Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 95.5%

Time: 9.2s

Alternatives: 17

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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq 10^{-227}:\\ \;\;\;\;\frac{t_0 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{t_0}{z}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 1e-227) (/ (* t_0 x) z) (/ x (/ 1.0 (/ t_0 z))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= 1e-227) {
		tmp = (t_0 * x) / z;
	} else {
		tmp = x / (1.0 / (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 (t_0 <= 1d-227) then
        tmp = (t_0 * x) / z
    else
        tmp = x / (1.0d0 / (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 (t_0 <= 1e-227) {
		tmp = (t_0 * x) / z;
	} else {
		tmp = x / (1.0 / (t_0 / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if t_0 <= 1e-227:
		tmp = (t_0 * x) / z
	else:
		tmp = x / (1.0 / (t_0 / z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= 1e-227)
		tmp = Float64(Float64(t_0 * x) / z);
	else
		tmp = Float64(x / Float64(1.0 / Float64(t_0 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (t_0 <= 1e-227)
		tmp = (t_0 * x) / z;
	else
		tmp = x / (1.0 / (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[t$95$0, 1e-227], N[(N[(t$95$0 * x), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(1.0 / N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 9.99999999999999945e-228

   1. Initial program 99.0%

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

   if 9.99999999999999945e-228 < (/.f64 (sin.f64 y) y)

   1. Initial program 96.6%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 84.3%

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

   3. Simplified 84.3%

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

      1. associate-/r/ 99.3%

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

      2. clear-num 99.1%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-227}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{\frac{\sin y}{y}}{z}}}\\ \end{array} \]

Alternative 2: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00152 \lor \neg \left(y \leq 2.55 \cdot 10^{-11}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00152) (not (<= y 2.55e-11)))
   (* (sin y) (/ x (* y z)))
   (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00152) || !(y <= 2.55e-11)) {
		tmp = sin(y) * (x / (y * z));
	} else {
		tmp = 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 <= (-0.00152d0)) .or. (.not. (y <= 2.55d-11))) then
        tmp = sin(y) * (x / (y * z))
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00152) || !(y <= 2.55e-11)) {
		tmp = Math.sin(y) * (x / (y * z));
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00152) or not (y <= 2.55e-11):
		tmp = math.sin(y) * (x / (y * z))
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00152) || !(y <= 2.55e-11))
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00152) || ~((y <= 2.55e-11)))
		tmp = sin(y) * (x / (y * z));
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00152], N[Not[LessEqual[y, 2.55e-11]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0015200000000000001 or 2.54999999999999992e-11 < y

   1. Initial program 95.2%

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

      1. associate-*l/ 92.7%

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

      2. times-frac 92.4%

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

      3. *-commutative 92.4%

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

      4. associate-*r/ 92.5%

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

      5. *-commutative 92.5%

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

   3. Simplified 92.5%

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

   if -0.0015200000000000001 < y < 2.54999999999999992e-11

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*r/ 76.4%

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

      5. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in y around 0 100.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00152 \lor \neg \left(y \leq 2.55 \cdot 10^{-11}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00152:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.00152)
   (* (/ (sin y) z) (/ x y))
   (if (<= y 2.55e-11) (/ x z) (* (sin y) (/ x (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.00152) {
		tmp = (sin(y) / z) * (x / y);
	} else if (y <= 2.55e-11) {
		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 <= (-0.00152d0)) then
        tmp = (sin(y) / z) * (x / y)
    else if (y <= 2.55d-11) 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 <= -0.00152) {
		tmp = (Math.sin(y) / z) * (x / y);
	} else if (y <= 2.55e-11) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.00152:
		tmp = (math.sin(y) / z) * (x / y)
	elif y <= 2.55e-11:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.00152)
		tmp = Float64(Float64(sin(y) / z) * Float64(x / y));
	elseif (y <= 2.55e-11)
		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 <= -0.00152)
		tmp = (sin(y) / z) * (x / y);
	elseif (y <= 2.55e-11)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.00152], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.55e-11], N[(x / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0015200000000000001

