Average Error: 0.8 → 0.6
Time: 2.2s
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 r146142 = a;
        double r146143 = exp(r146142);
        double r146144 = b;
        double r146145 = exp(r146144);
        double r146146 = r146143 + r146145;
        double r146147 = r146143 / r146146;
        return r146147;
}
↓
double f(double a, double b) {
        double r146148 = a;
        double r146149 = exp(r146148);
        double r146150 = b;
        double r146151 = exp(r146150);
        double r146152 = r146149 + r146151;
        double r146153 = log(r146152);
        double r146154 = r146148 - r146153;
        double r146155 = exp(r146154);
        return r146155;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.8
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.8

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

  3. Applied add-exp-log0.8

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

  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 2020035 +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))))