Average Error: 0.6 → 0.6
Time: 14.0s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{e^{a} + e^{b}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{e^{a}}{e^{a} + e^{b}}
double f(double a, double b) {
        double r77574 = a;
        double r77575 = exp(r77574);
        double r77576 = b;
        double r77577 = exp(r77576);
        double r77578 = r77575 + r77577;
        double r77579 = r77575 / r77578;
        return r77579;
}
↓
double f(double a, double b) {
        double r77580 = a;
        double r77581 = exp(r77580);
        double r77582 = b;
        double r77583 = exp(r77582);
        double r77584 = r77581 + r77583;
        double r77585 = r77581 / r77584;
        return r77585;
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 0.6

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

  3. Applied *-un-lft-identity0.6

    \[\leadsto \frac{e^{a}}{\color{blue}{1 \cdot \left(e^{a} + e^{b}\right)}}\]

  4. Applied *-un-lft-identity0.6

    \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{1 \cdot \left(e^{a} + e^{b}\right)}\]

  5. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{e^{a}}{e^{a} + e^{b}}}\]

  6. Simplified0.6

    \[\leadsto \color{blue}{1} \cdot \frac{e^{a}}{e^{a} + e^{b}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.7

    \[\leadsto 1 \cdot \color{blue}{\log \left(e^{\frac{e^{a}}{e^{a} + e^{b}}}\right)}\]

  9. Final simplification0.6

    \[\leadsto \frac{e^{a}}{e^{a} + e^{b}}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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