   1. Initial program 95.9%

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

      1. associate-*r/ 95.8%

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

      2. associate-/l/ 90.8%

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

      3. *-commutative 90.8%

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

      4. times-frac 96.0%

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

   3. Simplified 96.0%

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

   if -0.0015200000000000001 < y < 2.54999999999999992e-11

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*r/ 76.4%

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

      5. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in y around 0 100.0%

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

   if 2.54999999999999992e-11 < y

   1. Initial program 94.2%

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

      1. associate-*l/ 94.6%

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

      2. times-frac 94.5%

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

      3. *-commutative 94.5%

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

      4. associate-*r/ 94.7%

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

      5. *-commutative 94.7%

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

   3. Simplified 94.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00152:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 4: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00152:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.00152)
   (* (/ (sin y) z) (/ x y))
   (if (<= y 3e-17) (/ x z) (/ x (* y (/ z (sin y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.00152) {
		tmp = (sin(y) / z) * (x / y);
	} else if (y <= 3e-17) {
		tmp = x / z;
	} else {
		tmp = x / (y * (z / sin(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 <= (-0.00152d0)) then
        tmp = (sin(y) / z) * (x / y)
    else if (y <= 3d-17) then
        tmp = x / z
    else
        tmp = x / (y * (z / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.00152) {
		tmp = (Math.sin(y) / z) * (x / y);
	} else if (y <= 3e-17) {
		tmp = x / z;
	} else {
		tmp = x / (y * (z / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.00152:
		tmp = (math.sin(y) / z) * (x / y)
	elif y <= 3e-17:
		tmp = x / z
	else:
		tmp = x / (y * (z / math.sin(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.00152)
		tmp = Float64(Float64(sin(y) / z) * Float64(x / y));
	elseif (y <= 3e-17)
		tmp = Float64(x / z);
	else
		tmp = Float64(x / Float64(y * Float64(z / sin(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.00152)
		tmp = (sin(y) / z) * (x / y);
	elseif (y <= 3e-17)
		tmp = x / z;
	else
		tmp = x / (y * (z / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.00152], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-17], N[(x / z), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0015200000000000001

   1. Initial program 95.9%

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

      1. associate-*r/ 95.8%

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

      2. associate-/l/ 90.8%

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

      3. *-commutative 90.8%

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

      4. times-frac 96.0%

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

   3. Simplified 96.0%

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

   if -0.0015200000000000001 < y < 3.00000000000000006e-17

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 78.3%

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

      3. *-commutative 78.3%

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

      4. associate-*r/ 75.8%

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

      5. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in y around 0 100.0%

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

   if 3.00000000000000006e-17 < y

   1. Initial program 94.5%

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

      1. associate-/l* 95.0%

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

      2. associate-/r/ 95.0%

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

   3. Simplified 95.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00152:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 5: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

2. Final simplification 97.4%

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

Alternative 6: 65.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{6}{y}}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.35)
   (* (/ x (* y z)) (/ 6.0 y))
   (if (<= y 1.06e+32)
     (/ (* x (+ 1.0 (* (* y y) -0.16666666666666666))) z)
     (/ (/ (* x (/ 6.0 y)) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.35) {
		tmp = (x / (y * z)) * (6.0 / y);
	} else if (y <= 1.06e+32) {
		tmp = (x * (1.0 + ((y * y) * -0.16666666666666666))) / z;
	} else {
		tmp = ((x * (6.0 / 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 (y <= (-2.35d0)) then
        tmp = (x / (y * z)) * (6.0d0 / y)
    else if (y <= 1.06d+32) then
        tmp = (x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))) / z
    else
        tmp = ((x * (6.0d0 / y)) / y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.35) {
		tmp = (x / (y * z)) * (6.0 / y);
	} else if (y <= 1.06e+32) {
		tmp = (x * (1.0 + ((y * y) * -0.16666666666666666))) / z;
	} else {
		tmp = ((x * (6.0 / y)) / y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.35:
		tmp = (x / (y * z)) * (6.0 / y)
	elif y <= 1.06e+32:
		tmp = (x * (1.0 + ((y * y) * -0.16666666666666666))) / z
	else:
		tmp = ((x * (6.0 / y)) / y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.35)
		tmp = Float64(Float64(x / Float64(y * z)) * Float64(6.0 / y));
	elseif (y <= 1.06e+32)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666))) / z);
	else
		tmp = Float64(Float64(Float64(x * Float64(6.0 / y)) / y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.35)
		tmp = (x / (y * z)) * (6.0 / y);
	elseif (y <= 1.06e+32)
		tmp = (x * (1.0 + ((y * y) * -0.16666666666666666))) / z;
	else
		tmp = ((x * (6.0 / y)) / y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.35], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e+32], N[(N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x * N[(6.0 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.35000000000000009

