Quotient of sum of exps

Average Error: 0.6 → 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 = 2.3826293734222903e+48
  2. b = -1.352832018891749e-22

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 r100535 = a;
        double r100536 = exp(r100535);
        double r100537 = b;
        double r100538 = exp(r100537);
        double r100539 = r100536 + r100538;
        double r100540 = r100536 / r100539;
        return r100540;
}

↓

double f(double a, double b) {
        double r100541 = a;
        double r100542 = exp(r100541);
        double r100543 = b;
        double r100544 = exp(r100543);
        double r100545 = r100542 + r100544;
        double r100546 = log(r100545);
        double r100547 = r100541 - r100546;
        double r100548 = exp(r100547);
        return r100548;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.6
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 0.6

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

3. Applied add-exp-log 0.6

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

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 2020042 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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