Average Error: 29.9 → 1.2
Time: 12.2s
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 (pow E 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(pow(((double) M_E), 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 29.9

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

  2. Taylor expanded in b around 0 1.3

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

  3. Simplified1.2

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

  4. Applied *-un-lft-identity_binary641.2

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

  5. Applied exp-prod_binary641.2

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

  6. Simplified1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2021275 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))