Quotient of sum of exps

Average Error: 0.6 → 0.6

Time: 7.5s

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

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

↓

\[\frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}\]

C

double f(double a, double b) {
        double r200387 = a;
        double r200388 = exp(r200387);
        double r200389 = b;
        double r200390 = exp(r200389);
        double r200391 = r200388 + r200390;
        double r200392 = r200388 / r200391;
        return r200392;
}

↓

double f(double a, double b) {
        double r200393 = a;
        double r200394 = exp(r200393);
        double r200395 = sqrt(r200394);
        double r200396 = b;
        double r200397 = exp(r200396);
        double r200398 = fma(r200395, r200395, r200397);
        double r200399 = r200394 / r200398;
        return r200399;
}

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 add-sqr-sqrt 0.6

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

4. Applied fma-def 0.6

   \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}}\]

5. Final simplification 0.6

   \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}\]

Reproduce

herbie shell --seed 2020047 +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))))