Average Error: 0.8 → 0.8
Time: 3.8s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\sqrt{e^{a}}}{\frac{e^{a} + e^{b}}{\sqrt{e^{a}}}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{\sqrt{e^{a}}}{\frac{e^{a} + e^{b}}{\sqrt{e^{a}}}}
double f(double a, double b) {
        double r154298 = a;
        double r154299 = exp(r154298);
        double r154300 = b;
        double r154301 = exp(r154300);
        double r154302 = r154299 + r154301;
        double r154303 = r154299 / r154302;
        return r154303;
}
↓
double f(double a, double b) {
        double r154304 = a;
        double r154305 = exp(r154304);
        double r154306 = sqrt(r154305);
        double r154307 = b;
        double r154308 = exp(r154307);
        double r154309 = r154305 + r154308;
        double r154310 = r154309 / r154306;
        double r154311 = r154306 / r154310;
        return r154311;
}

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

Derivation

  1. Initial program 0.8

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

  3. Applied add-sqr-sqrt0.8

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

  4. Applied associate-/l*0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2020083 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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