Average Error: 0.6 → 0.5
Time: 4.1s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \log \left(e^{a} + e^{b}\right)}\right)\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \log \left(e^{a} + e^{b}\right)}\right)\right)
double code(double a, double b) {
	return (exp(a) / (exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	return expm1(log1p(exp((a - log((exp(a) + exp(b)))))));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
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. Using strategy rm

  6. Applied expm1-log1p-u0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020056 +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))))