Quotient of sum of exps

Average Error: 0.7 → 0.6

Time: 10.6s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = -1.1098270921310924e+244
  2. b = -2.6683066503381667e+258

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 r119831 = a;
        double r119832 = exp(r119831);
        double r119833 = b;
        double r119834 = exp(r119833);
        double r119835 = r119832 + r119834;
        double r119836 = r119832 / r119835;
        return r119836;
}

↓

double f(double a, double b) {
        double r119837 = a;
        double r119838 = exp(r119837);
        double r119839 = b;
        double r119840 = exp(r119839);
        double r119841 = r119838 + r119840;
        double r119842 = log(r119841);
        double r119843 = r119837 - r119842;
        double r119844 = exp(r119843);
        return r119844;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.7
Target	0.0
Herbie	0.6

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

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 0.6

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

5. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020046 +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))))