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 r20075 = a;
        double r20076 = exp(r20075);
        double r20077 = b;
        double r20078 = exp(r20077);
        double r20079 = r20076 + r20078;
        double r20080 = r20076 / r20079;
        return r20080;
}

↓

double f(double a, double b) {
        double r20081 = 1.0;
        double r20082 = a;
        double r20083 = exp(r20082);
        double r20084 = b;
        double r20085 = exp(r20084);
        double r20086 = r20083 + r20085;
        double r20087 = r20086 / r20083;
        double r20088 = r20081 / r20087;
        return r20088;
}

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))))