Average Error: 29.8 → 1.2
Time: 17.8s
Precision: binary64
\[[a, b] = \mathsf{sort}([a, b]) \\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := e^{a} + 1\\ \mathsf{log1p}\left(e^{a}\right) + \left(1 + \left(0.5 - \frac{0.5}{t_0}\right) \cdot b\right) \cdot \frac{b}{t_0} \end{array} \]
\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := e^{a} + 1\\
\mathsf{log1p}\left(e^{a}\right) + \left(1 + \left(0.5 - \frac{0.5}{t_0}\right) \cdot b\right) \cdot \frac{b}{t_0}
\end{array}

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (+ (log1p (exp a)) (* (+ 1.0 (* (- 0.5 (/ 0.5 t_0)) b)) (/ b t_0)))))
double code(double a, double b) {
	return log(exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	return log1p(exp(a)) + ((1.0 + ((0.5 - (0.5 / t_0)) * b)) * (b / 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 1.4

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{b}^{2}}{1 + e^{a}} + \left(\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{\left(1 + e^{a}\right)}^{2}}} \]

  3. Simplified1.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \left(\frac{b}{1 + e^{a}} + \left(\frac{b}{1 + e^{a}} \cdot b\right) \cdot \left(0.5 - \frac{0.5}{1 + e^{a}}\right)\right)} \]

  4. Taylor expanded in a around inf 1.4

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{b}^{2}}{1 + e^{a}} + \left(\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{\left(1 + e^{a}\right)}^{2}}} \]

  5. Simplified1.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \left(\left(0.5 - \frac{0.5}{1 + e^{a}}\right) \cdot b + 1\right) \cdot \frac{b}{1 + e^{a}}} \]

  6. Final simplification1.2

    \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \left(1 + \left(0.5 - \frac{0.5}{e^{a} + 1}\right) \cdot b\right) \cdot \frac{b}{e^{a} + 1} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))