Quotient of sum of exps

Average Error: 0.7 → 0.8

Time: 11.0s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = 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 r92751 = a;
        double r92752 = exp(r92751);
        double r92753 = b;
        double r92754 = exp(r92753);
        double r92755 = r92752 + r92754;
        double r92756 = r92752 / r92755;
        return r92756;
}

↓

double f(double a, double b) {
        double r92757 = 1.0;
        double r92758 = a;
        double r92759 = exp(r92758);
        double r92760 = b;
        double r92761 = exp(r92760);
        double r92762 = r92759 + r92761;
        double r92763 = r92762 / r92759;
        double r92764 = r92757 / r92763;
        return r92764;
}

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 +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))))