Average Error: 53.1 → 0.2
Time: 4.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} t_0 := \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{if}\;x \leq -1.886837515345653:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 7.126106571810149 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, \mathsf{fma}\left(0.075, {x}^{5}, \mathsf{fma}\left(-0.044642857142857144, {x}^{7}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_0\\ \end{array} \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ x (hypot 1.0 x))))))
   (if (<= x -1.886837515345653)
     (log (/ -0.5 x))
     (if (<= x 7.126106571810149e-5)
       (fma
        -0.16666666666666666
        (pow x 3.0)
        (fma 0.075 (pow x 5.0) (fma -0.044642857142857144 (pow x 7.0) x)))
       (+ t_0 t_0)))))
double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}
↓
double code(double x) {
	double t_0 = log(sqrt((x + hypot(1.0, x))));
	double tmp;
	if (x <= -1.886837515345653) {
		tmp = log((-0.5 / x));
	} else if (x <= 7.126106571810149e-5) {
		tmp = fma(-0.16666666666666666, pow(x, 3.0), fma(0.075, pow(x, 5.0), fma(-0.044642857142857144, pow(x, 7.0), x)));
	} else {
		tmp = t_0 + t_0;
	}
	return tmp;
}

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end
↓
function code(x)
	t_0 = log(sqrt(Float64(x + hypot(1.0, x))))
	tmp = 0.0
	if (x <= -1.886837515345653)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 7.126106571810149e-5)
		tmp = fma(-0.16666666666666666, (x ^ 3.0), fma(0.075, (x ^ 5.0), fma(-0.044642857142857144, (x ^ 7.0), x)));
	else
		tmp = Float64(t_0 + t_0);
	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_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.886837515345653], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 7.126106571810149e-5], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision] + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + t$95$0), $MachinePrecision]]]]

\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
t_0 := \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\
\mathbf{if}\;x \leq -1.886837515345653:\\
\;\;\;\;\log \left(\frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 7.126106571810149 \cdot 10^{-5}:\\
\;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, \mathsf{fma}\left(0.075, {x}^{5}, \mathsf{fma}\left(-0.044642857142857144, {x}^{7}, x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 + t_0\\


\end{array}

Error

Target

Original53.1
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 < -1.88683751534565292

    1. Initial program 63.0

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

    2. Simplified63.0

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

    3. Taylor expanded in x around -inf 0.6

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

    if -1.88683751534565292 < x < 7.1261065718101487e-5

    1. Initial program 58.9

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

    2. Simplified58.9

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

    3. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, \mathsf{fma}\left(0.075, {x}^{5}, \mathsf{fma}\left(-0.044642857142857144, {x}^{7}, x\right)\right)\right)} \]

    if 7.1261065718101487e-5 < x

    1. Initial program 32.1

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

    2. Simplified0.1

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

    3. Applied egg-rr0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.886837515345653:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 7.126106571810149 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, \mathsf{fma}\left(0.075, {x}^{5}, \mathsf{fma}\left(-0.044642857142857144, {x}^{7}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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)))))