Average Error: 29.5 → 1.0
Time: 7.0s
Precision: binary64
\[[a, b]=\mathsf{sort}([a, b])\]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := 1 + e^{a}\\ \log t_0 + \frac{b}{t_0} \cdot \left(1 + b \cdot \left(0.5 - \frac{0.5}{t_0}\right)\right) \end{array} \]
\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := 1 + e^{a}\\
\log t_0 + \frac{b}{t_0} \cdot \left(1 + b \cdot \left(0.5 - \frac{0.5}{t_0}\right)\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))))
   (+ (log t_0) (* (/ b t_0) (+ 1.0 (* b (- 0.5 (/ 0.5 t_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 log(t_0) + ((b / t_0) * (1.0 + (b * (0.5 - (0.5 / 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 29.5

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

  2. Taylor expanded around 0 1.1

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

  3. Simplified1.0

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

  4. Taylor expanded around inf 1.1

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

  5. Simplified1.0

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

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2021202 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))