Average Error: 0.8 → 0.8
Time: 11.5s
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 r142806 = a;
        double r142807 = exp(r142806);
        double r142808 = b;
        double r142809 = exp(r142808);
        double r142810 = r142807 + r142809;
        double r142811 = r142807 / r142810;
        return r142811;
}
↓
double f(double a, double b) {
        double r142812 = a;
        double r142813 = exp(r142812);
        double r142814 = exp(r142813);
        double r142815 = 1.0;
        double r142816 = b;
        double r142817 = exp(r142816);
        double r142818 = r142813 + r142817;
        double r142819 = r142815 / r142818;
        double r142820 = pow(r142814, r142819);
        double r142821 = log(r142820);
        return r142821;
}

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-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-prod0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019350 +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))))