Average Error: 0.7 → 0.6
Time: 13.2s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(e^{a} + e^{b}\right)}
double f(double a, double b) {
        double r4829151 = a;
        double r4829152 = exp(r4829151);
        double r4829153 = b;
        double r4829154 = exp(r4829153);
        double r4829155 = r4829152 + r4829154;
        double r4829156 = r4829152 / r4829155;
        return r4829156;
}
↓
double f(double a, double b) {
        double r4829157 = a;
        double r4829158 = exp(r4829157);
        double r4829159 = b;
        double r4829160 = exp(r4829159);
        double r4829161 = r4829158 + r4829160;
        double r4829162 = log(r4829161);
        double r4829163 = r4829157 - r4829162;
        double r4829164 = exp(r4829163);
        return r4829164;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-exp-log0.7

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

  4. Applied div-exp0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019130 +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))))