   1. Initial program 95.9%

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

      1. associate-/l* 90.9%

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

      2. associate-/r/ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around 0 31.6%

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

   5. Taylor expanded in y around inf 31.7%

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

      1. *-commutative 31.7%

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

      2. associate-*l/ 31.7%

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

      3. unpow2 31.7%

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

      4. associate-*l* 31.6%

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

   7. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. times-frac 32.8%

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

   9. Applied egg-rr 32.8%

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

   if -2.35000000000000009 < y < 1.0600000000000001e32

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.3%

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

      1. unpow2 96.3%

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

   4. Simplified 96.3%

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

   if 1.0600000000000001e32 < y

   1. Initial program 93.2%

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

      1. associate-/l* 93.8%

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

      2. associate-/r/ 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in y around 0 30.0%

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

   5. Taylor expanded in y around inf 29.9%

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

      1. *-commutative 29.9%

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

      2. unpow2 29.9%

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

   7. Simplified 29.9%

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

      1. associate-*r/ 29.9%

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

      2. *-commutative 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r* 30.0%

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

      5. associate-/r* 30.0%

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

      6. associate-/r* 30.1%

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

      7. *-un-lft-identity 30.1%

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

      8. times-frac 30.1%

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

      9. /-rgt-identity 30.1%

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

   9. Applied egg-rr 30.1%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{6}{y}}{y}}{z}\\ \end{array} \]

Alternative 7: 65.7% accurate, 8.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.4) (not (<= y 2.4))) (* (/ x (* y z)) (/ 6.0 y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4) || !(y <= 2.4)) {
		tmp = (x / (y * z)) * (6.0 / y);
	} else {
		tmp = 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 <= (-2.4d0)) .or. (.not. (y <= 2.4d0))) then
        tmp = (x / (y * z)) * (6.0d0 / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4) || !(y <= 2.4)) {
		tmp = (x / (y * z)) * (6.0 / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.4) or not (y <= 2.4):
		tmp = (x / (y * z)) * (6.0 / y)
	else:
		tmp = x / z
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.4], N[Not[LessEqual[y, 2.4]], $MachinePrecision]], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991 or 2.39999999999999991 < y

   1. Initial program 95.1%

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

      1. associate-/l* 92.3%

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

      2. associate-/r/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in y around 0 29.9%

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

   5. Taylor expanded in y around inf 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-*l/ 30.0%

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

      3. unpow2 30.0%

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

      4. associate-*l* 29.9%

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

   7. Simplified 29.9%

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

      1. *-commutative 29.9%

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

      2. times-frac 30.6%

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

   9. Applied egg-rr 30.6%

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

   if -2.39999999999999991 < y < 2.39999999999999991

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-*r/ 77.1%

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

      5. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in y around 0 98.6%

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

4. Final simplification 63.6%

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

Alternative 8: 65.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2000000:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2000000.0)
   (/ y (/ (* y z) x))
   (if (<= y 1.46e+51) (/ x z) (* 6.0 (/ x (* z (* y y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2000000.0) {
		tmp = y / ((y * z) / x);
	} else if (y <= 1.46e+51) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2000000.0d0)) then
        tmp = y / ((y * z) / x)
    else if (y <= 1.46d+51) then
        tmp = x / z
    else
        tmp = 6.0d0 * (x / (z * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2000000.0) {
		tmp = y / ((y * z) / x);
	} else if (y <= 1.46e+51) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2000000.0:
		tmp = y / ((y * z) / x)
	elif y <= 1.46e+51:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (z * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2000000.0)
		tmp = Float64(y / Float64(Float64(y * z) / x));
	elseif (y <= 1.46e+51)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(x / Float64(z * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2000000.0)
		tmp = y / ((y * z) / x);
	elseif (y <= 1.46e+51)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (z * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2000000.0], N[(y / N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.46e+51], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e6

