Quotient of sum of exps

Average Error: 0.7 → 0.8

Time: 7.6s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = 8.776596381696226e+279
  2. b = -3.0751430059816334e-208

Math

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

↓

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

C

double f(double a, double b) {
        double r40850 = a;
        double r40851 = exp(r40850);
        double r40852 = b;
        double r40853 = exp(r40852);
        double r40854 = r40851 + r40853;
        double r40855 = r40851 / r40854;
        return r40855;
}

↓

double f(double a, double b) {
        double r40856 = 1.0;
        double r40857 = a;
        double r40858 = exp(r40857);
        double r40859 = b;
        double r40860 = exp(r40859);
        double r40861 = r40858 + r40860;
        double r40862 = r40861 / r40858;
        double r40863 = r40856 / r40862;
        return r40863;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.7
Target	0.0
Herbie	0.8

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

Derivation

1. Initial program 0.7

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

3. Applied clear-num 0.8

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

4. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020045 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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