Quotient of sum of exps

Average Error: 0.6 → 0.6

Time: 6.8s

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

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

↓

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

C

double f(double a, double b) {
        double r88297 = a;
        double r88298 = exp(r88297);
        double r88299 = b;
        double r88300 = exp(r88299);
        double r88301 = r88298 + r88300;
        double r88302 = r88298 / r88301;
        return r88302;
}

↓

double f(double a, double b) {
        double r88303 = exp(1.0);
        double r88304 = a;
        double r88305 = exp(r88304);
        double r88306 = b;
        double r88307 = exp(r88306);
        double r88308 = r88305 + r88307;
        double r88309 = log(r88308);
        double r88310 = r88304 - r88309;
        double r88311 = pow(r88303, r88310);
        return r88311;
}

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-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. Using strategy rm

6. Applied *-un-lft-identity 0.5

   \[\leadsto e^{\color{blue}{1 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)}}\]

7. Applied exp-prod 0.6

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(a - \log \left(e^{a} + e^{b}\right)\right)}}\]

8. Simplified 0.6

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

9. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020042 +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))))