Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 97.8%

Time: 9.0s

Alternatives: 12

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: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e-14) (/ x (/ z (/ (sin y) y))) (/ (/ (* x (sin y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-14) {
		tmp = x / (z / (sin(y) / y));
	} else {
		tmp = ((x * sin(y)) / 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 (x <= 1d-14) then
        tmp = x / (z / (sin(y) / y))
    else
        tmp = ((x * sin(y)) / y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-14) {
		tmp = x / (z / (Math.sin(y) / y));
	} else {
		tmp = ((x * Math.sin(y)) / y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1e-14:
		tmp = x / (z / (math.sin(y) / y))
	else:
		tmp = ((x * math.sin(y)) / y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e-14)
		tmp = Float64(x / Float64(z / Float64(sin(y) / y)));
	else
		tmp = Float64(Float64(Float64(x * sin(y)) / y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1e-14)
		tmp = x / (z / (sin(y) / y));
	else
		tmp = ((x * sin(y)) / y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e-14], N[(x / N[(z / N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999999e-15

   1. Initial program 96.2%

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

      1. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   if 9.99999999999999999e-15 < x

   1. Initial program 99.5%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. associate-*l/ 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \end{array} \]

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= z 5e-49) (* x (/ t_0 z)) (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z <= 5e-49) {
		tmp = x * (t_0 / z);
	} else {
		tmp = t_0 * (x / 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= 5d-49) then
        tmp = x * (t_0 / z)
    else
        tmp = t_0 * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= 5e-49) {
		tmp = x * (t_0 / z);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= 5e-49:
		tmp = x * (t_0 / z)
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= 5e-49)
		tmp = Float64(x * Float64(t_0 / z));
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= 5e-49)
		tmp = x * (t_0 / z);
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, 5e-49], N[(x * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999999e-49

   1. Initial program 95.9%

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

      1. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   if 4.9999999999999999e-49 < z

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= z 4e+42) (/ x (/ z t_0)) (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z <= 4e+42) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= 4d+42) then
        tmp = x / (z / t_0)
    else
        tmp = t_0 * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= 4e+42) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= 4e+42:
		tmp = x / (z / t_0)
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= 4e+42)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= 4e+42)
		tmp = x / (z / t_0);
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, 4e+42], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 4.00000000000000018e42

   1. Initial program 96.3%

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

      1. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   if 4.00000000000000018e42 < z

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 5.5e-61) (/ x (/ z t_0)) (/ (* x t_0) z))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 5.5e-61) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= 5.5d-61) then
        tmp = x / (z / t_0)
    else
        tmp = (x * t_0) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 5.5e-61) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 5.5e-61:
		tmp = x / (z / t_0)
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 5.5e-61)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x * t_0) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 5.5e-61)
		tmp = x / (z / t_0);
	else
		tmp = (x * t_0) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 5.5e-61], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999997e-61

   1. Initial program 96.0%

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

      1. associate-/l* 98.5%

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

   3. Simplified 98.5%

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

   if 5.4999999999999997e-61 < x

   1. Initial program 99.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 5: 95.7% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.1%

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

   1. associate-*r/ 95.6%

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

3. Simplified 95.6%

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

4. Final simplification 95.6%

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

Alternative 6: 62.2% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e+79) (/ x z) (* (- y) (/ (/ x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+79) {
		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 <= 4.5d+79) 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 <= 4.5e+79) {
		tmp = x / z;
	} else {
		tmp = -y * ((x / y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e+79:
		tmp = x / z
	else:
		tmp = -y * ((x / y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e+79)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(-y) * Float64(Float64(x / y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e+79)
		tmp = x / z;
	else
		tmp = -y * ((x / y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e+79], N[(x / z), $MachinePrecision], N[((-y) * N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999994e79

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 66.8%

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

   if 4.49999999999999994e79 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 92.9%

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

      2. associate-/l/ 92.8%

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

      3. associate-/r* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in y around 0 13.0%

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

      1. associate-/l/ 25.5%

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

      2. div-inv 25.5%

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

   6. Applied egg-rr 25.5%

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

      1. *-commutative 25.5%

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

      2. un-div-inv 25.5%

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

      3. *-commutative 25.5%

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

      4. associate-/r* 13.0%

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

      5. add-sqr-sqrt 13.0%

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

      6. sqrt-unprod 6.5%

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

      7. sqr-neg 6.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 14.1%

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

      10. add-sqr-sqrt 14.1%

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

      11. sqrt-unprod 31.4%

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

      12. sqr-neg 31.4%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 13.0%

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

      15. associate-/r/ 23.0%

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

      16. associate-/l/ 35.2%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 8.5%

