Hyperbolic arcsine

Average Error: 53.0 → 0.2

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9508788240618709:\\ \;\;\;\;\frac{0.09375}{{x}^{4}} + \left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{0.25}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 0.0009794900121172633:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.9508788240618709)
   (+
    (/ 0.09375 (pow x 4.0))
    (- (+ (log (/ -1.0 x)) (log 0.5)) (/ 0.25 (* x x))))
   (if (<= x 0.0009794900121172633)
     (fma (pow x 3.0) -0.16666666666666666 x)
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

↓

double code(double x) {
	double tmp;
	if (x <= -0.9508788240618709) {
		tmp = (0.09375 / pow(x, 4.0)) + ((log((-1.0 / x)) + log(0.5)) - (0.25 / (x * x)));
	} else if (x <= 0.0009794900121172633) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -0.9508788240618709)
		tmp = Float64(Float64(0.09375 / (x ^ 4.0)) + Float64(Float64(log(Float64(-1.0 / x)) + log(0.5)) - Float64(0.25 / Float64(x * x))));
	elseif (x <= 0.0009794900121172633)
		tmp = fma((x ^ 3.0), -0.16666666666666666, x);
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -0.9508788240618709], N[(N[(0.09375 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Log[0.5], $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0009794900121172633], N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Error

Bits error versus x

Target

Original	53.0
Target	45.1
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -0.95087882406187085

   1. Initial program 63.0

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

   2. Simplified 63.0

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

   3. Taylor expanded in x around -inf 0.4

      \[\leadsto \color{blue}{\left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

   4. Simplified 0.4

      \[\leadsto \color{blue}{\frac{0.09375}{{x}^{4}} + \left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{0.25}{x \cdot x}\right)} \]

   if -0.95087882406187085 < x < 9.7949001211726332e-4

   1. Initial program 58.8

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

   2. Simplified 58.8

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

   3. Taylor expanded in x around 0 0.2

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

   4. Simplified 0.2

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

   if 9.7949001211726332e-4 < x

   1. Initial program 31.5

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

   2. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9508788240618709:\\ \;\;\;\;\frac{0.09375}{{x}^{4}} + \left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{0.25}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 0.0009794900121172633:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(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)))))