   1. Initial program 95.8%

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

      1. associate-*r/ 95.7%

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

      2. associate-/l/ 90.6%

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

      3. *-commutative 90.6%

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

      4. times-frac 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in y around 0 26.0%

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

      1. frac-times 18.6%

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

      2. *-commutative 18.6%

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

      3. associate-/l* 33.1%

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

   6. Applied egg-rr 33.1%

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

   if -2e6 < y < 1.4600000000000001e51

   1. Initial program 100.0%

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

      1. associate-*l/ 99.9%

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

      2. times-frac 81.3%

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

      3. *-commutative 81.3%

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

      4. associate-*r/ 79.2%

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

      5. *-commutative 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in y around 0 90.3%

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

   if 1.4600000000000001e51 < y

   1. Initial program 92.3%

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

      1. associate-/l* 93.0%

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

      2. associate-/r/ 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in y around 0 31.3%

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

   5. Taylor expanded in y around inf 31.3%

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

      1. *-commutative 31.3%

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

      2. unpow2 31.3%

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

   7. Simplified 31.3%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2000000:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 9: 65.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -100000.0)
   (/ y (/ (* y z) x))
   (if (<= y 2.4) (/ x z) (* 6.0 (/ (/ x y) (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -100000.0) {
		tmp = y / ((y * z) / x);
	} else if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((x / 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 (y <= (-100000.0d0)) then
        tmp = y / ((y * z) / x)
    else if (y <= 2.4d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * ((x / y) / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -100000.0) {
		tmp = y / ((y * z) / x);
	} else if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((x / y) / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -100000.0:
		tmp = y / ((y * z) / x)
	elif y <= 2.4:
		tmp = x / z
	else:
		tmp = 6.0 * ((x / y) / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -100000.0)
		tmp = Float64(y / Float64(Float64(y * z) / x));
	elseif (y <= 2.4)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(x / y) / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -100000.0)
		tmp = y / ((y * z) / x);
	elseif (y <= 2.4)
		tmp = x / z;
	else
		tmp = 6.0 * ((x / y) / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -100000.0], N[(y / N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(x / y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e5

   1. Initial program 95.8%

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

      1. associate-*r/ 95.7%

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

      2. associate-/l/ 90.6%

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

      3. *-commutative 90.6%

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

      4. times-frac 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in y around 0 26.0%

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

      1. frac-times 18.6%

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

      2. *-commutative 18.6%

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

      3. associate-/l* 33.1%

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

   6. Applied egg-rr 33.1%

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

   if -1e5 < y < 2.39999999999999991

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r/ 77.5%

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

      5. *-commutative 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around 0 97.1%

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

   if 2.39999999999999991 < y

   1. Initial program 93.8%

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

      1. associate-/l* 94.4%

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

      2. associate-/r/ 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 27.4%

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

   5. Taylor expanded in y around inf 27.4%

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

      1. *-commutative 27.4%

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

      2. unpow2 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in x around 0 27.4%

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

      1. rem-square-sqrt 13.0%

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

      2. unpow2 13.0%

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

      3. times-frac 13.0%

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

      4. associate-/r* 13.1%

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

      5. times-frac 13.1%

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

      6. associate-*l/ 13.1%

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

      7. rem-square-sqrt 27.5%

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

   10. Simplified 27.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 10: 65.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2000000:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{6}{y}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2000000.0)
   (/ y (/ (* y z) x))
   (if (<= y 2.4) (/ x z) (* x (/ (/ 6.0 y) (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2000000.0) {
		tmp = y / ((y * z) / x);
	} else if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = x * ((6.0 / 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 (y <= (-2000000.0d0)) then
        tmp = y / ((y * z) / x)
    else if (y <= 2.4d0) then
        tmp = x / z
    else
        tmp = x * ((6.0d0 / y) / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2000000.0) {
		tmp = y / ((y * z) / x);
	} else if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = x * ((6.0 / y) / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2000000.0:
		tmp = y / ((y * z) / x)
	elif y <= 2.4:
		tmp = x / z
	else:
		tmp = x * ((6.0 / y) / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2000000.0)
		tmp = Float64(y / Float64(Float64(y * z) / x));
	elseif (y <= 2.4)
		tmp = Float64(x / z);
	else
		tmp = Float64(x * Float64(Float64(6.0 / y) / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2000000.0)
		tmp = y / ((y * z) / x);
	elseif (y <= 2.4)
		tmp = x / z;
	else
		tmp = x * ((6.0 / y) / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2000000.0], N[(y / N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(x * N[(N[(6.0 / y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e6

