Average Error: 0.7 → 0.7
Time: 3.7s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)
double code(double a, double b) {
	return (exp(a) / (exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	return log1p(expm1((exp(a) / (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

Original0.7
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. Final simplification0.7

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

Reproduce

herbie shell --seed 2020053 +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))))