Average Error: 53.0 → 0.1
Time: 4.6s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\log \left({\left(\sqrt{\mathsf{fma}\left(0.125, {x}^{-3}, \mathsf{fma}\left(-0.0625, {x}^{-5}, \frac{-0.5}{x}\right)\right)}\right)}^{2}\right)\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (log
    (pow
     (sqrt (fma 0.125 (pow x -3.0) (fma -0.0625 (pow x -5.0) (/ -0.5 x))))
     2.0))
   (if (<= x 0.00072)
     (fma -0.16666666666666666 (pow x 3.0) x)
     (log (+ x (hypot 1.0 x))))))
double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = log(pow(sqrt(fma(0.125, pow(x, -3.0), fma(-0.0625, pow(x, -5.0), (-0.5 / x)))), 2.0));
	} else if (x <= 0.00072) {
		tmp = fma(-0.16666666666666666, pow(x, 3.0), x);
	} else {
		tmp = log((x + 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 <= -1.05)
		tmp = log((sqrt(fma(0.125, (x ^ -3.0), fma(-0.0625, (x ^ -5.0), Float64(-0.5 / x)))) ^ 2.0));
	elseif (x <= 0.00072)
		tmp = fma(-0.16666666666666666, (x ^ 3.0), x);
	else
		tmp = log(Float64(x + 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, -1.05], N[Log[N[Power[N[Sqrt[N[(0.125 * N[Power[x, -3.0], $MachinePrecision] + N[(-0.0625 * N[Power[x, -5.0], $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00072], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.05:\\
\;\;\;\;\log \left({\left(\sqrt{\mathsf{fma}\left(0.125, {x}^{-3}, \mathsf{fma}\left(-0.0625, {x}^{-5}, \frac{-0.5}{x}\right)\right)}\right)}^{2}\right)\\

\mathbf{elif}\;x \leq 0.00072:\\
\;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, x\right)\\

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


\end{array}

Error

Bits error versus x

Target

Original53.0
Target45.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.05000000000000004

    1. Initial program 63.1

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

    2. Simplified63.1

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

    3. Taylor expanded in x around -inf 0.1

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} + \left(\frac{-0.5}{x} + \frac{-0.0625}{{x}^{5}}\right)\right)} \]

    5. Applied egg-rr0.1

      \[\leadsto \log \color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(0.125, {x}^{-3}, \mathsf{fma}\left(-0.0625, {x}^{-5}, \frac{-0.5}{x}\right)\right)}\right)}^{2}\right)} \]

    if -1.05000000000000004 < x < 7.20000000000000045e-4

    1. Initial program 59.1

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

    2. Simplified59.1

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

    3. Taylor expanded in x around 0 0.1

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

    4. Simplified0.1

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

    if 7.20000000000000045e-4 < x

    1. Initial program 31.2

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

    2. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\log \left({\left(\sqrt{\mathsf{fma}\left(0.125, {x}^{-3}, \mathsf{fma}\left(-0.0625, {x}^{-5}, \frac{-0.5}{x}\right)\right)}\right)}^{2}\right)\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022169 
(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)))))