Hyperbolic arcsine

Percentage Accurate: 17.9% → 99.7%

Time: 8.4s

Alternatives: 8

Speedup: 207.0×

Specification

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x + 1}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 17.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x + 1}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.16666666666666666 + \left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.12)
   (log (- (/ 0.125 (pow x 3.0)) (/ 0.5 x)))
   (if (<= x 1.05)
     (+
      x
      (+
       (* (pow x 3.0) -0.16666666666666666)
       (+ (* -0.044642857142857144 (pow x 7.0)) (* 0.075 (pow x 5.0)))))
     (log (+ x (+ x (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.12) {
		tmp = log(((0.125 / pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 1.05) {
		tmp = x + ((pow(x, 3.0) * -0.16666666666666666) + ((-0.044642857142857144 * pow(x, 7.0)) + (0.075 * pow(x, 5.0))));
	} else {
		tmp = log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.12d0)) then
        tmp = log(((0.125d0 / (x ** 3.0d0)) - (0.5d0 / x)))
    else if (x <= 1.05d0) then
        tmp = x + (((x ** 3.0d0) * (-0.16666666666666666d0)) + (((-0.044642857142857144d0) * (x ** 7.0d0)) + (0.075d0 * (x ** 5.0d0))))
    else
        tmp = log((x + (x + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.12) {
		tmp = Math.log(((0.125 / Math.pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 1.05) {
		tmp = x + ((Math.pow(x, 3.0) * -0.16666666666666666) + ((-0.044642857142857144 * Math.pow(x, 7.0)) + (0.075 * Math.pow(x, 5.0))));
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.12:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) - (0.5 / x)))
	elif x <= 1.05:
		tmp = x + ((math.pow(x, 3.0) * -0.16666666666666666) + ((-0.044642857142857144 * math.pow(x, 7.0)) + (0.075 * math.pow(x, 5.0))))
	else:
		tmp = math.log((x + (x + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.12)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) - Float64(0.5 / x)));
	elseif (x <= 1.05)
		tmp = Float64(x + Float64(Float64((x ^ 3.0) * -0.16666666666666666) + Float64(Float64(-0.044642857142857144 * (x ^ 7.0)) + Float64(0.075 * (x ^ 5.0)))));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.12)
		tmp = log(((0.125 / (x ^ 3.0)) - (0.5 / x)));
	elseif (x <= 1.05)
		tmp = x + (((x ^ 3.0) * -0.16666666666666666) + ((-0.044642857142857144 * (x ^ 7.0)) + (0.075 * (x ^ 5.0))));
	else
		tmp = log((x + (x + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.12], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(x + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1200000000000001

   1. Initial program 5.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.9%

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r/ 98.9%

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

      4. metadata-eval 98.9%

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

   5. Simplified 98.9%

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

   if -1.1200000000000001 < x < 1.05000000000000004

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.16666666666666666 \cdot {x}^{3} + \left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right)\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.16666666666666666 + \left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (- (/ 0.125 (pow x 3.0)) (/ 0.5 x)))
   (if (<= x 1.0)
     (+ x (+ (* (pow x 3.0) -0.16666666666666666) (* 0.075 (pow x 5.0))))
     (log (+ x (+ x (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log(((0.125 / pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 1.0) {
		tmp = x + ((pow(x, 3.0) * -0.16666666666666666) + (0.075 * pow(x, 5.0)));
	} else {
		tmp = log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.15d0)) then
        tmp = log(((0.125d0 / (x ** 3.0d0)) - (0.5d0 / x)))
    else if (x <= 1.0d0) then
        tmp = x + (((x ** 3.0d0) * (-0.16666666666666666d0)) + (0.075d0 * (x ** 5.0d0)))
    else
        tmp = log((x + (x + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = Math.log(((0.125 / Math.pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 1.0) {
		tmp = x + ((Math.pow(x, 3.0) * -0.16666666666666666) + (0.075 * Math.pow(x, 5.0)));
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) - (0.5 / x)))
	elif x <= 1.0:
		tmp = x + ((math.pow(x, 3.0) * -0.16666666666666666) + (0.075 * math.pow(x, 5.0)))
	else:
		tmp = math.log((x + (x + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) - Float64(0.5 / x)));
	elseif (x <= 1.0)
		tmp = Float64(x + Float64(Float64((x ^ 3.0) * -0.16666666666666666) + Float64(0.075 * (x ^ 5.0))));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = log(((0.125 / (x ^ 3.0)) - (0.5 / x)));
	elseif (x <= 1.0)
		tmp = x + (((x ^ 3.0) * -0.16666666666666666) + (0.075 * (x ^ 5.0)));
	else
		tmp = log((x + (x + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.0], N[(x + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 5.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.9%

