Average Error: 29.8 → 1.1
Time: 6.2s
Precision: binary64
\[[a, b]=\mathsf{sort}([a, b])\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ t_0 + \frac{b}{e^{t_0}} \end{array} \]
\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := \mathsf{log1p}\left(e^{a}\right)\\
t_0 + \frac{b}{e^{t_0}}
\end{array}

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (log1p (exp a)))) (+ t_0 (/ b (exp t_0)))))
double code(double a, double b) {
	return log(exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double t_0 = log1p(exp(a));
	return t_0 + (b / exp(t_0));
}

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 29.8

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

  2. Taylor expanded in b around 0 1.2

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

  3. Simplified1.1

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

  4. Applied add-exp-log_binary641.1

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

  5. Simplified1.1

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

  6. Final simplification1.1

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

Reproduce

herbie shell --seed 2021224 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))