Average Error: 0.6 → 0.5
Time: 17.8s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(e^{a} + e^{b}\right)}
double f(double a, double b) {
        double r4041688 = a;
        double r4041689 = exp(r4041688);
        double r4041690 = b;
        double r4041691 = exp(r4041690);
        double r4041692 = r4041689 + r4041691;
        double r4041693 = r4041689 / r4041692;
        return r4041693;
}
↓
double f(double a, double b) {
        double r4041694 = a;
        double r4041695 = exp(r4041694);
        double r4041696 = b;
        double r4041697 = exp(r4041696);
        double r4041698 = r4041695 + r4041697;
        double r4041699 = log(r4041698);
        double r4041700 = r4041694 - r4041699;
        double r4041701 = exp(r4041700);
        return r4041701;
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

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Target

Original0.6
Target0.0
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2019158 +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))))