Average Error: 0.6 → 0.5
Time: 1.5s
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 r158407 = a;
        double r158408 = exp(r158407);
        double r158409 = b;
        double r158410 = exp(r158409);
        double r158411 = r158408 + r158410;
        double r158412 = r158408 / r158411;
        return r158412;
}
↓
double f(double a, double b) {
        double r158413 = a;
        double r158414 = exp(r158413);
        double r158415 = b;
        double r158416 = exp(r158415);
        double r158417 = r158414 + r158416;
        double r158418 = log(r158417);
        double r158419 = r158413 - r158418;
        double r158420 = exp(r158419);
        return r158420;
}

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

Original0.6
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 2020021 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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