Quotient of sum of exps

Average Error: 0.6 → 0.6

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

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

↓

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

C

double f(double a, double b) {
        double r101048 = a;
        double r101049 = exp(r101048);
        double r101050 = b;
        double r101051 = exp(r101050);
        double r101052 = r101049 + r101051;
        double r101053 = r101049 / r101052;
        return r101053;
}

↓

double f(double a, double b) {
        double r101054 = exp(1.0);
        double r101055 = a;
        double r101056 = exp(r101055);
        double r101057 = b;
        double r101058 = exp(r101057);
        double r101059 = r101056 + r101058;
        double r101060 = log(r101059);
        double r101061 = r101055 - r101060;
        double r101062 = pow(r101054, r101061);
        return r101062;
}

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

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

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