Average Error: 29.5 → 0.9
Time: 16.9s
Precision: binary64
\[[a, b] = \mathsf{sort}([a, b]) \\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\mathsf{log1p}\left(e^{a}\right) - \frac{b}{-1 - e^{a}} \]
\log \left(e^{a} + e^{b}\right)
↓
\mathsf{log1p}\left(e^{a}\right) - \frac{b}{-1 - e^{a}}

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b) :precision binary64 (- (log1p (exp a)) (/ b (- -1.0 (exp a)))))
double code(double a, double b) {
	return log(exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return log1p(exp(a)) - (b / (-1.0 - exp(a)));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.5

    \[\log \left(e^{a} + e^{b}\right) \]

  2. Taylor expanded in b around 0 1.1

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

  3. Simplified0.9

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

  4. Applied frac-2neg_binary640.9

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

  5. Simplified0.9

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

  6. Final simplification0.9

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

Reproduce

herbie shell --seed 2022097 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))