Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 95.5%

Time: 7.9s

Alternatives: 13

Speedup: 1.0×

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: 96.0% 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

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e+155) (* (/ (sin y) y) (/ x z)) (/ (* x (sin y)) (* y z))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+155) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = (x * sin(y)) / (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 <= 1.9d+155) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = (x * sin(y)) / (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 <= 1.9e+155) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = (x * Math.sin(y)) / (y * z);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 1.9e+155:
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = (x * math.sin(y)) / (y * z)
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e+155)
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(Float64(x * sin(y)) / Float64(y * z));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.9e+155)
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = (x * sin(y)) / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1.9e+155], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e155

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

   if 1.9e155 < y

   1. Initial program 94.9%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 99.4%

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

      3. associate-/l/ 94.9%

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

      4. associate-/r/ 95.1%

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

      5. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \end{array} \]

Alternative 2: 95.7% 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 97.1%

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

      1. associate-/l* 96.9%

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

      2. associate-/r/ 83.2%

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

      3. associate-/l/ 76.1%

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

      4. associate-/r/ 82.3%

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

      5. associate-/r* 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in y around 0 70.2%

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

   if 2e-8 < y

   1. Initial program 97.7%

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

      1. associate-/l* 90.4%

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

      2. associate-/r/ 90.4%

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

      3. associate-/l/ 97.7%

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

      4. associate-/r/ 97.8%

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

      5. associate-/r* 90.6%

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

   3. Simplified 90.6%

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

4. Final simplification 75.2%

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

Alternative 3: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.2e+156) (* (/ (sin y) y) (/ x z)) (* (sin y) (/ x (* y z)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.2e+156) {
		tmp = (sin(y) / y) * (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 <= 2.2d+156) then
        tmp = (sin(y) / y) * (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 <= 2.2e+156) {
		tmp = (Math.sin(y) / y) * (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 <= 2.2e+156:
		tmp = (math.sin(y) / y) * (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 <= 2.2e+156)
		tmp = Float64(Float64(sin(y) / y) * 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 <= 2.2e+156)
		tmp = (sin(y) / y) * (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, 2.2e+156], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000004e156

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

   if 2.20000000000000004e156 < y

   1. Initial program 94.9%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 99.4%

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

      3. associate-/l/ 94.9%

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

      4. associate-/r/ 95.1%

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

      5. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 4: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Final simplification 97.3%

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

Alternative 5: 66.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4) (/ x z) (* 6.0 (* (/ 1.0 y) (/ (/ x z) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / 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 <= 2.4d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * ((1.0d0 / y) * ((x / z) / 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 <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2.4:
		tmp = x / z
	else:
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(1.0 / y) * Float64(Float64(x / z) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x / z;
	else
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(1.0 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 97.1%

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

      1. associate-/l* 96.9%

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

      2. associate-/r/ 83.3%

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

      3. associate-/l/ 76.2%

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

      4. associate-/r/ 82.4%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 70.2%

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

   if 2.39999999999999991 < y

   1. Initial program 97.7%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around 0 37.7%

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

      1. *-commutative 37.7%

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

      2. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y around inf 37.7%

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

      1. unpow2 37.7%

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

      2. *-commutative 37.7%

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

   9. Simplified 37.7%

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

       1. associate-/r* 37.5%

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

       2. *-un-lft-identity 37.5%

          \[\leadsto 6 \cdot \frac{\color{blue}{1 \cdot \frac{x}{z}}}{y \cdot y} \]

       3. times-frac 37.7%

          \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]

   11. Applied egg-rr 37.7%

       \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \]

Alternative 6: 66.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.62 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.62e+16)
   (* (/ x z) (+ 1.0 (* (* y y) -0.16666666666666666)))
   (* 6.0 (* (/ 1.0 y) (/ (/ x z) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.62e+16) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / 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 <= 1.62d+16) then
        tmp = (x / z) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = 6.0d0 * ((1.0d0 / y) * ((x / z) / 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 <= 1.62e+16) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 1.62e+16:
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.62e+16)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 * Float64(Float64(1.0 / y) * Float64(Float64(x / z) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.62e+16)
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1.62e+16], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(1.0 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.62e16

