Average Error: 29.5 → 1.1
Time: 20.5s
Precision: binary64
\[[a, b] = \mathsf{sort}([a, b]) \\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ t_1 := \sqrt{t_0}\\ t_0 + \frac{b}{{\left(e^{t_1}\right)}^{t_1}} \end{array} \]
\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := \mathsf{log1p}\left(e^{a}\right)\\
t_1 := \sqrt{t_0}\\
t_0 + \frac{b}{{\left(e^{t_1}\right)}^{t_1}}
\end{array}

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (log1p (exp a))) (t_1 (sqrt t_0)))
   (+ t_0 (/ b (pow (exp t_1) t_1)))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	double t_0 = log1p(exp(a));
	double t_1 = sqrt(t_0);
	return t_0 + (b / pow(exp(t_1), t_1));
}

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 1.2

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

  3. Simplified1.1

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

  4. Applied egg-rr1.1

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

  5. Final simplification1.1

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

Reproduce

herbie shell --seed 2022125 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))