Hyperbolic arcsine

Average Error: 52.8 → 0.2

Time: 42.8s

Precision: binary64

Cost: 20164

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;{x}^{5} \cdot 0.075 + \left(x + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + x \cdot 2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 1.0)
     (+ (* (pow x 5.0) 0.075) (+ x (* (pow x 3.0) -0.16666666666666666)))
     (log (+ (/ 0.5 x) (* x 2.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log(((0.125 / pow(x, 3.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0)))));
	} else if (x <= 1.0) {
		tmp = (pow(x, 5.0) * 0.075) + (x + (pow(x, 3.0) * -0.16666666666666666));
	} else {
		tmp = log(((0.5 / x) + (x * 2.0)));
	}
	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 <= (-1.1d0)) then
        tmp = log(((0.125d0 / (x ** 3.0d0)) - ((0.5d0 / x) + (0.0625d0 / (x ** 5.0d0)))))
    else if (x <= 1.0d0) then
        tmp = ((x ** 5.0d0) * 0.075d0) + (x + ((x ** 3.0d0) * (-0.16666666666666666d0)))
    else
        tmp = log(((0.5d0 / x) + (x * 2.0d0)))
    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 <= -1.1) {
		tmp = Math.log(((0.125 / Math.pow(x, 3.0)) - ((0.5 / x) + (0.0625 / Math.pow(x, 5.0)))));
	} else if (x <= 1.0) {
		tmp = (Math.pow(x, 5.0) * 0.075) + (x + (Math.pow(x, 3.0) * -0.16666666666666666));
	} else {
		tmp = Math.log(((0.5 / x) + (x * 2.0)));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) - ((0.5 / x) + (0.0625 / math.pow(x, 5.0)))))
	elif x <= 1.0:
		tmp = (math.pow(x, 5.0) * 0.075) + (x + (math.pow(x, 3.0) * -0.16666666666666666))
	else:
		tmp = math.log(((0.5 / x) + (x * 2.0)))
	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 <= -1.1)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) - Float64(Float64(0.5 / x) + Float64(0.0625 / (x ^ 5.0)))));
	elseif (x <= 1.0)
		tmp = Float64(Float64((x ^ 5.0) * 0.075) + Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)));
	else
		tmp = log(Float64(Float64(0.5 / x) + Float64(x * 2.0)));
	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 <= -1.1)
		tmp = log(((0.125 / (x ^ 3.0)) - ((0.5 / x) + (0.0625 / (x ^ 5.0)))));
	elseif (x <= 1.0)
		tmp = ((x ^ 5.0) * 0.075) + (x + ((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(((0.5 / x) + (x * 2.0)));
	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_] := If[LessEqual[x, -1.1], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] + N[(0.0625 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[Power[x, 5.0], $MachinePrecision] * 0.075), $MachinePrecision] + N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(0.5 / x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	52.8
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 < -1.1000000000000001

   1. Initial program 62.7

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

   2. Taylor expanded in x around -inf 0.3

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

   3. Simplified 0.3

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)} \]
      Proof

      [Start] 0.3	\[ \log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational_best-simplify-55 [=>] 0.3	\[ \log \left(\color{blue}{1 \cdot \frac{0.125}{{x}^{3}}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational_best-simplify-1 [=>] 0.3	\[ \log \left(\color{blue}{\frac{0.125}{{x}^{3}} \cdot 1} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational_best-simplify-7 [=>] 0.3	\[ \log \left(\color{blue}{\frac{0.125}{{x}^{3}}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational_best-simplify-3 [<=] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right) \]
      rational_best-simplify-55 [=>] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \left(\color{blue}{1 \cdot \frac{0.5}{x}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]
      rational_best-simplify-1 [=>] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \left(\color{blue}{\frac{0.5}{x} \cdot 1} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]
      rational_best-simplify-7 [=>] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \left(\color{blue}{\frac{0.5}{x}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]
      rational_best-simplify-55 [=>] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \color{blue}{1 \cdot \frac{0.0625}{{x}^{5}}}\right)\right) \]
      rational_best-simplify-1 [=>] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \color{blue}{\frac{0.0625}{{x}^{5}} \cdot 1}\right)\right) \]
      rational_best-simplify-7 [=>] 0.3	\[ \log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \color{blue}{\frac{0.0625}{{x}^{5}}}\right)\right) \]

   if -1.1000000000000001 < x < 1

   1. Initial program 58.6

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

   2. Taylor expanded in x around 0 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ -0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right) \]
      rational_best-simplify-3 [=>] 0.1	\[ -0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right)} \]
      rational_best-simplify-47 [=>] 0.1	\[ \color{blue}{0.075 \cdot {x}^{5} + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \]
      rational_best-simplify-1 [=>] 0.1	\[ \color{blue}{{x}^{5} \cdot 0.075} + \left(x + -0.16666666666666666 \cdot {x}^{3}\right) \]
      rational_best-simplify-1 [=>] 0.1	\[ {x}^{5} \cdot 0.075 + \left(x + \color{blue}{{x}^{3} \cdot -0.16666666666666666}\right) \]

   if 1 < x

   1. Initial program 31.6

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

   2. Taylor expanded in x around inf 0.2

      \[\leadsto \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} \]

   3. Simplified 0.2

      \[\leadsto \log \color{blue}{\left(\frac{0.5}{x} + x \cdot 2\right)} \]
      Proof

      [Start] 0.2	\[ \log \left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) \]
      rational_best-simplify-3 [=>] 0.2	\[ \log \color{blue}{\left(0.5 \cdot \frac{1}{x} + 2 \cdot x\right)} \]
      rational_best-simplify-55 [=>] 0.2	\[ \log \left(\color{blue}{1 \cdot \frac{0.5}{x}} + 2 \cdot x\right) \]
      rational_best-simplify-1 [=>] 0.2	\[ \log \left(\color{blue}{\frac{0.5}{x} \cdot 1} + 2 \cdot x\right) \]
      rational_best-simplify-7 [=>] 0.2	\[ \log \left(\color{blue}{\frac{0.5}{x}} + 2 \cdot x\right) \]
      rational_best-simplify-1 [=>] 0.2	\[ \log \left(\frac{0.5}{x} + \color{blue}{x \cdot 2}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;{x}^{5} \cdot 0.075 + \left(x + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + x \cdot 2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;{x}^{5} \cdot 0.075 + \left(x + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + x \cdot 2\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13444

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + x \cdot 2\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + x \cdot 2\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-1.5}{x} - \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + x \cdot 2\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6
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 7
Error	26.4
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]

Alternative 8
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 9
Error	30.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023099 
(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)))))