Hyperbolic arcsine

Average Error: 53.0 → 0.3

Time: 7.2s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left(0.075 \cdot {x}^{5} + -0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0072)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 1.3)
     (+ x (+ (* 0.075 (pow x 5.0)) (* -0.16666666666666666 (pow x 3.0))))
     (log (+ x x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.0072) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 1.3) {
		tmp = x + ((0.075 * pow(x, 5.0)) + (-0.16666666666666666 * pow(x, 3.0)));
	} else {
		tmp = log((x + 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 <= -0.0072) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 1.3) {
		tmp = x + ((0.075 * Math.pow(x, 5.0)) + (-0.16666666666666666 * Math.pow(x, 3.0)));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.0072:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 1.3:
		tmp = x + ((0.075 * math.pow(x, 5.0)) + (-0.16666666666666666 * math.pow(x, 3.0)))
	else:
		tmp = math.log((x + 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 <= -0.0072)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(Float64(0.075 * (x ^ 5.0)) + Float64(-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = log(Float64(x + 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 <= -0.0072)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 1.3)
		tmp = x + ((0.075 * (x ^ 5.0)) + (-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log((x + 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, -0.0072], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.3], N[(x + N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.0
Target	45.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -0.0071999999999999998

   1. Initial program 62.7

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

   2. Simplified 62.7

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

      [Start] 62.7	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 62.7	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 62.7	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Applied egg-rr 62.7

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

   4. Simplified 0.2

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

      [Start] 62.7	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      div-sub [<=] 62.2	\[ \log \color{blue}{\left(\frac{x \cdot x - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      +-commutative [=>] 62.2	\[ \log \left(\frac{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate--r+ [=>] 31.7	\[ \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 0.2	\[ \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 0.2	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 0.2	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 0.2	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 0.2	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 0.2	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right) \]
      sub-neg [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   5. Applied egg-rr 0.2

      \[\leadsto \color{blue}{0 + \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]

   6. Simplified 0.2

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

      [Start] 0.2	\[ 0 + \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right) \]
      +-lft-identity [=>] 0.2	\[ \color{blue}{\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      log-rec [=>] 0.2	\[ \color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)} \]

   if -0.0071999999999999998 < x < 1.30000000000000004

   1. Initial program 58.9

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

   2. Simplified 58.9

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

      [Start] 58.9	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 58.9	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 58.9	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around 0 0.1

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

   4. Applied egg-rr 32.2

      \[\leadsto \color{blue}{{\left(\sqrt{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}\right)}^{2}} \]

   5. Applied egg-rr 0.1

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

   6. Taylor expanded in x around 0 0.1

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

   if 1.30000000000000004 < x

   1. Initial program 31.7

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

   2. Simplified 0.1

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

      [Start] 31.7	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 31.7	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 0.1	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around inf 0.7

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

   4. Simplified 0.7

      \[\leadsto \log \color{blue}{\left(x + x\right)} \]
      Proof

      [Start] 0.7	\[ \log \left(2 \cdot x\right) \]
      count-2 [<=] 0.7	\[ \log \color{blue}{\left(x + x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left(0.075 \cdot {x}^{5} + -0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00096:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6
Error	15.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Error	30.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023011 
(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)))))