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r/ 98.9%

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

      4. metadata-eval 98.9%

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

   5. Simplified 98.9%

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

   if -1.1499999999999999 < x < 1

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

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

   if 1 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (- (/ 0.125 (pow x 3.0)) (/ 0.5 x)))
   (if (<= x 0.95)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ x (+ x (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log(((0.125 / pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 0.95) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Fortran

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)))
    else if (x <= 0.95d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log((x + (x + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

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)));
	} else if (x <= 0.95) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) - (0.5 / x)))
	elif x <= 0.95:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x + (x + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) - Float64(0.5 / x)));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log(((0.125 / (x ^ 3.0)) - (0.5 / x)));
	elseif (x <= 0.95)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x + (x + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 5.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.9%

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r/ 98.9%

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

      4. metadata-eval 98.9%

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

   5. Simplified 98.9%

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

   if -1.1000000000000001 < x < 0.94999999999999996

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

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

   if 0.94999999999999996 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 0.95)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ x (+ x (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 0.95d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log((x + (x + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 0.95:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x + (x + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 0.95)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x + (x + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 5.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.4%

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

   if -1.26000000000000001 < x < 0.94999999999999996

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

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

   if 0.94999999999999996 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 1.3) (+ x (* (pow x 3.0) -0.16666666666666666)) (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.3d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 1.3:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 1.3)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 5.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.4%

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

   if -1.26000000000000001 < x < 1.30000000000000004

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

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

   if 1.30000000000000004 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

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

Alternative 6: 99.2% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26) (log (/ -0.5 x)) (if (<= x 1.3) x (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.3d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 1.3:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 1.3)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 5.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.4%

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

   if -1.26000000000000001 < x < 1.30000000000000004

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.3%

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

   if 1.30000000000000004 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.3) x (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 7.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.6%

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

   if 1.30000000000000004 < x

   1. Initial program 54.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 17.5%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 56.9%

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

4. Final simplification 56.9%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot x + 1}\\ \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) 1.0))))
   (if (< x 0.0) (log (/ -1.0 (- x t_0))) (log (+ x t_0)))))

C

double code(double x) {
	double t_0 = sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = log((-1.0 / (x - t_0)));
	} else {
		tmp = log((x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + 1.0d0))
    if (x < 0.0d0) then
        tmp = log(((-1.0d0) / (x - t_0)))
    else
        tmp = log((x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = Math.log((-1.0 / (x - t_0)));
	} else {
		tmp = Math.log((x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(((x * x) + 1.0))
	tmp = 0
	if x < 0.0:
		tmp = math.log((-1.0 / (x - t_0)))
	else:
		tmp = math.log((x + t_0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(Float64(x * x) + 1.0))
	tmp = 0.0
	if (x < 0.0)
		tmp = log(Float64(-1.0 / Float64(x - t_0)));
	else
		tmp = log(Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(((x * x) + 1.0));
	tmp = 0.0;
	if (x < 0.0)
		tmp = log((-1.0 / (x - t_0)));
	else
		tmp = log((x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[Less[x, 0.0], N[Log[N[(-1.0 / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + t$95$0), $MachinePrecision]], $MachinePrecision]]]

Reproduce

herbie shell --seed 2024039 
(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)))))