Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.6%

Time: 7.5s

Alternatives: 10

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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq -4 \cdot 10^{+35} \lor \neg \left(x \leq 2 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= x -4e+35) (not (<= x 2e-74))) (/ (* x t_0) z) (/ x (/ z t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((x <= -4e+35) || !(x <= 2e-74)) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x / (z / t_0);
	}
	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 <= (-4d+35)) .or. (.not. (x <= 2d-74))) then
        tmp = (x * t_0) / z
    else
        tmp = x / (z / t_0)
    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 <= -4e+35) || !(x <= 2e-74)) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if (x <= -4e+35) or not (x <= 2e-74):
		tmp = (x * t_0) / z
	else:
		tmp = x / (z / t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((x <= -4e+35) || !(x <= 2e-74))
		tmp = Float64(Float64(x * t_0) / z);
	else
		tmp = Float64(x / Float64(z / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if ((x <= -4e+35) || ~((x <= 2e-74)))
		tmp = (x * t_0) / z;
	else
		tmp = x / (z / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[x, -4e+35], N[Not[LessEqual[x, 2e-74]], $MachinePrecision]], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999999e35 or 1.99999999999999992e-74 < x

   1. Initial program 99.8%

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

   if -3.9999999999999999e35 < x < 1.99999999999999992e-74

   1. Initial program 91.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+35} \lor \neg \left(x \leq 2 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 2: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0003:\\ \;\;\;\;\frac{x + x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0003)
   (/ (+ x (* x (* y (* y -0.16666666666666666)))) z)
   (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0003) {
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / 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 <= 0.0003d0) then
        tmp = (x + (x * (y * (y * (-0.16666666666666666d0))))) / 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 <= 0.0003) {
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.0003:
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0003)
		tmp = Float64(Float64(x + Float64(x * Float64(y * Float64(y * -0.16666666666666666)))) / 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 <= 0.0003)
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.0003], N[(N[(x + N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 < 2.99999999999999974e-4

   1. Initial program 95.4%

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

   2. Taylor expanded in y around 0 70.1%

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

      1. unpow2 70.1%

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

   4. Simplified 70.1%

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

      1. distribute-rgt-in 70.1%

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

      2. *-un-lft-identity 70.1%

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

      3. associate-*r* 70.1%

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

   6. Applied egg-rr 70.1%

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

   if 2.99999999999999974e-4 < y

   1. Initial program 97.9%

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

      1. associate-*l/ 95.6%

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

      2. times-frac 84.7%

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

      3. *-commutative 84.7%

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

      4. associate-*r/ 84.7%

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

      5. *-commutative 84.7%

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

   3. Simplified 84.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0003:\\ \;\;\;\;\frac{x + x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 3: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-8) (/ x z) (* (/ (sin y) z) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-8) {
		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 <= 2.4d-8) 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 <= 2.4e-8) {
		tmp = x / z;
	} else {
		tmp = (Math.sin(y) / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-8:
		tmp = x / z
	else:
		tmp = (math.sin(y) / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-8)
		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 <= 2.4e-8)
		tmp = x / z;
	else
		tmp = (sin(y) / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-8], 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 < 2.39999999999999998e-8

   1. Initial program 95.4%

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

   2. Taylor expanded in y around 0 74.7%

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

   if 2.39999999999999998e-8 < y

   1. Initial program 97.9%

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

      1. associate-*r/ 97.9%

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

      2. associate-/l/ 84.9%

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

      3. *-commutative 84.9%

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

      4. times-frac 98.0%

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

   3. Simplified 98.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+135) (/ x (/ z (/ (sin y) y))) (* (/ (sin y) z) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+135) {
		tmp = x / (z / (sin(y) / y));
	} 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 <= 2d+135) then
        tmp = x / (z / (sin(y) / y))
    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 <= 2e+135) {
		tmp = x / (z / (Math.sin(y) / y));
	} else {
		tmp = (Math.sin(y) / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+135:
		tmp = x / (z / (math.sin(y) / y))
	else:
		tmp = (math.sin(y) / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+135)
		tmp = Float64(x / Float64(z / Float64(sin(y) / y)));
	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 <= 2e+135)
		tmp = x / (z / (sin(y) / y));
	else
		tmp = (sin(y) / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+135], N[(x / N[(z / N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e135

