Hyperbolic arcsine

Average Error: 52.9 → 0.4

Time: 13.3s

Precision: binary64

Cost: 27204

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\log \left(0.078125 \cdot \frac{0.5}{{x}^{7}} + \left(-\left(\frac{0.5}{x} + 0.125 \cdot \left(\frac{0.5}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.9)
   (log
    (+
     (* 0.078125 (/ 0.5 (pow x 7.0)))
     (- (+ (/ 0.5 x) (* 0.125 (- (/ 0.5 (pow x 5.0)) (/ 1.0 (pow x 3.0))))))))
   (if (<= x 0.8) x (log (+ (/ 0.5 x) (+ x x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.9) {
		tmp = log(((0.078125 * (0.5 / pow(x, 7.0))) + -((0.5 / x) + (0.125 * ((0.5 / pow(x, 5.0)) - (1.0 / pow(x, 3.0)))))));
	} else if (x <= 0.8) {
		tmp = x;
	} else {
		tmp = log(((0.5 / x) + (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 <= (-0.9d0)) then
        tmp = log(((0.078125d0 * (0.5d0 / (x ** 7.0d0))) + -((0.5d0 / x) + (0.125d0 * ((0.5d0 / (x ** 5.0d0)) - (1.0d0 / (x ** 3.0d0)))))))
    else if (x <= 0.8d0) then
        tmp = x
    else
        tmp = log(((0.5d0 / x) + (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 <= -0.9) {
		tmp = Math.log(((0.078125 * (0.5 / Math.pow(x, 7.0))) + -((0.5 / x) + (0.125 * ((0.5 / Math.pow(x, 5.0)) - (1.0 / Math.pow(x, 3.0)))))));
	} else if (x <= 0.8) {
		tmp = x;
	} else {
		tmp = Math.log(((0.5 / x) + (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 <= -0.9:
		tmp = math.log(((0.078125 * (0.5 / math.pow(x, 7.0))) + -((0.5 / x) + (0.125 * ((0.5 / math.pow(x, 5.0)) - (1.0 / math.pow(x, 3.0)))))))
	elif x <= 0.8:
		tmp = x
	else:
		tmp = math.log(((0.5 / x) + (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 <= -0.9)
		tmp = log(Float64(Float64(0.078125 * Float64(0.5 / (x ^ 7.0))) + Float64(-Float64(Float64(0.5 / x) + Float64(0.125 * Float64(Float64(0.5 / (x ^ 5.0)) - Float64(1.0 / (x ^ 3.0))))))));
	elseif (x <= 0.8)
		tmp = x;
	else
		tmp = log(Float64(Float64(0.5 / x) + 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 <= -0.9)
		tmp = log(((0.078125 * (0.5 / (x ^ 7.0))) + -((0.5 / x) + (0.125 * ((0.5 / (x ^ 5.0)) - (1.0 / (x ^ 3.0)))))));
	elseif (x <= 0.8)
		tmp = x;
	else
		tmp = log(((0.5 / x) + (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, -0.9], N[Log[N[(N[(0.078125 * N[(0.5 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(0.5 / x), $MachinePrecision] + N[(0.125 * N[(N[(0.5 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.8], x, N[Log[N[(N[(0.5 / x), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	52.9
Target	45.1
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 < -0.900000000000000022

   1. Initial program 63.0

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

   2. Taylor expanded in x around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0390625 \cdot \frac{1}{{x}^{7}}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

   3. Simplified 0.2

      \[\leadsto \log \color{blue}{\left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0390625 \cdot \frac{1}{{x}^{7}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)} \]
      Proof

      [Start] 0.2	\[ \log \left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0390625 \cdot \frac{1}{{x}^{7}}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational.json-simplify-1 [<=] 0.2	\[ \log \left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0390625 \cdot \frac{1}{{x}^{7}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right) \]

   4. Applied egg-rr 0.2

      \[\leadsto \log \color{blue}{\left(0.078125 \cdot \frac{0.5}{{x}^{7}} + \left(-\left(\frac{0.5}{x} + 0.125 \cdot \left(\frac{0.5}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\right)\right)\right)} \]

   if -0.900000000000000022 < x < 0.80000000000000004

   1. Initial program 58.4

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

   2. Taylor expanded in x around 0 0.3

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

   3. Taylor expanded in x around 0 0.6

      \[\leadsto \color{blue}{x} \]

   if 0.80000000000000004 < x

   1. Initial program 31.8

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

   2. Taylor expanded in x around inf 0.3

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

   3. Applied egg-rr 0.3

      \[\leadsto \log \color{blue}{\left(\left(\left(x + x\right) + \frac{0.5}{x}\right) - 0\right)} \]

   4. Simplified 0.3

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

      [Start] 0.3	\[ \log \left(\left(\left(x + x\right) + \frac{0.5}{x}\right) - 0\right) \]
      rational.json-simplify-5 [=>] 0.3	\[ \log \color{blue}{\left(\left(x + x\right) + \frac{0.5}{x}\right)} \]
      rational.json-simplify-1 [=>] 0.3	\[ \log \color{blue}{\left(\frac{0.5}{x} + \left(x + x\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\log \left(0.078125 \cdot \frac{0.5}{{x}^{7}} + \left(-\left(\frac{0.5}{x} + 0.125 \cdot \left(\frac{0.5}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	20356

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\log \left(-\left(\frac{0.5}{x} + 0.125 \cdot \left(\frac{0.5}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{{x}^{3}} \cdot -0.125 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	6856

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

Alternative 5
Error	26.7
Cost	6724

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

Alternative 6
Error	15.8
Cost	6724

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

Alternative 7
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023067 
(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)))))