Hyperbolic arcsine

Percentage Accurate: 18.3% → 99.1%

Time: 4.3s

Alternatives: 5

Speedup: 20.3×

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: 18.3% 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: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 0.8) (* 1.0 x) (log (+ (- x (/ -0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.8) {
		tmp = 1.0 * x;
	} else {
		tmp = log(((x - (-0.5 / x)) + x));
	}
	return tmp;
}

Fortran

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 <= 0.8d0) then
        tmp = 1.0d0 * x
    else
        tmp = log(((x - ((-0.5d0) / x)) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.8) {
		tmp = 1.0 * x;
	} else {
		tmp = Math.log(((x - (-0.5 / x)) + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 0.8:
		tmp = 1.0 * x
	else:
		tmp = math.log(((x - (-0.5 / x)) + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.8)
		tmp = Float64(1.0 * x);
	else
		tmp = log(Float64(Float64(x - Float64(-0.5 / x)) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 0.8)
		tmp = 1.0 * x;
	else
		tmp = log(((x - (-0.5 / x)) + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.8], N[(1.0 * x), $MachinePrecision], N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.8%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.30000000000000004 < x < 0.80000000000000004

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      10. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto x \cdot 1 \]

   if 0.80000000000000004 < x

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \log \left(x + \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower--.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. *-inverses N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3) (log (/ -0.5 x)) (if (<= x 1.25) (* 1.0 x) (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = 1.0 * x;
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Fortran

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 <= 1.25d0) then
        tmp = 1.0d0 * x
    else
        tmp = log((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = 1.0 * x;
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = 1.0 * x
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(1.0 * x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = 1.0 * x;
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(1.0 * x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.8%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      10. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto x \cdot 1 \]

   if 1.25 < x

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25) (* 1.0 x) (log (* 2.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 * x;
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = 1.0d0 * x
    else
        tmp = log((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 * x;
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = 1.0 * x
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 * x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = 1.0 * x;
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(1.0 * x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      10. metadata-eval 65.5

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 67.0%

       \[\leadsto x \cdot 1 \]

   if 1.25 < x

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.55) (* 1.0 x) (log (+ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 1.0 * x;
	} else {
		tmp = log((1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = 1.0d0 * x
    else
        tmp = log((1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 1.0 * x;
	} else {
		tmp = Math.log((1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = 1.0 * x
	else:
		tmp = math.log((1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(1.0 * x);
	else
		tmp = log(Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = 1.0 * x;
	else
		tmp = log((1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(1.0 * x), $MachinePrecision], N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

      10. metadata-eval 65.5

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 67.0%

       \[\leadsto x \cdot 1 \]

   if 1.55000000000000004 < x

   1. Initial program 57.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 31.2

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

   5. Applied rewrites 31.2%

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.0 x))

C

double code(double x) {
	return 1.0 * x;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 * x;
}

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 * x;
end

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1 \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1 \]

   5. *-rgt-identity N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{x} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

   8. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{-1}{6}, x\right) \]

   10. metadata-eval 47.4

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \]

5. Applied rewrites 47.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 47.4%

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

8. Taylor expanded in x around 0

   \[\leadsto x \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 49.8%

    \[\leadsto x \cdot 1 \]

11. Final simplification 49.8%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Developer Target 1: 30.4% accurate, 0.9× 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 2024332 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))