Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 96.1%

Time: 8.1s

Alternatives: 11

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: 96.1% 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: 96.1% 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]

Derivation

1. Initial program 97.1%

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

2. Final simplification 97.1%

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

Alternative 2: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e-9) (/ x z) (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e-9) {
		tmp = x / z;
	} else {
		tmp = sin(y) * (x / (y * z));
	}
	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) :: tmp
    if (y <= 6.5d-9) then
        tmp = x / z
    else
        tmp = sin(y) * (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e-9) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.5e-9:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e-9)
		tmp = Float64(x / z);
	else
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e-9)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e-9], 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 < 6.5000000000000003e-9

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r/ 76.5%

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

      5. *-commutative 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in y around 0 71.5%

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

   if 6.5000000000000003e-9 < y

   1. Initial program 95.7%

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

      1. associate-*l/ 85.4%

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

      2. times-frac 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-*r/ 90.4%

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

      5. *-commutative 90.4%

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

   3. Simplified 90.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 3: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00025:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00025)
   (/ (* x (+ 1.0 (* -0.16666666666666666 (* y y)))) z)
   (* (/ (sin y) z) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00025) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = (sin(y) / z) * (x / y);
	}
	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) :: tmp
    if (y <= 0.00025d0) then
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))) / z
    else
        tmp = (sin(y) / z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00025) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = (Math.sin(y) / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00025:
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z
	else:
		tmp = (math.sin(y) / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00025)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))) / z);
	else
		tmp = Float64(Float64(sin(y) / z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00025)
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	else
		tmp = (sin(y) / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00025], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5000000000000001e-4

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 68.9%

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

      1. unpow2 68.9%

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

   4. Simplified 68.9%

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

   if 2.5000000000000001e-4 < y

   1. Initial program 95.6%

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

      1. associate-*r/ 95.6%

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

      2. associate-/l/ 90.2%

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

      3. *-commutative 90.2%

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

      4. times-frac 95.6%

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

   3. Simplified 95.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00025:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 59.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{6}{z}}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e+21)
   (/ (* x (+ 1.0 (* -0.16666666666666666 (* y y)))) z)
   (/ (/ 6.0 z) (* y (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+21) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = (6.0 / z) / (y * (y / x));
	}
	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) :: tmp
    if (y <= 7.5d+21) then
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))) / z
    else
        tmp = (6.0d0 / z) / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+21) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = (6.0 / z) / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.5e+21:
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z
	else:
		tmp = (6.0 / z) / (y * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e+21)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))) / z);
	else
		tmp = Float64(Float64(6.0 / z) / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.5e+21)
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	else
		tmp = (6.0 / z) / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.5e+21], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(6.0 / z), $MachinePrecision] / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5e21

   1. Initial program 97.7%

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

   2. Taylor expanded in y around 0 68.1%

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

      1. unpow2 68.1%

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

   4. Simplified 68.1%

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

   if 7.5e21 < y

   1. Initial program 95.2%

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

      1. associate-/l* 89.1%

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

      2. associate-/r/ 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in y around 0 38.1%

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

   5. Taylor expanded in y around inf 38.1%

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

      1. *-commutative 38.1%

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

      2. unpow2 38.1%

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

   7. Simplified 38.1%

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

   8. Taylor expanded in x around 0 38.1%

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

      1. unpow2 38.1%

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

      2. associate-/r* 38.1%

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

      3. associate-*r/ 38.1%

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

      4. associate-*l/ 38.1%

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

      5. associate-*r/ 37.8%

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

      6. associate-/l* 38.1%

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

      7. associate-*l/ 38.2%

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

   10. Simplified 38.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{6}{z}}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 5: 61.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00056) (/ x z) (* 6.0 (/ x (* z (* y y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	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) :: tmp
    if (y <= 0.00056d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * (x / (z * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00056:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (z * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00056)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(x / Float64(z * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00056)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (z * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00056], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999995e-4

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 5.5999999999999995e-4 < y

   1. Initial program 95.6%

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

      1. associate-/l* 90.2%

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

      2. associate-/r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 37.5%

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

   5. Taylor expanded in y around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

   7. Simplified 37.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 6: 61.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00056) (/ x z) (* (/ 6.0 z) (/ x (* y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (6.0 / z) * (x / (y * y));
	}
	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) :: tmp
    if (y <= 0.00056d0) then
        tmp = x / z
    else
        tmp = (6.0d0 / z) * (x / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (6.0 / z) * (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00056:
		tmp = x / z
	else:
		tmp = (6.0 / z) * (x / (y * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00056)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(6.0 / z) * Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00056)
		tmp = x / z;
	else
		tmp = (6.0 / z) * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00056], N[(x / z), $MachinePrecision], N[(N[(6.0 / z), $MachinePrecision] * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999995e-4

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 5.5999999999999995e-4 < y

   1. Initial program 95.6%

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

      1. associate-/l* 90.2%

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

      2. associate-/r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 37.5%

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

   5. Taylor expanded in y around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-*l/ 37.5%

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

      3. unpow2 37.5%

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

      4. associate-*l* 37.5%

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

   7. Simplified 37.5%

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

      1. associate-*r* 37.5%

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

      2. times-frac 37.5%

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

   9. Applied egg-rr 37.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \end{array} \]

Alternative 7: 62.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00056) (/ x z) (* (/ x (* y z)) (/ 6.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	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) :: tmp
    if (y <= 0.00056d0) then
        tmp = x / z
    else
        tmp = (x / (y * z)) * (6.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00056:
		tmp = x / z
	else:
		tmp = (x / (y * z)) * (6.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00056)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / Float64(y * z)) * Float64(6.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00056)
		tmp = x / z;
	else
		tmp = (x / (y * z)) * (6.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00056], N[(x / z), $MachinePrecision], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999995e-4