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

      19. sqr-neg 8.5%

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

      20. sqrt-unprod 36.3%

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

      21. add-sqr-sqrt 36.3%

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

   8. Applied egg-rr 35.2%

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

      1. frac-2neg 35.2%

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

      2. distribute-rgt-neg-out 35.2%

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

      3. div-inv 35.2%

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

      4. associate-/r* 35.2%

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

      5. clear-num 35.2%

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

      6. add-sqr-sqrt 19.3%

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

      7. sqrt-unprod 36.3%

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

      8. sqr-neg 36.3%

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

      9. sqrt-unprod 17.4%

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

      10. add-sqr-sqrt 36.3%

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

   10. Applied egg-rr 36.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 7: 62.4% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x} \cdot \left(-z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.3e+79) (/ x z) (/ y (* (/ y x) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e+79) {
		tmp = x / z;
	} else {
		tmp = y / ((y / x) * -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 <= 4.3d+79) then
        tmp = x / z
    else
        tmp = y / ((y / x) * -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e+79) {
		tmp = x / z;
	} else {
		tmp = y / ((y / x) * -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.3e+79:
		tmp = x / z
	else:
		tmp = y / ((y / x) * -z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.3e+79)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(Float64(y / x) * Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.3e+79)
		tmp = x / z;
	else
		tmp = y / ((y / x) * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.3e+79], N[(x / z), $MachinePrecision], N[(y / N[(N[(y / x), $MachinePrecision] * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3000000000000003e79

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 66.8%

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

   if 4.3000000000000003e79 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 92.9%

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

      2. associate-/l/ 92.8%

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

      3. associate-/r* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in y around 0 13.0%

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

      1. associate-/l/ 25.5%

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

      2. div-inv 25.5%

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

   6. Applied egg-rr 25.5%

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

      1. *-commutative 25.5%

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

      2. un-div-inv 25.5%

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

      3. *-commutative 25.5%

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

      4. associate-/r* 13.0%

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

      5. add-sqr-sqrt 13.0%

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

      6. sqrt-unprod 6.5%

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

      7. sqr-neg 6.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 14.1%

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

      10. add-sqr-sqrt 14.1%

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

      11. sqrt-unprod 31.4%

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

      12. sqr-neg 31.4%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 13.0%

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

      15. associate-/r/ 23.0%

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

      16. distribute-frac-neg 23.0%

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

      17. frac-2neg 23.0%

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

      18. distribute-frac-neg 23.0%

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

      19. remove-double-neg 23.0%

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

      20. associate-/l/ 35.2%

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

   8. Applied egg-rr 36.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x} \cdot \left(-z\right)}\\ \end{array} \]

Alternative 8: 60.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.5) (/ x z) (* x (/ y (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.5) {
		tmp = x / z;
	} else {
		tmp = x * (y / (z * 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 <= 0.5d0) then
        tmp = x / z
    else
        tmp = x * (y / (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.5) {
		tmp = x / z;
	} else {
		tmp = x * (y / (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.5:
		tmp = x / z
	else:
		tmp = x * (y / (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(x * Float64(y / Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.5)
		tmp = x / z;
	else
		tmp = x * (y / (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.5], N[(x / z), $MachinePrecision], N[(x * N[(y / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.5

   1. Initial program 98.6%

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

   2. Taylor expanded in y around 0 69.9%

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

   if 0.5 < y

   1. Initial program 93.3%

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

      1. associate-*r/ 94.7%

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

      2. associate-/l/ 94.7%

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

      3. associate-/r* 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in y around 0 19.8%

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

      1. *-commutative 19.8%

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

      2. frac-2neg 19.8%

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

      3. associate-*l/ 19.7%

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

      4. distribute-neg-frac 19.7%

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

   6. Applied egg-rr 19.7%

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

      1. associate-/l* 27.2%

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

      2. associate-/r/ 19.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 9.2%

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

      5. sqr-neg 9.2%

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

      6. sqrt-unprod 13.9%

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

      7. add-sqr-sqrt 13.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 30.9%

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

      10. sqr-neg 30.9%

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

      11. sqrt-unprod 19.8%

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

      12. add-sqr-sqrt 19.8%

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

      13. associate-/r* 29.0%

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

      14. *-commutative 29.0%

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

   8. Applied egg-rr 29.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \]

Alternative 9: 62.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+28) (/ x z) (* y (/ (/ x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+28) {
		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 <= 2d+28) 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 <= 2e+28) {
		tmp = x / z;
	} else {
		tmp = y * ((x / y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+28:
		tmp = x / z
	else:
		tmp = y * ((x / y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+28)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(Float64(x / y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+28)
		tmp = x / z;
	else
		tmp = y * ((x / y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+28], N[(x / z), $MachinePrecision], N[(y * N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e28

