Average Error: 0.6 → 0.7
Time: 18.5s
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 r4096585 = a;
        double r4096586 = exp(r4096585);
        double r4096587 = b;
        double r4096588 = exp(r4096587);
        double r4096589 = r4096586 + r4096588;
        double r4096590 = r4096586 / r4096589;
        return r4096590;
}
↓
double f(double a, double b) {
        double r4096591 = a;
        double r4096592 = exp(r4096591);
        double r4096593 = exp(r4096592);
        double r4096594 = 1.0;
        double r4096595 = b;
        double r4096596 = exp(r4096595);
        double r4096597 = r4096592 + r4096596;
        double r4096598 = r4096594 / r4096597;
        double r4096599 = pow(r4096593, r4096598);
        double r4096600 = log(r4096599);
        return r4096600;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-exp-log0.6

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

  4. Applied div-exp0.5

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

  6. Applied add-log-exp0.6

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

  7. Simplified0.7

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

  9. Applied div-inv0.7

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

  10. Applied exp-prod0.7

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

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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