Average Error: 30.1 → 0.9
Time: 15.1s
Precision: binary64
\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := e^{a} + 1\\ \mathsf{log1p}\left(e^{a}\right) + \mathsf{fma}\left(b \cdot b, \frac{0.5}{t_0} + \frac{-0.5}{{t_0}^{2}}, \frac{b}{t_0}\right) \end{array} \]
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (+
    (log1p (exp a))
    (fma (* b b) (+ (/ 0.5 t_0) (/ -0.5 (pow t_0 2.0))) (/ b t_0)))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	return log1p(exp(a)) + fma((b * b), ((0.5 / t_0) + (-0.5 / pow(t_0, 2.0))), (b / t_0));
}

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end
↓
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	return Float64(log1p(exp(a)) + fma(Float64(b * b), Float64(Float64(0.5 / t_0) + Float64(-0.5 / (t_0 ^ 2.0))), Float64(b / t_0)))
end

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(0.5 / t$95$0), $MachinePrecision] + N[(-0.5 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := e^{a} + 1\\
\mathsf{log1p}\left(e^{a}\right) + \mathsf{fma}\left(b \cdot b, \frac{0.5}{t_0} + \frac{-0.5}{{t_0}^{2}}, \frac{b}{t_0}\right)
\end{array}

Error

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded in b around 0 1.0

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2022197 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))