Hyperbolic arcsine

Average Error: 53.1 → 0.1

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\log \left(\mathsf{fma}\left(0.125, {x}^{-3}, \mathsf{fma}\left(-0.0625, {x}^{-5}, \frac{-0.5}{x}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.021:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (log (fma 0.125 (pow x -3.0) (fma -0.0625 (pow x -5.0) (/ -0.5 x))))
   (if (<= x 0.021)
     (fma
      (pow x 3.0)
      -0.16666666666666666
      (fma (pow x 7.0) -0.044642857142857144 (fma 0.075 (pow x 5.0) 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 <= -1.06) {
		tmp = log(fma(0.125, pow(x, -3.0), fma(-0.0625, pow(x, -5.0), (-0.5 / x))));
	} else if (x <= 0.021) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, fma(pow(x, 7.0), -0.044642857142857144, fma(0.075, pow(x, 5.0), 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 <= -1.06)
		tmp = log(fma(0.125, (x ^ -3.0), fma(-0.0625, (x ^ -5.0), Float64(-0.5 / x))));
	elseif (x <= 0.021)
		tmp = fma((x ^ 3.0), -0.16666666666666666, fma((x ^ 7.0), -0.044642857142857144, fma(0.075, (x ^ 5.0), 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, -1.06], N[Log[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], If[LessEqual[x, 0.021], N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + N[(N[Power[x, 7.0], $MachinePrecision] * -0.044642857142857144 + N[(0.075 * N[Power[x, 5.0], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Error

Bits error versus x

Target

Original	53.1
Target	45.4
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if x < -1.0600000000000001

   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 0.2

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

   4. Simplified 0.2

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

   5. Applied egg-rr 0.2

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

   if -1.0600000000000001 < x < 0.0210000000000000013

   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 0.1

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

   4. Simplified 0.1

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

   5. Taylor expanded in x around 0 0.1

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

   6. Simplified 0.1

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

   if 0.0210000000000000013 < x

   1. Initial program 31.9

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

   2. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\log \left(\mathsf{fma}\left(0.125, {x}^{-3}, \mathsf{fma}\left(-0.0625, {x}^{-5}, \frac{-0.5}{x}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.021:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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)))))