Average Error: 0.6 → 0.5
Time: 13.5s
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 r103277 = a;
        double r103278 = exp(r103277);
        double r103279 = b;
        double r103280 = exp(r103279);
        double r103281 = r103278 + r103280;
        double r103282 = r103278 / r103281;
        return r103282;
}
↓
double f(double a, double b) {
        double r103283 = a;
        double r103284 = exp(r103283);
        double r103285 = b;
        double r103286 = exp(r103285);
        double r103287 = r103284 + r103286;
        double r103288 = log(r103287);
        double r103289 = r103283 - r103288;
        double r103290 = exp(r103289);
        return r103290;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.0
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-exp-log0.6

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

  4. Applied div-exp0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019195 +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))))