Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 95.5%

Time: 6.8s

Alternatives: 8

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \frac{\sin y}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \frac{\sin y}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-8) (/ x z) (* (sin y) (/ x (* y z)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-8) {
		tmp = x / z;
	} else {
		tmp = sin(y) * (x / (y * z));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d-8) then
        tmp = x / z
    else
        tmp = sin(y) * (x / (y * z))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-8) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2e-8:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-8)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2e-8], N[(x / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-8

   1. Initial program 96.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      2. times-frac 83.8%

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

      3. *-commutative 83.8%

         \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

      4. associate-*r/ 81.3%

         \[\leadsto \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]

      5. *-commutative 81.3%

         \[\leadsto \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{\sin y \cdot \frac{x}{y \cdot z}} \]

   4. Taylor expanded in y around 0 77.0%

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

   if 2e-8 < y

   1. Initial program 92.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 91.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      2. times-frac 97.0%

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

      3. *-commutative 97.0%

         \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

      4. associate-*r/ 97.2%

         \[\leadsto \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]

      5. *-commutative 97.2%

         \[\leadsto \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\sin y \cdot \frac{x}{y \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 2: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \frac{x}{\frac{y}{\sin y} \cdot z} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (/ x (* (/ y (sin y)) z)))

C

y = abs(y);
double code(double x, double y, double z) {
	return x / ((y / sin(y)) * z);
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / ((y / sin(y)) * z)
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	return x / ((y / Math.sin(y)) * z);
}

Python

y = abs(y)
def code(x, y, z):
	return x / ((y / math.sin(y)) * z)

Julia

y = abs(y)
function code(x, y, z)
	return Float64(x / Float64(Float64(y / sin(y)) * z))
end

MATLAB

y = abs(y)
function tmp = code(x, y, z)
	tmp = x / ((y / sin(y)) * z);
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x / N[(N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Step-by-step derivation

   1. associate-/l* 98.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
4. Step-by-step derivation

   1. clear-num 97.8%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}} \]

   2. associate-/r/ 97.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\sin y}{y}} \cdot z}} \]

   3. clear-num 98.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

5. Applied egg-rr 98.0%

   \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

6. Final simplification 98.0%

   \[\leadsto \frac{x}{\frac{y}{\sin y} \cdot z} \]

Alternative 3: 65.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0255:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{6}{z}}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0255) (/ x z) (* x (/ (/ 6.0 z) (* y y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0255) {
		tmp = x / z;
	} else {
		tmp = x * ((6.0 / z) / (y * y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.0255d0) then
        tmp = x / z
    else
        tmp = x * ((6.0d0 / z) / (y * y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0255) {
		tmp = x / z;
	} else {
		tmp = x * ((6.0 / z) / (y * y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 0.0255:
		tmp = x / z
	else:
		tmp = x * ((6.0 / z) / (y * y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0255)
		tmp = Float64(x / z);
	else
		tmp = Float64(x * Float64(Float64(6.0 / z) / Float64(y * y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0255)
		tmp = x / z;
	else
		tmp = x * ((6.0 / z) / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 0.0255], N[(x / z), $MachinePrecision], N[(x * N[(N[(6.0 / z), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0254999999999999984

   1. Initial program 96.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      2. times-frac 83.9%

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

      3. *-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

      4. associate-*r/ 81.4%

         \[\leadsto \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]

      5. *-commutative 81.4%

         \[\leadsto \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\sin y \cdot \frac{x}{y \cdot z}} \]

   4. Taylor expanded in y around 0 76.8%

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

   if 0.0254999999999999984 < y

   1. Initial program 92.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Step-by-step derivation

      1. associate-/l* 96.9%

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}} \]

      2. associate-/r/ 96.9%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\sin y}{y}} \cdot z}} \]

      3. clear-num 97.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   5. Applied egg-rr 97.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

   6. Taylor expanded in y around 0 32.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \cdot z} \]
   7. Step-by-step derivation

      1. *-commutative 32.0%

         \[\leadsto \frac{x}{\left(1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}\right) \cdot z} \]

      2. unpow2 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in y around inf 32.0%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2} \cdot z}} \]
   10. Step-by-step derivation