   1. Initial program 95.8%

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

      1. associate-*r/ 95.7%

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

      2. associate-/l/ 90.6%

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

      3. *-commutative 90.6%

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

      4. times-frac 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in y around 0 26.0%

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

      1. frac-times 18.6%

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

      2. *-commutative 18.6%

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

      3. associate-/l* 33.1%

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

   6. Applied egg-rr 33.1%

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

   if -2e6 < y < 2.39999999999999991

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r/ 77.5%

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

      5. *-commutative 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around 0 97.1%

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

   if 2.39999999999999991 < y

   1. Initial program 93.8%

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

      1. associate-/l* 94.4%

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

      2. associate-/r/ 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 27.4%

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

   5. Taylor expanded in y around inf 27.4%

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

      1. *-commutative 27.4%

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

      2. unpow2 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in x around 0 27.4%

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

      1. associate-*r/ 27.4%

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

      2. *-commutative 27.4%

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

      3. unpow2 27.4%

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

      4. associate-*r* 27.4%

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

      5. associate-*r/ 27.4%

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

      6. associate-/r* 27.5%

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

   10. Simplified 27.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2000000:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{6}{y}}{y \cdot z}\\ \end{array} \]

Alternative 11: 65.4% accurate, 8.2× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4)
   (* (/ x (* y z)) (/ 6.0 y))
   (if (<= y 2.4) (/ x z) (/ (/ (* x (/ 6.0 y)) y) z))))

C

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4) {
		tmp = (x / (y * z)) * (6.0 / y);
	} else if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = ((x * (6.0 / y)) / y) / z;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(N[(N[(x * N[(6.0 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999991

   1. Initial program 95.9%

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

      1. associate-/l* 90.9%

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

      2. associate-/r/ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around 0 31.6%

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

   5. Taylor expanded in y around inf 31.7%

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

      1. *-commutative 31.7%

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

      2. associate-*l/ 31.7%

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

      3. unpow2 31.7%

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

      4. associate-*l* 31.6%

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

   7. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. times-frac 32.8%

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

   9. Applied egg-rr 32.8%

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

   if -2.39999999999999991 < y < 2.39999999999999991

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-*r/ 77.1%

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

      5. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in y around 0 98.6%

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

   if 2.39999999999999991 < y

   1. Initial program 93.8%

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

      1. associate-/l* 94.4%

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

      2. associate-/r/ 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 27.4%

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

   5. Taylor expanded in y around inf 27.4%

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

      1. *-commutative 27.4%

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

      2. unpow2 27.4%

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

   7. Simplified 27.4%

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

      1. associate-*r/ 27.4%

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

      2. *-commutative 27.4%

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

      3. *-commutative 27.4%

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

      4. associate-*r* 27.4%

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

      5. associate-/r* 27.5%

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

      6. associate-/r* 27.5%

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

      7. *-un-lft-identity 27.5%

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

      8. times-frac 27.5%

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

      9. /-rgt-identity 27.5%

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

   9. Applied egg-rr 27.5%

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

4. Final simplification 63.6%

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

Alternative 12: 65.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -100000 \lor \neg \left(y \leq 4.3 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -100000.0) (not (<= y 4.3e-44))) (/ y (* z (/ y x))) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -100000.0) || !(y <= 4.3e-44)) {
		tmp = y / (z * (y / x));
	} else {
		tmp = 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 <= (-100000.0d0)) .or. (.not. (y <= 4.3d-44))) then
        tmp = y / (z * (y / x))
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -100000.0) || !(y <= 4.3e-44)) {
		tmp = y / (z * (y / x));
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -100000.0) or not (y <= 4.3e-44):
		tmp = y / (z * (y / x))
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -100000.0) || !(y <= 4.3e-44))
		tmp = Float64(y / Float64(z * Float64(y / x)));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -100000.0) || ~((y <= 4.3e-44)))
		tmp = y / (z * (y / x));
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -100000.0], N[Not[LessEqual[y, 4.3e-44]], $MachinePrecision]], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e5 or 4.30000000000000013e-44 < y

