Average Error: 0.5 → 0.5
Time: 8.7s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{e^{a} + e^{b}}\]
double f(double a, double b) {
        double r18763146 = a;
        double r18763147 = exp(r18763146);
        double r18763148 = b;
        double r18763149 = exp(r18763148);
        double r18763150 = r18763147 + r18763149;
        double r18763151 = r18763147 / r18763150;
        return r18763151;
}
↓
double f(double a, double b) {
        double r18763152 = a;
        double r18763153 = exp(r18763152);
        double r18763154 = b;
        double r18763155 = exp(r18763154);
        double r18763156 = r18763153 + r18763155;
        double r18763157 = r18763153 / r18763156;
        return r18763157;
}


\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{e^{a}}{e^{a} + e^{b}}

Error

Bits error versus a

Bits error versus b

Target

Original0.5
Target0.0
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around inf 0.5

    \[\leadsto \frac{e^{a}}{\color{blue}{e^{b} + e^{a}}}\]

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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