       1. associate-*r/ 32.0%

          \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2} \cdot z}} \]

       2. unpow2 32.0%

          \[\leadsto \frac{6 \cdot x}{\color{blue}{\left(y \cdot y\right)} \cdot z} \]

       3. associate-*l* 32.1%

          \[\leadsto \frac{6 \cdot x}{\color{blue}{y \cdot \left(y \cdot z\right)}} \]

   11. Simplified 32.1%

       \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot \left(y \cdot z\right)}} \]

   12. Taylor expanded in x around 0 32.0%

       \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2} \cdot z}} \]
   13. Step-by-step derivation

       1. associate-*r/ 32.0%

          \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2} \cdot z}} \]

       2. *-commutative 32.0%

          \[\leadsto \frac{\color{blue}{x \cdot 6}}{{y}^{2} \cdot z} \]

       3. unpow2 32.0%

          \[\leadsto \frac{x \cdot 6}{\color{blue}{\left(y \cdot y\right)} \cdot z} \]

       4. times-frac 32.0%

          \[\leadsto \color{blue}{\frac{x}{y \cdot y} \cdot \frac{6}{z}} \]

       5. associate-*l/ 31.9%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{6}{z}}{y \cdot y}} \]

       6. associate-*r/ 32.0%

          \[\leadsto \color{blue}{x \cdot \frac{\frac{6}{z}}{y \cdot y}} \]

   14. Simplified 32.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{6}{z}}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0255:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{6}{z}}{y \cdot y}\\ \end{array} \]

Alternative 4: 65.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0255:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{6}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0255) (/ x z) (* (/ x y) (/ 6.0 (* y z)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0255) {
		tmp = x / z;
	} else {
		tmp = (x / y) * (6.0 / (y * z));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.0255d0) then
        tmp = x / z
    else
        tmp = (x / y) * (6.0d0 / (y * z))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0255) {
		tmp = x / z;
	} else {
		tmp = (x / y) * (6.0 / (y * z));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 0.0255:
		tmp = x / z
	else:
		tmp = (x / y) * (6.0 / (y * z))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0255)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / y) * Float64(6.0 / Float64(y * z)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0255)
		tmp = x / z;
	else
		tmp = (x / y) * (6.0 / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 0.0255], N[(x / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(6.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0254999999999999984

   1. Initial program 96.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

      2. times-frac 83.9%

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

      3. *-commutative 83.9%

         \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

      4. associate-*r/ 81.4%

         \[\leadsto \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]

      5. *-commutative 81.4%

         \[\leadsto \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\sin y \cdot \frac{x}{y \cdot z}} \]

   4. Taylor expanded in y around 0 76.8%

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

   if 0.0254999999999999984 < y

   1. Initial program 92.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Step-by-step derivation

      1. associate-/l* 96.9%

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}} \]

      2. associate-/r/ 96.9%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\sin y}{y}} \cdot z}} \]

      3. clear-num 97.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

   5. Applied egg-rr 97.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

   6. Taylor expanded in y around 0 32.0%

      \[\leadsto \frac{x}{\color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \cdot z} \]
   7. Step-by-step derivation

      1. *-commutative 32.0%

         \[\leadsto \frac{x}{\left(1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}\right) \cdot z} \]

      2. unpow2 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in y around inf 32.0%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2} \cdot z}} \]
   10. Step-by-step derivation

       1. associate-*r/ 32.0%

          \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2} \cdot z}} \]

       2. unpow2 32.0%

          \[\leadsto \frac{6 \cdot x}{\color{blue}{\left(y \cdot y\right)} \cdot z} \]

       3. associate-*l* 32.1%

          \[\leadsto \frac{6 \cdot x}{\color{blue}{y \cdot \left(y \cdot z\right)}} \]

   11. Simplified 32.1%

       \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot \left(y \cdot z\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 32.1%

          \[\leadsto \frac{\color{blue}{x \cdot 6}}{y \cdot \left(y \cdot z\right)} \]

       2. times-frac 32.1%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{6}{y \cdot z}} \]

