Average Error: 29.6 → 1.2
Time: 13.7s
Precision: binary64
\[[a, b]=\mathsf{sort}([a, b])\]
\[\log \left(e^{a} + e^{b}\right)\]
↓
\[\left(\log \left(e^{a} + 1\right) + \left(0.5 \cdot \frac{{b}^{2}}{e^{a} + 1} + \frac{b}{e^{a} + 1}\right)\right) - 0.5 \cdot \frac{{b}^{2}}{{\left(e^{a} + 1\right)}^{2}}\]
\log \left(e^{a} + e^{b}\right)
↓
\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}}
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (-
  (+
   (log (+ (exp a) 1.0))
   (+ (* 0.5 (/ (pow b 2.0) (+ (exp a) 1.0))) (/ b (+ (exp a) 1.0))))
  (* 0.5 (/ (pow b 2.0) (pow (+ (exp a) 1.0) 2.0)))))
double code(double a, double b) {
	return log(exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return (log(exp(a) + 1.0) + ((0.5 * (pow(b, 2.0) / (exp(a) + 1.0))) + (b / (exp(a) + 1.0)))) - (0.5 * (pow(b, 2.0) / pow((exp(a) + 1.0), 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 29.6

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

  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. Final simplification1.2

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

Reproduce

herbie shell --seed 2021050 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))