Average Error: 0.6 → 0.6
Time: 16.6s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a} \cdot \frac{1}{e^{b} + e^{a}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a} \cdot \frac{1}{e^{b} + e^{a}}
double f(double a, double b) {
        double r120667 = a;
        double r120668 = exp(r120667);
        double r120669 = b;
        double r120670 = exp(r120669);
        double r120671 = r120668 + r120670;
        double r120672 = r120668 / r120671;
        return r120672;
}
↓
double f(double a, double b) {
        double r120673 = a;
        double r120674 = exp(r120673);
        double r120675 = 1.0;
        double r120676 = b;
        double r120677 = exp(r120676);
        double r120678 = r120677 + r120674;
        double r120679 = r120675 / r120678;
        double r120680 = r120674 * r120679;
        return r120680;
}

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

Derivation

  1. Initial program 0.6

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

  3. Applied 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. Using strategy rm

  6. Applied sub-neg0.5

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

  7. Applied exp-sum0.6

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

  8. Simplified0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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