       3. *-commutative 32.1%

          \[\leadsto \frac{x}{y} \cdot \frac{6}{\color{blue}{z \cdot y}} \]

   13. Applied egg-rr 32.1%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{6}{z \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0255:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{6}{y \cdot z}\\ \end{array} \]

Alternative 5: 65.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \frac{x}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (/ x (* z (+ 1.0 (* (* y y) 0.16666666666666666)))))

C

y = abs(y);
double code(double x, double y, double z) {
	return x / (z * (1.0 + ((y * y) * 0.16666666666666666)));
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z * (1.0d0 + ((y * y) * 0.16666666666666666d0)))
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	return x / (z * (1.0 + ((y * y) * 0.16666666666666666)));
}

Python

y = abs(y)
def code(x, y, z):
	return x / (z * (1.0 + ((y * y) * 0.16666666666666666)))

Julia

y = abs(y)
function code(x, y, z)
	return Float64(x / Float64(z * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))))
end

MATLAB

y = abs(y)
function tmp = code(x, y, z)
	tmp = x / (z * (1.0 + ((y * y) * 0.16666666666666666)));
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x / N[(z * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Step-by-step derivation

   1. associate-/l* 98.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
4. Step-by-step derivation

   1. clear-num 97.8%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}} \]

   2. associate-/r/ 97.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\sin y}{y}} \cdot z}} \]

   3. clear-num 98.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

5. Applied egg-rr 98.0%

   \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

6. Taylor expanded in y around 0 67.5%

   \[\leadsto \frac{x}{\color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \cdot z} \]
7. Step-by-step derivation

   1. *-commutative 67.5%

      \[\leadsto \frac{x}{\left(1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}\right) \cdot z} \]

   2. unpow2 67.5%

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

8. Simplified 67.5%

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

9. Final simplification 67.5%

   \[\leadsto \frac{x}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \]

Alternative 6: 65.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \frac{x}{z \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (/ x (* z (+ 1.0 (* y (* y -0.16666666666666666))))))

C

y = abs(y);
double code(double x, double y, double z) {
	return x / (z * (1.0 + (y * (y * -0.16666666666666666))));
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z * (1.0d0 + (y * (y * (-0.16666666666666666d0)))))
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	return x / (z * (1.0 + (y * (y * -0.16666666666666666))));
}

Python

y = abs(y)
def code(x, y, z):
	return x / (z * (1.0 + (y * (y * -0.16666666666666666))))

Julia

y = abs(y)
function code(x, y, z)
	return Float64(x / Float64(z * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)))))
end

MATLAB

y = abs(y)
function tmp = code(x, y, z)
	tmp = x / (z * (1.0 + (y * (y * -0.16666666666666666))));
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x / N[(z * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Step-by-step derivation

   1. associate-/l* 98.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
4. Step-by-step derivation

   1. clear-num 97.8%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}} \]

   2. associate-/r/ 97.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\sin y}{y}} \cdot z}} \]

   3. clear-num 98.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y}} \cdot z} \]

5. Applied egg-rr 98.0%

   \[\leadsto \frac{x}{\color{blue}{\frac{y}{\sin y} \cdot z}} \]

6. Taylor expanded in y around 0 67.5%

   \[\leadsto \frac{x}{\color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \cdot z} \]
7. Step-by-step derivation

   1. *-commutative 67.5%

      \[\leadsto \frac{x}{\left(1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}\right) \cdot z} \]

   2. unpow2 67.5%

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

8. Simplified 67.5%

   \[\leadsto \frac{x}{\color{blue}{\left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \cdot z} \]
9. Step-by-step derivation

   1. expm1-log1p-u 67.5%

      \[\leadsto \frac{x}{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)}\right) \cdot z} \]

   2. expm1-udef 67.5%

      \[\leadsto \frac{x}{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)} - 1\right)}\right) \cdot z} \]

   3. log1p-udef 67.5%

      \[\leadsto \frac{x}{\left(1 + \left(e^{\color{blue}{\log \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}} - 1\right)\right) \cdot z} \]