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 5.5999999999999995e-4 < y

   1. Initial program 95.6%

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

      1. associate-/l* 90.2%

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

      2. associate-/r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 37.5%

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

   5. Taylor expanded in y around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-*l/ 37.5%

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

      3. unpow2 37.5%

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

      4. associate-*l* 37.5%

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

   7. Simplified 37.5%

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

      1. *-commutative 37.5%

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

      2. times-frac 37.6%

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

   9. Applied egg-rr 37.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \]

Alternative 8: 61.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\frac{6}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00056) (/ x z) (* (/ x y) (/ (/ 6.0 y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (x / y) * ((6.0 / y) / z);
	}
	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) :: tmp
    if (y <= 0.00056d0) then
        tmp = x / z
    else
        tmp = (x / y) * ((6.0d0 / y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (x / y) * ((6.0 / y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00056:
		tmp = x / z
	else:
		tmp = (x / y) * ((6.0 / y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00056)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / y) * Float64(Float64(6.0 / y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00056)
		tmp = x / z;
	else
		tmp = (x / y) * ((6.0 / y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00056], N[(x / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(N[(6.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999995e-4

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 5.5999999999999995e-4 < y

   1. Initial program 95.6%

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

      1. associate-/l* 90.2%

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

      2. associate-/r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 37.5%

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

   5. Taylor expanded in y around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

      3. associate-*r* 37.5%

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

      4. *-commutative 37.5%

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

      5. associate-*l* 37.5%

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

      6. *-commutative 37.5%

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

      7. associate-*l* 37.5%

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

      8. *-commutative 37.5%

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

   7. Simplified 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. times-frac 37.6%

         \[\leadsto \color{blue}{\frac{1}{y \cdot z} \cdot \frac{x}{y \cdot 0.16666666666666666}} \]

      3. un-div-inv 37.6%

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

      4. metadata-eval 37.6%

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

      5. div-inv 37.6%

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

      6. clear-num 37.6%

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

      7. associate-*l/ 37.6%

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

      8. *-un-lft-identity 37.6%

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

      9. *-commutative 37.6%

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

      10. *-commutative 37.6%

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

      11. times-frac 37.6%

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

   9. Applied egg-rr 37.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\frac{6}{y}}{z}\\ \end{array} \]

Alternative 9: 61.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{6}{z}}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00056) (/ x z) (/ (/ 6.0 z) (* y (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (6.0 / z) / (y * (y / x));
	}
	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) :: tmp
    if (y <= 0.00056d0) then
        tmp = x / z
    else
        tmp = (6.0d0 / z) / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00056) {
		tmp = x / z;
	} else {
		tmp = (6.0 / z) / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00056:
		tmp = x / z
	else:
		tmp = (6.0 / z) / (y * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00056)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(6.0 / z) / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00056)
		tmp = x / z;
	else
		tmp = (6.0 / z) / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00056], N[(x / z), $MachinePrecision], N[(N[(6.0 / z), $MachinePrecision] / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999995e-4

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 5.5999999999999995e-4 < y

   1. Initial program 95.6%

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

      1. associate-/l* 90.2%

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

      2. associate-/r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 37.5%

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

   5. Taylor expanded in y around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in x around 0 37.5%

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

      1. unpow2 37.5%

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

      2. associate-/r* 37.5%

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

      3. associate-*r/ 37.5%

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

      4. associate-*l/ 37.5%

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

      5. associate-*r/ 37.2%

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

      6. associate-/l* 37.5%

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

      7. associate-*l/ 37.6%

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

   10. Simplified 37.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00056:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{6}{z}}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 10: 61.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.6e-9) (/ x z) (* y (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-9) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	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) :: tmp
    if (y <= 6.6d-9) then
        tmp = x / z
    else
        tmp = y * (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-9) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.6e-9:
		tmp = x / z
	else:
		tmp = y * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.6e-9)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.6e-9)
		tmp = x / z;
	else
		tmp = y * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.6e-9], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.60000000000000037e-9

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. times-frac 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r/ 76.5%

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

      5. *-commutative 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in y around 0 71.5%

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

   if 6.60000000000000037e-9 < y

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. associate-/l/ 90.3%

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

      3. *-commutative 90.3%

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

      4. times-frac 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in y around 0 27.6%

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

      1. *-commutative 27.6%

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

      2. clear-num 29.0%

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

      3. frac-times 37.6%

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

      4. *-un-lft-identity 37.6%

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

   6. Applied egg-rr 37.6%

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

      1. div-inv 37.6%

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

      2. associate-*l/ 38.1%

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

      3. clear-num 38.1%

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

   8. Applied egg-rr 38.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 11: 58.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x / 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 / z
end function

Java

public static double code(double x, double y, double z) {
	return x / z;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. associate-*l/ 94.3%

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

   2. times-frac 81.2%

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

   3. *-commutative 81.2%

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

   4. associate-*r/ 80.4%

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

   5. *-commutative 80.4%

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

3. Simplified 80.4%

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

4. Taylor expanded in y around 0 57.2%

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

5. Final simplification 57.2%

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

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 2023240 
(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))