Average Error: 0.7 → 0.9
Time: 5.4s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left(\mathsf{fma}\left(e^{b}, e^{b} - e^{a}, {\left(e^{a}\right)}^{2}\right)\right)}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left(\mathsf{fma}\left(e^{b}, e^{b} - e^{a}, {\left(e^{a}\right)}^{2}\right)\right)}\right)
double f(double a, double b) {
        double r152600 = a;
        double r152601 = exp(r152600);
        double r152602 = b;
        double r152603 = exp(r152602);
        double r152604 = r152601 + r152603;
        double r152605 = r152601 / r152604;
        return r152605;
}
↓
double f(double a, double b) {
        double r152606 = a;
        double r152607 = exp(r152606);
        double r152608 = 3.0;
        double r152609 = pow(r152607, r152608);
        double r152610 = b;
        double r152611 = exp(r152610);
        double r152612 = pow(r152611, r152608);
        double r152613 = r152609 + r152612;
        double r152614 = r152607 / r152613;
        double r152615 = exp(r152614);
        double r152616 = r152611 - r152607;
        double r152617 = 2.0;
        double r152618 = pow(r152607, r152617);
        double r152619 = fma(r152611, r152616, r152618);
        double r152620 = pow(r152615, r152619);
        double r152621 = log(r152620);
        return r152621;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.7
Target0.0
Herbie0.9
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied flip3-+17.6

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

  4. Applied associate-/r/17.6

    \[\leadsto \color{blue}{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}} \cdot \left(e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)\right)}\]
  5. Using strategy rm

  6. Applied add-log-exp17.8

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

  7. Simplified0.9

    \[\leadsto \log \color{blue}{\left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left(\mathsf{fma}\left(e^{b}, e^{b} - e^{a}, {\left(e^{a}\right)}^{2}\right)\right)}\right)}\]

  8. Final simplification0.9

    \[\leadsto \log \left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left(\mathsf{fma}\left(e^{b}, e^{b} - e^{a}, {\left(e^{a}\right)}^{2}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2020046 +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))))