Hyperbolic arcsine

Average Error: 52.7 → 0.2

Time: 8.8s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x + \left(0.075 \cdot {x}^{5} + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -26628.147070873067)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 0.0007358342361412555)
     (+ x (+ (* 0.075 (pow x 5.0)) (* (pow x 3.0) -0.16666666666666666)))
     (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 <= -26628.147070873067) {
		tmp = log(((0.125 / pow(x, 3.0)) + (-0.5 / x)));
	} else if (x <= 0.0007358342361412555) {
		tmp = x + ((0.075 * pow(x, 5.0)) + (pow(x, 3.0) * -0.16666666666666666));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -26628.147070873067) {
		tmp = Math.log(((0.125 / Math.pow(x, 3.0)) + (-0.5 / x)));
	} else if (x <= 0.0007358342361412555) {
		tmp = x + ((0.075 * Math.pow(x, 5.0)) + (Math.pow(x, 3.0) * -0.16666666666666666));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

↓

def code(x):
	tmp = 0
	if x <= -26628.147070873067:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) + (-0.5 / x)))
	elif x <= 0.0007358342361412555:
		tmp = x + ((0.075 * math.pow(x, 5.0)) + (math.pow(x, 3.0) * -0.16666666666666666))
	else:
		tmp = math.log((x + math.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 <= -26628.147070873067)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) + Float64(-0.5 / x)));
	elseif (x <= 0.0007358342361412555)
		tmp = Float64(x + Float64(Float64(0.075 * (x ^ 5.0)) + Float64((x ^ 3.0) * -0.16666666666666666)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -26628.147070873067)
		tmp = log(((0.125 / (x ^ 3.0)) + (-0.5 / x)));
	elseif (x <= 0.0007358342361412555)
		tmp = x + ((0.075 * (x ^ 5.0)) + ((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = 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, -26628.147070873067], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0007358342361412555], N[(x + N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	52.7
Target	44.8
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 < -26628.1470708730667

   1. Initial program 63.7

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

   2. Simplified 63.7

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 26 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around -inf 0.0

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

   4. Simplified 0.0

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)} \]
      Proof
      (+.f64 (/.f64 1/8 (pow.f64 x 3)) (/.f64 -1/2 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/8 1)) (pow.f64 x 3)) (/.f64 -1/2 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3)))) (/.f64 -1/2 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (*.f64 1/2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error

   if -26628.1470708730667 < x < 7.3583423614125552e-4

   1. Initial program 58.4

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

   2. Simplified 58.4

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 26 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.4

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

   4. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, 0.075 \cdot {x}^{5}\right), 1, x\right)} \]

   5. Applied egg-rr 0.4

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

   6. Taylor expanded in x around 0 0.4

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

   if 7.3583423614125552e-4 < x

   1. Initial program 30.5

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 26 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x + \left(0.075 \cdot {x}^{5} + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13444

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;-\log \left(\frac{x}{-0.5}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;-\log \left(\frac{x}{-0.5}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;-\log \left(\frac{x}{-0.5}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 5
Error	0.7
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 6
Error	0.7
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;-\log \left(\frac{x}{-0.5}\right)\\ \mathbf{elif}\;x \leq 0.0007358342361412555:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 7
Error	15.6
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq -26628.147070873067:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022315 
(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)))))