   4. add-exp-log 67.5%

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

   5. +-commutative 67.5%

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

   6. add-sqr-sqrt 67.5%

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

   7. sqrt-unprod 67.5%

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

   8. swap-sqr 67.5%

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

   9. metadata-eval 67.5%

      \[\leadsto \frac{x}{\left(1 + \left(\left(\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{0.027777777777777776}} + 1\right) - 1\right)\right) \cdot z} \]

   10. metadata-eval 67.5%

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

   11. swap-sqr 67.5%

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

   12. associate-*r* 67.5%

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

   13. associate-*r* 67.5%

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

   14. sqrt-unprod 25.0%

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

   15. add-sqr-sqrt 67.8%

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

   16. fma-def 67.8%

       \[\leadsto \frac{x}{\left(1 + \left(\color{blue}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)} - 1\right)\right) \cdot z} \]

10. Applied egg-rr 67.8%

    \[\leadsto \frac{x}{\left(1 + \color{blue}{\left(\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) - 1\right)}\right) \cdot z} \]
11. Step-by-step derivation

    1. fma-udef 67.8%

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

    2. associate--l+ 67.8%

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

    3. metadata-eval 67.8%

       \[\leadsto \frac{x}{\left(1 + \left(y \cdot \left(y \cdot -0.16666666666666666\right) + \color{blue}{0}\right)\right) \cdot z} \]

12. Simplified 67.8%

    \[\leadsto \frac{x}{\left(1 + \color{blue}{\left(y \cdot \left(y \cdot -0.16666666666666666\right) + 0\right)}\right) \cdot z} \]

13. Final simplification 67.8%

    \[\leadsto \frac{x}{z \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]

Alternative 7: 57.5% accurate, 21.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \frac{1}{\frac{z}{x}} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (/ 1.0 (/ z x)))

C

y = abs(y);
double code(double x, double y, double z) {
	return 1.0 / (z / x);
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (z / x)
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	return 1.0 / (z / x);
}

Python

y = abs(y)
def code(x, y, z):
	return 1.0 / (z / x)

Julia

y = abs(y)
function code(x, y, z)
	return Float64(1.0 / Float64(z / x))
end

MATLAB

y = abs(y)
function tmp = code(x, y, z)
	tmp = 1.0 / (z / x);
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(1.0 / N[(z / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Step-by-step derivation

   1. associate-/l* 98.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
4. Step-by-step derivation

   1. clear-num 96.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{\frac{\sin y}{y}}}{x}}} \]

   2. associate-/r/ 97.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{\frac{\sin y}{y}}} \cdot x} \]

   3. clear-num 98.1%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z}} \cdot x \]

   4. associate-/l/ 90.9%

      \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

   5. *-commutative 90.9%

      \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

5. Applied egg-rr 90.9%

   \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z} \cdot x} \]

6. Taylor expanded in y around 0 58.8%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot x \]
7. Step-by-step derivation

   1. associate-*l/ 59.0%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{z}} \]

   2. *-un-lft-identity 59.0%

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

   3. clear-num 59.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \]

8. Applied egg-rr 59.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \]

9. Final simplification 59.1%

   \[\leadsto \frac{1}{\frac{z}{x}} \]

Alternative 8: 57.7% accurate, 35.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \frac{x}{z} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (/ x z))

C

y = abs(y);
double code(double x, double y, double z) {
	return x / z;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / z
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	return x / z;
}

Python

y = abs(y)
def code(x, y, z):
	return x / z

Julia

y = abs(y)
function code(x, y, z)
	return Float64(x / z)
end

MATLAB

y = abs(y)
function tmp = code(x, y, z)
	tmp = x / z;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x / z), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Step-by-step derivation

   1. associate-*l/ 96.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]

   2. times-frac 87.6%

      \[\leadsto \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]

   3. *-commutative 87.6%

      \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]

   4. associate-*r/ 85.9%

      \[\leadsto \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]

   5. *-commutative 85.9%

      \[\leadsto \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

3. Simplified 85.9%

   \[\leadsto \color{blue}{\sin y \cdot \frac{x}{y \cdot z}} \]

4. Taylor expanded in y around 0 59.0%

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

5. Final simplification 59.0%

   \[\leadsto \frac{x}{z} \]

Developer target: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2023252 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))