Average Error: 29.3 → 0.9
Time: 17.8s
Precision: binary64
\[[a, b] = \mathsf{sort}([a, b]) \\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := 1 + e^{a}\\ \left(0.5 \cdot \frac{{b}^{2}}{t_0} + \left(e^{\log \left(\mathsf{log1p}\left(e^{a}\right)\right)} + \frac{b}{t_0}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{t_0}^{2}} \end{array} \]
\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := 1 + e^{a}\\
\left(0.5 \cdot \frac{{b}^{2}}{t_0} + \left(e^{\log \left(\mathsf{log1p}\left(e^{a}\right)\right)} + \frac{b}{t_0}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{t_0}^{2}}
\end{array}

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))))
   (-
    (+ (* 0.5 (/ (pow b 2.0) t_0)) (+ (exp (log (log1p (exp a)))) (/ b t_0)))
    (* 0.5 (/ (pow b 2.0) (pow t_0 2.0))))))
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);
	return ((0.5 * (pow(b, 2.0) / t_0)) + (exp(log(log1p(exp(a)))) + (b / t_0))) - (0.5 * (pow(b, 2.0) / pow(t_0, 2.0)));
}

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 29.3

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

  2. Taylor expanded in b around 0 1.0

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

  3. Applied egg-rr0.9

    \[\leadsto \left(0.5 \cdot \frac{{b}^{2}}{1 + e^{a}} + \left(\color{blue}{e^{\log \left(\mathsf{log1p}\left(e^{a}\right)\right)}} + \frac{b}{1 + e^{a}}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{\left(1 + e^{a}\right)}^{2}} \]

  4. Final simplification0.9

    \[\leadsto \left(0.5 \cdot \frac{{b}^{2}}{1 + e^{a}} + \left(e^{\log \left(\mathsf{log1p}\left(e^{a}\right)\right)} + \frac{b}{1 + e^{a}}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{\left(1 + e^{a}\right)}^{2}} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))