Average Error: 0.7 → 0.7
Time: 4.1s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left({\left(e^{\frac{1}{e^{a} + e^{b}}}\right)}^{\left(e^{a}\right)}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{\frac{1}{e^{a} + e^{b}}}\right)}^{\left(e^{a}\right)}\right)
double f(double a, double b) {
        double r126474 = a;
        double r126475 = exp(r126474);
        double r126476 = b;
        double r126477 = exp(r126476);
        double r126478 = r126475 + r126477;
        double r126479 = r126475 / r126478;
        return r126479;
}
↓
double f(double a, double b) {
        double r126480 = 1.0;
        double r126481 = a;
        double r126482 = exp(r126481);
        double r126483 = b;
        double r126484 = exp(r126483);
        double r126485 = r126482 + r126484;
        double r126486 = r126480 / r126485;
        double r126487 = exp(r126486);
        double r126488 = pow(r126487, r126482);
        double r126489 = log(r126488);
        return r126489;
}

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-exp-log0.7

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

  4. Applied div-exp0.7

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

  6. Applied log1p-expm1-u0.7

    \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(a - \log \left(e^{a} + e^{b}\right)\right)\right)}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.8

    \[\leadsto \color{blue}{\log \left(e^{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(a - \log \left(e^{a} + e^{b}\right)\right)\right)}}\right)}\]

  9. Simplified0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2020025 +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))))