Average Error: 0.7 → 0.7
Time: 4.1s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)
double f(double a, double b) {
        double r124578 = a;
        double r124579 = exp(r124578);
        double r124580 = b;
        double r124581 = exp(r124580);
        double r124582 = r124579 + r124581;
        double r124583 = r124579 / r124582;
        return r124583;
}
↓
double f(double a, double b) {
        double r124584 = a;
        double r124585 = exp(r124584);
        double r124586 = exp(r124585);
        double r124587 = 1.0;
        double r124588 = b;
        double r124589 = exp(r124588);
        double r124590 = r124585 + r124589;
        double r124591 = r124587 / r124590;
        double r124592 = pow(r124586, r124591);
        double r124593 = log(r124592);
        return r124593;
}

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

Derivation

  1. Initial program 0.7

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

  3. Applied add-log-exp0.9

    \[\leadsto \color{blue}{\log \left(e^{\frac{e^{a}}{e^{a} + e^{b}}}\right)}\]
  4. Using strategy rm

  5. Applied div-inv0.9

    \[\leadsto \log \left(e^{\color{blue}{e^{a} \cdot \frac{1}{e^{a} + e^{b}}}}\right)\]

  6. Applied exp-prod0.7

    \[\leadsto \log \color{blue}{\left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)}\]

  7. Final simplification0.7

    \[\leadsto \log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)\]

Reproduce

herbie shell --seed 2020060 +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))))