Average Error: 52.8 → 0.2
Time: 4.4s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00091:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\\ \end{array} \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 0.00091)
     (fma (pow x 3.0) -0.16666666666666666 x)
     (log (+ x (expm1 (log1p (hypot 1.0 x))))))))
double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log(((0.125 / pow(x, 3.0)) + (-0.5 / x)));
	} else if (x <= 0.00091) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else {
		tmp = log((x + expm1(log1p(hypot(1.0, x)))));
	}
	return tmp;
}

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end
↓
function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) + Float64(-0.5 / x)));
	elseif (x <= 0.00091)
		tmp = fma((x ^ 3.0), -0.16666666666666666, x);
	else
		tmp = log(Float64(x + expm1(log1p(hypot(1.0, x)))));
	end
	return tmp
end

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := If[LessEqual[x, -1.1], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00091], N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[Log[N[(x + N[(Exp[N[Log[1 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 0.00091:\\
\;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\\


\end{array}

Error

Bits error versus x

Target

Original52.8
Target45.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.1000000000000001

    1. Initial program 62.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

    2. Simplified62.7

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

    3. Taylor expanded in x around -inf 0.4

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} \]

    4. Simplified0.4

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)} \]

    if -1.1000000000000001 < x < 9.1e-4

    1. Initial program 58.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

    2. Simplified58.7

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

    3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{x - 0.16666666666666666 \cdot {x}^{3}} \]

    4. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]

    if 9.1e-4 < x

    1. Initial program 31.3

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

    2. Simplified0.1

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

    3. Applied egg-rr0.2

      \[\leadsto \log \left(x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00091:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))