   1. Initial program 95.9%

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

      1. associate-/l* 97.4%

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

   3. Simplified 97.4%

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

   if 1.99999999999999992e135 < y

   1. Initial program 96.9%

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

      1. associate-*r/ 96.9%

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

      2. associate-/l/ 77.8%

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

      3. *-commutative 77.8%

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

      4. times-frac 97.0%

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

   3. Simplified 97.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 59.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+48)
   (/ x (/ z (+ 1.0 (* -0.16666666666666666 (* y y)))))
   (* (/ x (* y z)) (/ 6.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+48) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.2d+48) then
        tmp = x / (z / (1.0d0 + ((-0.16666666666666666d0) * (y * y))))
    else
        tmp = (x / (y * z)) * (6.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+48) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.2e+48:
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))))
	else:
		tmp = (x / (y * z)) * (6.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e+48)
		tmp = Float64(x / Float64(z / Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))));
	else
		tmp = Float64(Float64(x / Float64(y * z)) * Float64(6.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e+48)
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	else
		tmp = (x / (y * z)) * (6.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e+48], N[(x / N[(z / N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000005e48

   1. Initial program 95.7%

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

      1. associate-/l* 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in y around 0 66.6%

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

      1. unpow2 66.6%

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

   6. Simplified 66.6%

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

   if 8.2000000000000005e48 < y

   1. Initial program 97.5%

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

      1. associate-/l* 80.9%

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

      2. associate-/r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in y around 0 43.4%

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

   5. Taylor expanded in y around inf 43.4%

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

      1. *-commutative 43.4%

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

      2. *-commutative 43.4%

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

   7. Simplified 43.4%

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

   8. Taylor expanded in x around 0 43.4%

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

      1. associate-*r/ 43.4%

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

      2. *-commutative 43.4%

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

      3. times-frac 43.5%

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

      4. metadata-eval 43.5%

         \[\leadsto \frac{\color{blue}{\frac{1}{0.16666666666666666}}}{z} \cdot \frac{x}{{y}^{2}} \]

      5. associate-/r* 43.5%

         \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 \cdot z}} \cdot \frac{x}{{y}^{2}} \]

      6. unpow2 43.5%

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

      7. associate-/r* 43.5%

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

      8. times-frac 43.4%

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

      9. associate-*l/ 43.4%

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

      10. associate-/r* 43.4%

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

      11. associate-*l/ 45.2%

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

      12. associate-/r* 45.2%

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

      13. metadata-eval 45.2%

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

      14. times-frac 45.2%

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

      15. associate-*r/ 45.2%

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

      16. *-lft-identity 45.2%

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

      17. associate-*l/ 45.2%

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

      18. associate-*r* 45.2%

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

      19. *-commutative 45.2%

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

      20. associate-*l/ 45.2%

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

      21. metadata-eval 45.2%

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

   10. Simplified 45.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \]

Alternative 6: 59.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x + x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+48)
   (/ (+ x (* x (* y (* y -0.16666666666666666)))) z)
   (* (/ x (* y z)) (/ 6.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+48) {
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z;
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.2d+48) then
        tmp = (x + (x * (y * (y * (-0.16666666666666666d0))))) / z
    else
        tmp = (x / (y * z)) * (6.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+48) {
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z;
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.2e+48:
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z
	else:
		tmp = (x / (y * z)) * (6.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e+48)
		tmp = Float64(Float64(x + Float64(x * Float64(y * Float64(y * -0.16666666666666666)))) / z);
	else
		tmp = Float64(Float64(x / Float64(y * z)) * Float64(6.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e+48)
		tmp = (x + (x * (y * (y * -0.16666666666666666)))) / z;
	else
		tmp = (x / (y * z)) * (6.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e+48], N[(N[(x + N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000005e48

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0 66.6%

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

      1. unpow2 66.6%

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

   4. Simplified 66.6%

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

      1. distribute-rgt-in 66.7%

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

      2. *-un-lft-identity 66.7%

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

      3. associate-*r* 66.7%

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

   6. Applied egg-rr 66.7%

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

   if 8.2000000000000005e48 < y

   1. Initial program 97.5%

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

      1. associate-/l* 80.9%

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

      2. associate-/r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in y around 0 43.4%

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

   5. Taylor expanded in y around inf 43.4%

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

      1. *-commutative 43.4%

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

      2. *-commutative 43.4%

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

   7. Simplified 43.4%

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

   8. Taylor expanded in x around 0 43.4%

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

      1. associate-*r/ 43.4%

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

      2. *-commutative 43.4%

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

      3. times-frac 43.5%

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

      4. metadata-eval 43.5%

         \[\leadsto \frac{\color{blue}{\frac{1}{0.16666666666666666}}}{z} \cdot \frac{x}{{y}^{2}} \]

