Average Error: 0.7 → 0.7
Time: 3.8s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{b} + e^{a}}\right)}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{b} + e^{a}}\right)}\right)
double f(double a, double b) {
        double r180488 = a;
        double r180489 = exp(r180488);
        double r180490 = b;
        double r180491 = exp(r180490);
        double r180492 = r180489 + r180491;
        double r180493 = r180489 / r180492;
        return r180493;
}
↓
double f(double a, double b) {
        double r180494 = a;
        double r180495 = exp(r180494);
        double r180496 = exp(r180495);
        double r180497 = 1.0;
        double r180498 = b;
        double r180499 = exp(r180498);
        double r180500 = r180499 + r180495;
        double r180501 = r180497 / r180500;
        double r180502 = pow(r180496, r180501);
        double r180503 = log(r180502);
        return r180503;
}

Error

Bits error versus a

Bits error versus b

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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 *-un-lft-identity0.7

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

  4. Applied add-sqr-sqrt0.7

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

  5. Applied times-frac0.7

    \[\leadsto \color{blue}{\frac{\sqrt{e^{a}}}{1} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}}\]

  6. Simplified0.7

    \[\leadsto \color{blue}{\sqrt{e^{a}}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.8

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

  9. Simplified0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2020025 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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