Average Error: 29.6 → 1.1
Time: 7.5s
Precision: binary64
\[[a \; b]=\mathsf{sort}([a \; b])\]
\[\log \left(e^{a} + e^{b}\right)\]
↓
\[\frac{b}{1 + e^{a}} + \left(\left(b \cdot \frac{b}{1 + e^{a}}\right) \cdot \left(0.5 + \frac{-0.5}{1 + e^{a}}\right) + \log \left(1 + e^{a}\right)\right)\]
\log \left(e^{a} + e^{b}\right)
↓
\frac{b}{1 + e^{a}} + \left(\left(b \cdot \frac{b}{1 + e^{a}}\right) \cdot \left(0.5 + \frac{-0.5}{1 + e^{a}}\right) + \log \left(1 + e^{a}\right)\right)
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (+
  (/ b (+ 1.0 (exp a)))
  (+
   (* (* b (/ b (+ 1.0 (exp a)))) (+ 0.5 (/ -0.5 (+ 1.0 (exp a)))))
   (log (+ 1.0 (exp a))))))
double code(double a, double b) {
	return log(exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return (b / (1.0 + exp(a))) + (((b * (b / (1.0 + exp(a)))) * (0.5 + (-0.5 / (1.0 + exp(a))))) + log(1.0 + 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 29.6

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

  2. Taylor expanded around 0 1.2

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

  3. Simplified1.1

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2021025 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))