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*r/ 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in y around 0 69.8%

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

      1. unpow2 69.8%

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

   6. Simplified 69.8%

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

   if 1.62e16 < y

   1. Initial program 97.6%

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

      1. associate-/l* 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in y around 0 39.6%

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

      1. *-commutative 39.6%

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

      2. unpow2 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in y around inf 39.6%

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

      1. unpow2 39.6%

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

      2. *-commutative 39.6%

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

   9. Simplified 39.6%

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

       1. associate-/r* 39.4%

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

       2. *-un-lft-identity 39.4%

          \[\leadsto 6 \cdot \frac{\color{blue}{1 \cdot \frac{x}{z}}}{y \cdot y} \]

       3. times-frac 39.5%

          \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]

   11. Applied egg-rr 39.5%

       \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.62 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \]

Alternative 7: 66.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4) (/ x z) (* 6.0 (/ x (* y (* y z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (y * (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 <= 2.4d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * (x / (y * (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 <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (y * (y * z)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2.4:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (y * (y * z)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(x / Float64(y * Float64(y * z))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (y * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 97.1%

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

      1. associate-/l* 96.9%

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

      2. associate-/r/ 83.3%

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

      3. associate-/l/ 76.2%

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

      4. associate-/r/ 82.4%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 70.2%

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

   if 2.39999999999999991 < y

   1. Initial program 97.7%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around 0 37.7%

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

      1. *-commutative 37.7%

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

      2. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y around inf 37.7%

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

      1. unpow2 37.7%

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

      2. *-commutative 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in z around 0 37.7%

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

       1. unpow2 37.7%

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

       2. associate-*l* 37.7%

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

   12. Simplified 37.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 8: 66.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot z}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4) (/ x z) (* 6.0 (/ (/ x (* y z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((x / (y * z)) / 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 <= 2.4d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * ((x / (y * z)) / 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 <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((x / (y * z)) / y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2.4:
		tmp = x / z
	else:
		tmp = 6.0 * ((x / (y * z)) / y)
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(x / Float64(y * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x / z;
	else
		tmp = 6.0 * ((x / (y * z)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 97.1%

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

      1. associate-/l* 96.9%

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

      2. associate-/r/ 83.3%

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

      3. associate-/l/ 76.2%

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

      4. associate-/r/ 82.4%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 70.2%

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

   if 2.39999999999999991 < y

   1. Initial program 97.7%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around 0 37.7%

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

      1. *-commutative 37.7%

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

      2. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y around inf 37.7%

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

      1. unpow2 37.7%

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

      2. *-commutative 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in x around 0 37.7%

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

       1. unpow2 37.7%

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

       2. *-commutative 37.7%

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

       3. associate-/r* 37.5%

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

       4. *-lft-identity 37.5%

          \[\leadsto 6 \cdot \frac{\color{blue}{1 \cdot \frac{x}{z}}}{y \cdot y} \]

       5. times-frac 37.7%

          \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]

       6. associate-*l/ 37.7%

          \[\leadsto 6 \cdot \color{blue}{\frac{1 \cdot \frac{\frac{x}{z}}{y}}{y}} \]

       7. *-lft-identity 37.7%

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

       8. associate-/l/ 37.7%

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

   12. Simplified 37.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot z}}{y}\\ \end{array} \]

Alternative 9: 66.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

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 + (0.16666666666666666d0 * (y * y))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. associate-/l* 95.3%

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

3. Simplified 95.3%

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

4. Taylor expanded in y around 0 65.6%

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

   1. *-commutative 65.6%

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

   2. unpow2 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in z around 0 65.6%

