Quotient of sum of exps

Average Error: 0.7 → 0.5

Time: 9.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. a = 7.130456307924188e+301
  2. b = -1.4864914298183363e-09

Math

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

↓

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

C

double f(double a, double b) {
        double r146713 = a;
        double r146714 = exp(r146713);
        double r146715 = b;
        double r146716 = exp(r146715);
        double r146717 = r146714 + r146716;
        double r146718 = r146714 / r146717;
        return r146718;
}

↓

double f(double a, double b) {
        double r146719 = a;
        double r146720 = exp(r146719);
        double r146721 = b;
        double r146722 = exp(r146721);
        double r146723 = r146720 + r146722;
        double r146724 = log(r146723);
        double r146725 = r146719 - r146724;
        double r146726 = exp(r146725);
        return r146726;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.7
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 0.7

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

3. Applied add-exp-log 0.7

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

4. Applied div-exp 0.5

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

5. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019351 +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))))