Quotient of sum of exps

Average Error: 0.7 → 0.5

Time: 8.1s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = 7.130456307924188e+301
  2. b = -1.4864914298183363e-09

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 r143602 = a;
        double r143603 = exp(r143602);
        double r143604 = b;
        double r143605 = exp(r143604);
        double r143606 = r143603 + r143605;
        double r143607 = r143603 / r143606;
        return r143607;
}

↓

double f(double a, double b) {
        double r143608 = a;
        double r143609 = exp(r143608);
        double r143610 = b;
        double r143611 = exp(r143610);
        double r143612 = r143609 + r143611;
        double r143613 = log(r143612);
        double r143614 = r143608 - r143613;
        double r143615 = exp(r143614);
        return r143615;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.7
Target	0.0
Herbie	0.5

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

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 0.5

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

5. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019351 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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