Hyperbolic arcsine

Average Error: 53.3 → 0.4

Time: 8.1s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.05)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (+ (log 2.0) (+ (log x) (/ 0.25 (* x x)))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(2.0) + (log(x) + (0.25 / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.05d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log(2.0d0) + (log(x) + (0.25d0 / (x * x)))
    end if
    code = tmp
end function

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 <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(2.0) + (Math.log(x) + (0.25 / (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 <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.05:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(2.0) + (math.log(x) + (0.25 / (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 <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.05)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(log(2.0) + Float64(log(x) + Float64(0.25 / 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 <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.05)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(2.0) + (log(x) + (0.25 / (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, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(N[Log[x], $MachinePrecision] + N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	53.3
Target	45.7
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 62.9

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

   2. Simplified 62.9

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

      [Start] 62.9	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 62.9	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 62.9	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around -inf 0.6

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

   if -1.30000000000000004 < x < 1.05000000000000004

   1. Initial program 58.7

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

   2. Simplified 58.7

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

      [Start] 58.7	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 58.7	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 58.7	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around 0 0.2

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

   if 1.05000000000000004 < x

   1. Initial program 33.1

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

   2. Simplified 0.0

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

      [Start] 33.1	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 33.1	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 0.0	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around inf 0.5

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   4. Simplified 0.5

      \[\leadsto \color{blue}{\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right)} \]
      Proof

      [Start] 0.5	\[ -1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + 0.25 \cdot \frac{1}{{x}^{2}}\right) \]
      associate-+r+ [=>] 0.5	\[ \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) + 0.25 \cdot \frac{1}{{x}^{2}}} \]
      +-commutative [<=] 0.5	\[ \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}} \]
      associate-+l+ [=>] 0.5	\[ \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]
      mul-1-neg [=>] 0.5	\[ \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) \]
      log-rec [=>] 0.5	\[ \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right) \]
      remove-double-neg [=>] 0.5	\[ \log 2 + \left(\color{blue}{\log x} + 0.25 \cdot \frac{1}{{x}^{2}}\right) \]
      associate-*r/ [=>] 0.5	\[ \log 2 + \left(\log x + \color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) \]
      metadata-eval [=>] 0.5	\[ \log 2 + \left(\log x + \frac{\color{blue}{0.25}}{{x}^{2}}\right) \]
      unpow2 [=>] 0.5	\[ \log 2 + \left(\log x + \frac{0.25}{\color{blue}{x \cdot x}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13256

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x \cdot x} + \log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \]

Alternative 4
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 5
Error	0.6
Cost	6856

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

Alternative 6
Error	26.3
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]

Alternative 7
Error	15.4
Cost	6724

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

Alternative 8
Error	26.3
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \]

Alternative 9
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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)))))