166524 = b;
        double r166525 = exp(r166524);
        double r166526 = r166523 + r166525;
        double r166527 = r166523 / r166526;
        return r166527;
}
↓
double f(double a, double b) {
        double r166528 = a;
        double r166529 = exp(r166528);
        double r166530 = b;
        double r166531 = exp(r166530);
        double r166532 = r166529 + r166531;
        double r166533 = log(r166532);
        double r166534 = r166528 - r166533;
        double r166535 = exp(r166534);
        return r166535;
}

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
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

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-exp0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020003 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1 (+ 1 (exp (- b a))))

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