Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 96.7%

Time: 8.3s

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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7.6e+58) (/ x (* z (/ y (sin y)))) (/ (sin y) (* y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.6e+58) {
		tmp = x / (z * (y / sin(y)));
	} else {
		tmp = sin(y) / (y * (z / 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 (z <= 7.6d+58) then
        tmp = x / (z * (y / sin(y)))
    else
        tmp = sin(y) / (y * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.6e+58) {
		tmp = x / (z * (y / Math.sin(y)));
	} else {
		tmp = Math.sin(y) / (y * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7.6e+58:
		tmp = x / (z * (y / math.sin(y)))
	else:
		tmp = math.sin(y) / (y * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7.6e+58)
		tmp = Float64(x / Float64(z * Float64(y / sin(y))));
	else
		tmp = Float64(sin(y) / Float64(y * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 7.6e+58)
		tmp = x / (z * (y / sin(y)));
	else
		tmp = sin(y) / (y * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7.6e+58], N[(x / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.5999999999999997e58

   1. Initial program 97.3%

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

      1. associate-/l* 98.4%

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

   3. Simplified 98.4%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 98.4%

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

      3. clear-num 98.4%

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

   5. Applied egg-rr 98.4%

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

   if 7.5999999999999997e58 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 88.1%

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

      1. associate-*l/ 90.0%

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

      2. associate-/r* 90.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l/ 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 2: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.65e-15) (/ x z) (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-15) {
		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 <= 2.65d-15) 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 <= 2.65e-15) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.65e-15:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.65e-15)
		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 <= 2.65e-15)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.65e-15], 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 < 2.6500000000000001e-15

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 75.4%

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

   if 2.6500000000000001e-15 < y

   1. Initial program 95.4%

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

      1. associate-*l/ 90.1%

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

      2. times-frac 92.9%

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

      3. *-commutative 92.9%

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

      4. associate-*r/ 92.9%

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

      5. *-commutative 92.9%

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

   3. Simplified 92.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.6e-14) (/ x z) (* (/ (sin y) z) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e-14) {
		tmp = x / 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 <= 5.6d-14) then
        tmp = x / 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 <= 5.6e-14) {
		tmp = x / z;
	} else {
		tmp = (Math.sin(y) / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.6e-14:
		tmp = x / z
	else:
		tmp = (math.sin(y) / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.6e-14)
		tmp = Float64(x / 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 <= 5.6e-14)
		tmp = x / z;
	else
		tmp = (sin(y) / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.6e-14], N[(x / 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 < 5.6000000000000001e-14

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 75.5%

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

   if 5.6000000000000001e-14 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 95.5%

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

      2. associate-/l/ 92.8%

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

      3. *-commutative 92.8%

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

      4. times-frac 95.4%

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

   3. Simplified 95.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (y / sin(y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

   1. associate-/l* 96.8%

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

3. Simplified 96.8%

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

   1. clear-num 96.8%

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

   2. associate-/r/ 96.8%

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

   3. clear-num 96.9%

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

5. Applied egg-rr 96.9%

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

6. Final simplification 96.9%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

2. Final simplification 97.8%

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

Alternative 6: 61.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.6e-14)
   (/ x z)
   (* (/ 1.0 y) (/ x (* z (* y 0.16666666666666666))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e-14) {
		tmp = x / z;
	} else {
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)));
	}
	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 <= 5.6d-14) then
        tmp = x / z
    else
        tmp = (1.0d0 / y) * (x / (z * (y * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e-14) {
		tmp = x / z;
	} else {
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.6e-14:
		tmp = x / z
	else:
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.6e-14)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(z * Float64(y * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.6e-14)
		tmp = x / z;
	else
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.6e-14], N[(x / z), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6000000000000001e-14

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 75.5%

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

   if 5.6000000000000001e-14 < y

   1. Initial program 95.4%

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

      1. associate-/l* 92.8%

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

   3. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. associate-/r/ 92.7%

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

      3. clear-num 92.8%

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

   5. Applied egg-rr 92.8%

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

   6. Taylor expanded in y around 0 35.7%

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

      1. unpow2 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in y around inf 35.7%

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

       1. unpow2 35.7%

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

       2. associate-*r* 35.7%

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

       3. *-commutative 35.7%

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

       4. associate-*r* 35.7%

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

       5. *-commutative 35.7%

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

       6. associate-*r* 35.8%

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

   11. Simplified 35.8%

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

       1. *-un-lft-identity 35.8%

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

       2. associate-*l* 35.7%

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

       3. *-commutative 35.7%

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

       4. associate-*l* 35.8%

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

       5. times-frac 37.3%

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

   13. Applied egg-rr 37.3%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{\left(y \cdot 0.16666666666666666\right) \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 7: 59.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1450000000:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1450000000.0)
   (/ (* x (+ 1.0 (* -0.16666666666666666 (* y y)))) z)
   (* (/ 1.0 y) (/ x (* z (* y 0.16666666666666666))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1450000000.0) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)));
	}
	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 <= 1450000000.0d0) then
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))) / z
    else
        tmp = (1.0d0 / y) * (x / (z * (y * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1450000000.0) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1450000000.0:
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z
	else:
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1450000000.0)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))) / z);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(z * Float64(y * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1450000000.0)
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	else
		tmp = (1.0 / y) * (x / (z * (y * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1450000000.0], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e9

