Average Error: 0.8 → 0.7
Time: 2.5s
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 r195206 = a;
        double r195207 = exp(r195206);
        double r195208 = b;
        double r195209 = exp(r195208);
        double r195210 = r195207 + r195209;
        double r195211 = r195207 / r195210;
        return r195211;
}
↓
double f(double a, double b) {
        double r195212 = a;
        double r195213 = exp(r195212);
        double r195214 = b;
        double r195215 = exp(r195214);
        double r195216 = r195213 + r195215;
        double r195217 = log(r195216);
        double r195218 = r195212 - r195217;
        double r195219 = exp(r195218);
        return r195219;
}

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

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-exp0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2020064 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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