Quotient of sum of exps

Average Error: 0.8 → 1.1

Time: 10.9s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = 2.4112655861716263e+231
  2. b = -5.670122129476186e-07

Math

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

↓

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

C

double f(double a, double b) {
        double r134736 = a;
        double r134737 = exp(r134736);
        double r134738 = b;
        double r134739 = exp(r134738);
        double r134740 = r134737 + r134739;
        double r134741 = r134737 / r134740;
        return r134741;
}

↓

double f(double a, double b) {
        double r134742 = a;
        double r134743 = exp(r134742);
        double r134744 = b;
        double r134745 = exp(r134744);
        double r134746 = r134743 + r134745;
        double r134747 = sqrt(r134746);
        double r134748 = r134743 / r134747;
        double r134749 = r134748 / r134747;
        return r134749;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.8
Target	0.0
Herbie	1.1

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

Derivation

1. Initial program 0.8

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

3. Applied add-sqr-sqrt 1.3

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

4. Applied associate-/r* 1.1

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

5. Final simplification 1.1

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

Reproduce

herbie shell --seed 2020043 +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))))