Quotient of sum of exps

Average Error: 0.6 → 0.5

Time: 13.4s

Precision: 64

• could not determine a ground truth for program body

  1. a = 3.9540437202843566e+32
  2. b = -1.7431458063348651e-140

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 r77786 = a;
        double r77787 = exp(r77786);
        double r77788 = b;
        double r77789 = exp(r77788);
        double r77790 = r77787 + r77789;
        double r77791 = r77787 / r77790;
        return r77791;
}

↓

double f(double a, double b) {
        double r77792 = a;
        double r77793 = exp(r77792);
        double r77794 = b;
        double r77795 = exp(r77794);
        double r77796 = r77793 + r77795;
        double r77797 = log(r77796);
        double r77798 = r77792 - r77797;
        double r77799 = exp(r77798);
        return r77799;
}

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 2019195 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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