      5. associate-/r* 43.5%

         \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 \cdot z}} \cdot \frac{x}{{y}^{2}} \]

      6. unpow2 43.5%

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

      7. associate-/r* 43.5%

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

      8. times-frac 43.4%

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

      9. associate-*l/ 43.4%

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

      10. associate-/r* 43.4%

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

      11. associate-*l/ 45.2%

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

      12. associate-/r* 45.2%

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

      13. metadata-eval 45.2%

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

      14. times-frac 45.2%

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

      15. associate-*r/ 45.2%

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

      16. *-lft-identity 45.2%

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

      17. associate-*l/ 45.2%

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

      18. associate-*r* 45.2%

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

      19. *-commutative 45.2%

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

      20. associate-*l/ 45.2%

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

      21. metadata-eval 45.2%

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

   10. Simplified 45.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x + x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \]

Alternative 7: 61.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e+89) (/ x z) (* (/ x (* y z)) (/ 6.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+89) {
		tmp = x / z;
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.2d+89) then
        tmp = x / z
    else
        tmp = (x / (y * z)) * (6.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+89) {
		tmp = x / z;
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.2e+89:
		tmp = x / z
	else:
		tmp = (x / (y * z)) * (6.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e+89)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / Float64(y * z)) * Float64(6.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e+89)
		tmp = x / z;
	else
		tmp = (x / (y * z)) * (6.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e+89], N[(x / z), $MachinePrecision], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999996e89

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 69.1%

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

   if 9.1999999999999996e89 < y

   1. Initial program 97.2%

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

      1. associate-/l* 80.5%

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

      2. associate-/r/ 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in y around 0 44.4%

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

   5. Taylor expanded in y around inf 44.4%

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

      1. *-commutative 44.4%

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

      2. *-commutative 44.4%

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

   7. Simplified 44.4%

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

   8. Taylor expanded in x around 0 44.5%

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

      1. associate-*r/ 44.5%

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

      2. *-commutative 44.5%

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

      3. times-frac 44.5%

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

      4. metadata-eval 44.5%

         \[\leadsto \frac{\color{blue}{\frac{1}{0.16666666666666666}}}{z} \cdot \frac{x}{{y}^{2}} \]

      5. associate-/r* 44.5%

         \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 \cdot z}} \cdot \frac{x}{{y}^{2}} \]

      6. unpow2 44.5%

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

      7. associate-/r* 44.5%

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

      8. times-frac 44.5%

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

      9. associate-*l/ 44.5%

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

      10. associate-/r* 44.5%

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

      11. associate-*l/ 46.4%

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

      12. associate-/r* 46.4%

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

      13. metadata-eval 46.4%

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

      14. times-frac 46.4%

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

      15. associate-*r/ 46.4%

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

      16. *-lft-identity 46.4%

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

      17. associate-*l/ 46.4%

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

      18. associate-*r* 46.4%

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

      19. *-commutative 46.4%

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

      20. associate-*l/ 46.4%

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

      21. metadata-eval 46.4%

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

   10. Simplified 46.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \]

Alternative 8: 61.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+77) (/ x z) (* y (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+77) {
		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+77) 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+77) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+77:
		tmp = x / z
	else:
		tmp = y * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+77)
		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 <= 2e+77)
		tmp = x / z;
	else
		tmp = y * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+77], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999997e77

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 69.7%

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

   if 1.99999999999999997e77 < y

   1. Initial program 97.4%

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

      1. associate-*l/ 94.3%

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

      2. times-frac 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r/ 81.6%

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

      5. *-commutative 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 43.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 9: 61.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.55e-20) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.55e-20) {
		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 <= 1.55d-20) 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 <= 1.55e-20) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.55e-20:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.55e-20)
		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 <= 1.55e-20)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.55e-20], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-20

   1. Initial program 95.3%

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

   2. Taylor expanded in y around 0 74.1%

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

   if 1.55e-20 < y

   1. Initial program 98.0%

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

      1. associate-*r/ 98.0%

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

      2. associate-/l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in y around 0 28.9%

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

      1. *-commutative 28.9%

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

      2. clear-num 28.9%

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

      3. frac-times 41.2%

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

      4. *-un-lft-identity 41.2%

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

   6. Applied egg-rr 41.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 10: 57.9% 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 96.1%

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

2. Taylor expanded in y around 0 58.8%

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

3. Final simplification 58.8%

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

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 2023238 
(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))