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 68.3%

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

      1. unpow2 68.3%

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

   4. Simplified 68.3%

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

   if 1.45e9 < y

   1. Initial program 95.1%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

      1. clear-num 92.3%

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

      2. associate-/r/ 92.3%

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

      3. clear-num 92.4%

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

   5. Applied egg-rr 92.4%

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

   6. Taylor expanded in y around 0 36.2%

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

      1. unpow2 36.2%

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

   8. Simplified 36.2%

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

   9. Taylor expanded in y around inf 36.2%

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

       1. unpow2 36.2%

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

       2. associate-*r* 36.2%

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

       3. *-commutative 36.2%

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

       4. associate-*r* 36.2%

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

       5. *-commutative 36.2%

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

       6. associate-*r* 36.3%

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

   11. Simplified 36.3%

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

       1. *-un-lft-identity 36.3%

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

       2. associate-*l* 36.2%

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

       3. *-commutative 36.2%

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

       4. associate-*l* 36.3%

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

       5. times-frac 37.8%

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

   13. Applied egg-rr 37.8%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{\left(y \cdot 0.16666666666666666\right) \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1450000000:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 8: 65.9% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (/ x (* z (+ 1.0 (* (* y y) 0.16666666666666666)))))

C

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

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 * (1.0d0 + ((y * y) * 0.16666666666666666d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

   1. associate-/l* 96.8%

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

3. Simplified 96.8%

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

   1. clear-num 96.8%

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

   2. associate-/r/ 96.8%

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

   3. clear-num 96.9%

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

5. Applied egg-rr 96.9%

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

6. Taylor expanded in y around 0 69.1%

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

   1. unpow2 69.1%

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

8. Simplified 69.1%

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

9. Final simplification 69.1%

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

Alternative 9: 61.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.65e-15) (/ x z) (* y (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-15) {
		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 <= 2.65d-15) 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 <= 2.65e-15) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.65e-15:
		tmp = x / z
	else:
		tmp = y * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.65e-15)
		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 <= 2.65e-15)
		tmp = x / z;
	else
		tmp = y * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.65e-15], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6500000000000001e-15

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 75.4%

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

   if 2.6500000000000001e-15 < y

   1. Initial program 95.4%

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

      1. associate-*l/ 90.1%

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

      2. times-frac 92.9%

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

      3. *-commutative 92.9%

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

      4. associate-*r/ 92.9%

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

      5. *-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in y around 0 36.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 10: 62.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4000000000.0) (/ x z) (/ y (* y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4000000000.0) {
		tmp = x / z;
	} else {
		tmp = y / (y * (z / 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 <= 4000000000.0d0) then
        tmp = x / z
    else
        tmp = y / (y * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4000000000.0) {
		tmp = x / z;
	} else {
		tmp = y / (y * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4000000000.0:
		tmp = x / z
	else:
		tmp = y / (y * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4000000000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(y * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4000000000.0)
		tmp = x / z;
	else
		tmp = y / (y * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4000000000.0], N[(x / z), $MachinePrecision], N[(y / N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e9

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 74.5%

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

   if 4e9 < y

   1. Initial program 95.1%

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

      1. associate-*l/ 89.4%

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

      2. times-frac 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*r/ 92.4%

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

      5. *-commutative 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in y around 0 35.7%

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

      1. clear-num 37.6%

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

      2. un-div-inv 37.6%

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

      3. associate-/l* 37.4%

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

      4. div-inv 37.4%

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

      5. clear-num 37.4%

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

   6. Applied egg-rr 37.4%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 11: 62.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 10000.0) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 10000.0) {
		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 <= 10000.0d0) 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 <= 10000.0) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 10000.0:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 10000.0)
		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 <= 10000.0)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 10000.0], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e4

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 75.7%

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

   if 1e4 < y

   1. Initial program 95.3%

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

      1. associate-*r/ 95.4%

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

      2. associate-/l/ 92.7%

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

      3. *-commutative 92.7%

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

      4. times-frac 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in y around 0 27.0%

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

      1. *-commutative 27.0%

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

      2. clear-num 27.0%

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

      3. frac-times 36.1%

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

      4. *-un-lft-identity 36.1%

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

   6. Applied egg-rr 36.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 12: 58.2% 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 97.8%

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

2. Taylor expanded in y around 0 60.6%

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

3. Final simplification 60.6%

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

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 2023257 
(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))