Hyperbolic arcsine

Average Error: 52.7 → 0.6

Time: 5.3s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3570739280.310989:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00020224963872625962:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -3570739280.310989)
   (log (/ -0.5 x))
   (if (<= x 0.00020224963872625962)
     (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
     (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 <= -3570739280.310989) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00020224963872625962) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + (x + (0.075 * pow(x, 5.0)));
	} 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 <= -3570739280.310989) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00020224963872625962) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + (x + (0.075 * Math.pow(x, 5.0)));
	} 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 <= -3570739280.310989:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00020224963872625962:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + (x + (0.075 * math.pow(x, 5.0)))
	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 <= -3570739280.310989)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00020224963872625962)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(x + Float64(0.075 * (x ^ 5.0))));
	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 <= -3570739280.310989)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00020224963872625962)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + (x + (0.075 * (x ^ 5.0)));
	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, -3570739280.310989], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00020224963872625962], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	52.7
Target	44.9
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if x < -3570739280.3109889

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 43 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

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

   if -3570739280.3109889 < x < 2.02249638726259621e-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)))))): 43 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 1.1

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

   if 2.02249638726259621e-4 < x

   1. Initial program 30.8

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 43 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 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3570739280.310989:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00020224963872625962:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -3570739280.310989:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00020224963872625962:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -3570739280.310989:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00020224963872625962:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -3570739280.310989:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00020224963872625962:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -3570739280.310989:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00020224963872625962:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 5
Error	15.7
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 0.00020224963872625962:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 6
Error	30.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022306 
(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)))))