Average Error: 29.4 → 0.9
Time: 6.5s
Precision: binary64
\[[a, b]=\mathsf{sort}([a, b])\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := 1 + e^{a}\\ t_1 := \frac{b}{t_0}\\ \left(t_1 + \left(b \cdot t_1\right) \cdot \left(0.5 - \frac{0.5}{t_0}\right)\right) + \mathsf{log1p}\left(e^{a}\right) \end{array} \]
\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := 1 + e^{a}\\
t_1 := \frac{b}{t_0}\\
\left(t_1 + \left(b \cdot t_1\right) \cdot \left(0.5 - \frac{0.5}{t_0}\right)\right) + \mathsf{log1p}\left(e^{a}\right)
\end{array}

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))) (t_1 (/ b t_0)))
   (+ (+ t_1 (* (* b t_1) (- 0.5 (/ 0.5 t_0)))) (log1p (exp a)))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	double t_1 = b / t_0;
	return (t_1 + ((b * t_1) * (0.5 - (0.5 / t_0)))) + log1p(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 29.4

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

  2. Taylor expanded in b around 0 1.2

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

  3. Simplified0.9

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

  4. Applied +-commutative_binary640.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2022160 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))