Average Error: 0.7 → 0.7
Time: 4.1s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right) \]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (expm1 (log1p (/ (exp a) (+ (exp a) (exp b))))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return expm1(log1p(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. Applied expm1-log1p-u_binary640.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2022055 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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