Hyperbolic arcsine

Average Error: 53.1 → 0.2

Time: 8.8s

Precision: binary64

Cost: 32968

Math

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

↓

\[\begin{array}{l} t_0 := x + \mathsf{hypot}\left(1, x\right)\\ \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \log t_0 + \log \left(\sqrt{t_0}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (hypot 1.0 x))))
   (if (<= x -1.25)
     (log (/ -0.5 x))
     (if (<= x 0.00115)
       (+ x (* (* x x) (* x -0.16666666666666666)))
       (+ (* 0.5 (log t_0)) (log (sqrt t_0)))))))

C

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

↓

double code(double x) {
	double t_0 = x + hypot(1.0, x);
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00115) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = (0.5 * log(t_0)) + log(sqrt(t_0));
	}
	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 t_0 = x + Math.hypot(1.0, x);
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00115) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = (0.5 * Math.log(t_0)) + Math.log(Math.sqrt(t_0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = x + math.hypot(1.0, x)
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00115:
		tmp = x + ((x * x) * (x * -0.16666666666666666))
	else:
		tmp = (0.5 * math.log(t_0)) + math.log(math.sqrt(t_0))
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(x + hypot(1.0, x))
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00115)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(x * -0.16666666666666666)));
	else
		tmp = Float64(Float64(0.5 * log(t_0)) + log(sqrt(t_0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = x + hypot(1.0, x);
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00115)
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	else
		tmp = (0.5 * log(t_0)) + log(sqrt(t_0));
	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_] := Block[{t$95$0 = N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00115], N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] + N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	53.1
Target	45.2
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 < -1.25

   1. Initial program 62.7

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

   2. Simplified 62.8

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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 [=>] 62.8	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around -inf 0.7

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

   if -1.25 < x < 0.00115

   1. Initial program 59.0

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

   2. Simplified 59.0

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

      [Start] 59.0	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 59.0	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 59.0	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around 0 0.1

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

   4. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) - 1\right)} + x \]

   5. Simplified 0.2

      \[\leadsto \color{blue}{\left(1 + \left(-0.16666666666666666 \cdot {x}^{3} - 1\right)\right)} + x \]
      Proof

      [Start] 0.2	\[ \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) - 1\right) + x \]
      associate--l+ [=>] 0.2	\[ \color{blue}{\left(1 + \left(-0.16666666666666666 \cdot {x}^{3} - 1\right)\right)} + x \]

   6. Applied egg-rr 0.1

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

   if 0.00115 < x

   1. Initial program 31.2

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

   2. Simplified 0.0

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

      [Start] 31.2	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 31.2	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 0.0	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{0.5 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)} + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00094:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 3
Error	16.2
Cost	6724

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

Alternative 4
Error	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022354 
(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)))))