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 69.1%

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

   if 1.99999999999999992e28 < y

   1. Initial program 92.6%

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

      1. associate-*r/ 94.2%

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

      2. associate-/l/ 94.1%

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

      3. associate-/r* 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in y around 0 17.0%

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

      1. associate-/l/ 27.1%

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

      2. div-inv 27.1%

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

   6. Applied egg-rr 27.1%

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

      1. *-commutative 27.1%

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

      2. un-div-inv 27.1%

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

      3. *-commutative 27.1%

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

      4. associate-/r* 17.0%

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

      5. add-sqr-sqrt 17.0%

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

      6. sqrt-unprod 11.8%

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

      7. sqr-neg 11.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 15.0%

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

      10. add-sqr-sqrt 15.0%

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

      11. sqrt-unprod 29.0%

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

      12. sqr-neg 29.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 17.0%

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

      15. associate-/r/ 25.2%

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

      16. associate-/l/ 35.0%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 10.5%

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

      19. sqr-neg 10.5%

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

      20. sqrt-unprod 33.0%

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

      21. add-sqr-sqrt 33.0%

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

   8. Applied egg-rr 35.0%

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

      1. associate-/r* 28.5%

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

      2. add-sqr-sqrt 10.6%

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

      3. sqrt-unprod 33.3%

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

      4. sqr-neg 33.3%

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

      5. sqrt-unprod 16.5%

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

      6. add-sqr-sqrt 26.5%

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

      7. associate-/r* 33.0%

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

      8. div-inv 33.0%

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

      9. *-commutative 33.0%

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

      10. associate-/r* 33.0%

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

      11. clear-num 33.0%

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

      12. add-sqr-sqrt 18.8%

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

      13. sqrt-unprod 33.3%

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

      14. sqr-neg 33.3%

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

      15. sqrt-unprod 14.7%

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

      16. add-sqr-sqrt 35.0%

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

   10. Applied egg-rr 35.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 10: 62.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+30) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+30) {
		tmp = x / z;
	} else {
		tmp = y / (z * (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 <= 3d+30) then
        tmp = x / z
    else
        tmp = y / (z * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+30) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+30:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+30)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(z * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e+30)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+30], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999978e30

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 68.7%

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

   if 2.99999999999999978e30 < y

   1. Initial program 92.5%

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

      1. associate-*r/ 94.1%

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

      2. associate-/l/ 94.0%

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

      3. associate-/r* 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in y around 0 17.3%

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

      1. associate-/l/ 27.5%

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

      2. div-inv 27.5%

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

   6. Applied egg-rr 27.5%

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

      1. *-commutative 27.5%

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

      2. un-div-inv 27.5%

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

      3. *-commutative 27.5%

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

      4. associate-/r* 17.3%

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

      5. add-sqr-sqrt 17.3%

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

      6. sqrt-unprod 12.0%

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

      7. sqr-neg 12.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 15.1%

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

      10. add-sqr-sqrt 15.1%

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

      11. sqrt-unprod 29.3%

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

      12. sqr-neg 29.3%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 17.3%

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

      15. associate-/r/ 25.5%

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

      16. associate-/l/ 35.5%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 10.5%

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

      19. sqr-neg 10.5%

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

      20. sqrt-unprod 33.3%

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

      21. add-sqr-sqrt 33.3%

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

   8. Applied egg-rr 35.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 11: 62.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+28) (/ x z) (/ y (/ (* z y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+28) {
		tmp = x / z;
	} else {
		tmp = y / ((z * 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 <= 2d+28) then
        tmp = x / z
    else
        tmp = y / ((z * y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+28) {
		tmp = x / z;
	} else {
		tmp = y / ((z * y) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+28:
		tmp = x / z
	else:
		tmp = y / ((z * y) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+28)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(Float64(z * y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+28)
		tmp = x / z;
	else
		tmp = y / ((z * y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+28], N[(x / z), $MachinePrecision], N[(y / N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e28

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 69.1%

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

   if 1.99999999999999992e28 < y

   1. Initial program 92.6%

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

      1. associate-*r/ 94.2%

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

   3. Simplified 94.2%

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

      1. *-commutative 94.2%

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

      2. associate-/l/ 94.1%

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

      3. associate-*l/ 94.1%

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

      4. *-commutative 94.1%

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

      5. associate-/l* 94.1%

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

   5. Applied egg-rr 94.1%

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

   6. Taylor expanded in y around 0 35.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \end{array} \]

Alternative 12: 58.7% 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. Taylor expanded in y around 0 55.3%

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

3. Final simplification 55.3%

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

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 2023318 
(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))