Average Error: 30.0 → 0.9
Time: 29.0s
Precision: binary64
\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := e^{a} + 1\\ \mathsf{log1p}\left(e^{a}\right) + \left(\frac{b}{t_0} + \frac{b \cdot b}{t_0} \cdot \left(0.5 + \frac{-0.5}{t_0}\right)\right) \end{array} \]
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (+ (log1p (exp a)) (+ (/ b t_0) (* (/ (* b b) t_0) (+ 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 = exp(a) + 1.0;
	return log1p(exp(a)) + ((b / t_0) + (((b * b) / t_0) * (0.5 + (-0.5 / t_0))));
}

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}
↓
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	return Math.log1p(Math.exp(a)) + ((b / t_0) + (((b * b) / t_0) * (0.5 + (-0.5 / t_0))));
}

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))
↓
def code(a, b):
	t_0 = math.exp(a) + 1.0
	return math.log1p(math.exp(a)) + ((b / t_0) + (((b * b) / t_0) * (0.5 + (-0.5 / 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)) + Float64(Float64(b / t_0) + Float64(Float64(Float64(b * b) / t_0) * Float64(0.5 + Float64(-0.5 / 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 / t$95$0), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.5 + N[(-0.5 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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

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 30.0

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

  2. Taylor expanded in b around 0 1.2

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2022155 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))