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

   1. unpow2 65.6%

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

9. Simplified 65.6%

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

10. Final simplification 65.6%

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

Alternative 10: 66.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

y = abs(y);
double code(double x, double y, double z) {
	return x / (z + (0.16666666666666666 * (y * (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 / (z + (0.16666666666666666d0 * (y * (y * z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. associate-/l* 95.3%

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

3. Simplified 95.3%

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

4. Taylor expanded in y around 0 65.6%

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

   1. *-commutative 65.6%

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

   2. unpow2 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in z around 0 65.6%

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

   1. unpow2 22.2%

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

   2. associate-*l* 22.3%

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

9. Simplified 65.6%

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

10. Final simplification 65.6%

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

Alternative 11: 60.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+101) (/ x z) (* (/ y z) (/ x y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+101) {
		tmp = x / z;
	} else {
		tmp = (y / z) * (x / 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 <= 4d+101) then
        tmp = x / z
    else
        tmp = (y / z) * (x / 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 <= 4e+101) {
		tmp = x / z;
	} else {
		tmp = (y / z) * (x / y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 4e+101:
		tmp = x / z
	else:
		tmp = (y / z) * (x / y)
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+101)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(y / z) * Float64(x / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+101)
		tmp = x / z;
	else
		tmp = (y / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 4e+101], N[(x / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999999e101

   1. Initial program 97.4%

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

      1. associate-/l* 95.4%

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

      2. associate-/r/ 83.4%

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

      3. associate-/l/ 78.7%

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

      4. associate-/r/ 84.3%

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

      5. associate-/r* 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in y around 0 65.1%

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

   if 3.9999999999999999e101 < y

   1. Initial program 96.6%

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

      1. associate-/l* 94.3%

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

      2. associate-/r/ 94.3%

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

      3. associate-/l/ 96.5%

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

      4. associate-/r/ 96.7%

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

      5. associate-/r* 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in x around 0 94.6%

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

   5. Taylor expanded in y around 0 23.7%

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

      1. *-commutative 23.7%

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

      2. *-commutative 23.7%

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

      3. times-frac 20.6%

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

   7. Applied egg-rr 20.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 12: 61.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e-8) (/ x z) (/ (* x y) (* y z))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e-8) {
		tmp = x / z;
	} else {
		tmp = (x * y) / (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 <= 3.2d-8) then
        tmp = x / z
    else
        tmp = (x * y) / (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 <= 3.2e-8) {
		tmp = x / z;
	} else {
		tmp = (x * y) / (y * z);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 3.2e-8:
		tmp = x / z
	else:
		tmp = (x * y) / (y * z)
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x * y) / Float64(y * z));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.2e-8)
		tmp = x / z;
	else
		tmp = (x * y) / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 3.2e-8], N[(x / z), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000002e-8

   1. Initial program 97.1%

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

      1. associate-/l* 96.9%

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

      2. associate-/r/ 83.2%

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

      3. associate-/l/ 76.1%

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

      4. associate-/r/ 82.3%

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

      5. associate-/r* 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in y around 0 70.2%

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

   if 3.2000000000000002e-8 < y

   1. Initial program 97.7%

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

      1. associate-/l* 90.4%

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

      2. associate-/r/ 90.4%

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

      3. associate-/l/ 97.7%

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

      4. associate-/r/ 97.8%

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

      5. associate-/r* 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 90.6%

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

   5. Taylor expanded in y around 0 24.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y \cdot z}\\ \end{array} \]

Alternative 13: 58.3% 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 97.3%

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

   1. associate-/l* 95.3%

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

   2. associate-/r/ 85.0%

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

   3. associate-/l/ 81.4%

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

   4. associate-/r/ 86.2%

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

   5. associate-/r* 83.7%

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

3. Simplified 83.7%

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

4. Taylor expanded in y around 0 56.9%

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

5. Final simplification 56.9%

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

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 2023293 
(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))