Average Error: 0.6 → 0.6
Time: 33.2s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\sqrt{e^{a}}}{e^{a} + e^{b}} \cdot \sqrt{e^{a}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{\sqrt{e^{a}}}{e^{a} + e^{b}} \cdot \sqrt{e^{a}}
double f(double a, double b) {
        double r20108969 = a;
        double r20108970 = exp(r20108969);
        double r20108971 = b;
        double r20108972 = exp(r20108971);
        double r20108973 = r20108970 + r20108972;
        double r20108974 = r20108970 / r20108973;
        return r20108974;
}
↓
double f(double a, double b) {
        double r20108975 = a;
        double r20108976 = exp(r20108975);
        double r20108977 = sqrt(r20108976);
        double r20108978 = b;
        double r20108979 = exp(r20108978);
        double r20108980 = r20108976 + r20108979;
        double r20108981 = r20108977 / r20108980;
        double r20108982 = r20108981 * r20108977;
        return r20108982;
}

Error

Bits error versus a

Bits error versus b

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Results

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Target

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

Derivation

  1. Initial program 0.6

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

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

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied times-frac0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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