Quotient of sum of exps

Average Error: 0.6 → 0.6

Time: 12.8s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = -2.3686796720332175e+206
  2. b = -5.202796025710632e+132

Math

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

↓

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

C

double f(double a, double b) {
        double r101873 = a;
        double r101874 = exp(r101873);
        double r101875 = b;
        double r101876 = exp(r101875);
        double r101877 = r101874 + r101876;
        double r101878 = r101874 / r101877;
        return r101878;
}

↓

double f(double a, double b) {
        double r101879 = a;
        double r101880 = exp(r101879);
        double r101881 = 1.0;
        double r101882 = b;
        double r101883 = exp(r101882);
        double r101884 = r101880 + r101883;
        double r101885 = r101881 / r101884;
        double r101886 = r101880 * r101885;
        return r101886;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.6
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 0.6

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

3. Applied div-inv 0.6

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

4. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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