Average Error: 0.8 → 0.7
Time: 18.4s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[{e}^{\left(a - \log \left(e^{a} + e^{b}\right)\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
{e}^{\left(a - \log \left(e^{a} + e^{b}\right)\right)}
double f(double a, double b) {
        double r27823506 = a;
        double r27823507 = exp(r27823506);
        double r27823508 = b;
        double r27823509 = exp(r27823508);
        double r27823510 = r27823507 + r27823509;
        double r27823511 = r27823507 / r27823510;
        return r27823511;
}
↓
double f(double a, double b) {
        double r27823512 = exp(1.0);
        double r27823513 = a;
        double r27823514 = exp(r27823513);
        double r27823515 = b;
        double r27823516 = exp(r27823515);
        double r27823517 = r27823514 + r27823516;
        double r27823518 = log(r27823517);
        double r27823519 = r27823513 - r27823518;
        double r27823520 = pow(r27823512, r27823519);
        return r27823520;
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

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Target

Original0.8
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.8

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

  3. Applied add-exp-log0.8

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

  4. Applied div-exp0.7

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

  6. Applied *-un-lft-identity0.7

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

  7. Applied *-un-lft-identity0.7

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

  8. Applied distribute-lft-out--0.7

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

  9. Applied exp-prod0.7

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

  10. Simplified0.7

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

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2019124 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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