Average Error: 0.7 → 0.7
Time: 4.6s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)\right)
double f(double a, double b) {
        double r108168 = a;
        double r108169 = exp(r108168);
        double r108170 = b;
        double r108171 = exp(r108170);
        double r108172 = r108169 + r108171;
        double r108173 = r108169 / r108172;
        return r108173;
}
↓
double f(double a, double b) {
        double r108174 = 1.0;
        double r108175 = a;
        double r108176 = exp(r108175);
        double r108177 = b;
        double r108178 = exp(r108177);
        double r108179 = r108176 + r108178;
        double r108180 = r108179 / r108176;
        double r108181 = r108174 / r108180;
        double r108182 = expm1(r108181);
        double r108183 = log1p(r108182);
        return r108183;
}

Error

Bits error versus a

Bits error versus b

Try it out

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 log1p-expm1-u0.7

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)}\]
  4. Using strategy rm

  5. Applied clear-num0.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}}\right)\right)\]

  6. Final simplification0.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)\right)\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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