   1. Initial program 95.5%

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

      1. associate-*r/ 95.6%

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

      2. associate-/l/ 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. times-frac 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in y around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. clear-num 30.3%

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

      3. frac-times 37.3%

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

      4. *-un-lft-identity 37.3%

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

   6. Applied egg-rr 37.3%

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

   if -1e5 < y < 4.30000000000000013e-44

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. times-frac 76.7%

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

      3. *-commutative 76.7%

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

      4. associate-*r/ 74.0%

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

      5. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in y around 0 98.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100000 \lor \neg \left(y \leq 4.3 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 13: 65.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -100000 \lor \neg \left(y \leq 10^{+51}\right):\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -100000.0) (not (<= y 1e+51))) (/ y (/ (* y z) x)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -100000.0) || !(y <= 1e+51)) {
		tmp = y / ((y * z) / x);
	} else {
		tmp = 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 <= (-100000.0d0)) .or. (.not. (y <= 1d+51))) then
        tmp = y / ((y * z) / x)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -100000.0) || !(y <= 1e+51)) {
		tmp = y / ((y * z) / x);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -100000.0) or not (y <= 1e+51):
		tmp = y / ((y * z) / x)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -100000.0) || !(y <= 1e+51))
		tmp = Float64(y / Float64(Float64(y * z) / x));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -100000.0) || ~((y <= 1e+51)))
		tmp = y / ((y * z) / x);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -100000.0], N[Not[LessEqual[y, 1e+51]], $MachinePrecision]], N[(y / N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e5 or 1e51 < y

   1. Initial program 94.5%

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

      1. associate-*r/ 94.6%

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

      2. associate-/l/ 91.4%

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

      3. *-commutative 91.4%

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

      4. times-frac 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in y around 0 23.5%

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

      1. frac-times 17.3%

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

      2. *-commutative 17.3%

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

      3. associate-/l* 32.2%

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

   6. Applied egg-rr 32.2%

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

   if -1e5 < y < 1e51

   1. Initial program 100.0%

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

      1. associate-*l/ 99.9%

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

      2. times-frac 81.3%

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

      3. *-commutative 81.3%

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

      4. associate-*r/ 79.2%

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

      5. *-commutative 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in y around 0 90.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100000 \lor \neg \left(y \leq 10^{+51}\right):\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 14: 59.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{\frac{y \cdot z}{y}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-83)
   (/ x (/ (* y z) y))
   (if (<= x 1.6e-148) (* (/ x y) (/ y z)) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-83) {
		tmp = x / ((y * z) / y);
	} else if (x <= 1.6e-148) {
		tmp = (x / y) * (y / z);
	} else {
		tmp = 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 (x <= (-1.4d-83)) then
        tmp = x / ((y * z) / y)
    else if (x <= 1.6d-148) then
        tmp = (x / y) * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-83) {
		tmp = x / ((y * z) / y);
	} else if (x <= 1.6e-148) {
		tmp = (x / y) * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e-83:
		tmp = x / ((y * z) / y)
	elif x <= 1.6e-148:
		tmp = (x / y) * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-83)
		tmp = Float64(x / Float64(Float64(y * z) / y));
	elseif (x <= 1.6e-148)
		tmp = Float64(Float64(x / y) * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-83)
		tmp = x / ((y * z) / y);
	elseif (x <= 1.6e-148)
		tmp = (x / y) * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e-83], N[(x / N[(N[(y * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-148], N[(N[(x / y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e-83

   1. Initial program 99.7%

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

      1. associate-/l* 95.0%

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

      2. associate-/r/ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in y around 0 38.5%

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

      1. associate-*l/ 47.9%

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

      2. *-commutative 47.9%

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

   6. Applied egg-rr 47.9%

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

   if -1.4e-83 < x < 1.59999999999999997e-148

   1. Initial program 91.6%

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

      1. associate-*r/ 71.9%

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

      2. associate-/l/ 79.7%

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

      3. *-commutative 79.7%

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

      4. times-frac 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in y around 0 80.7%

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

   if 1.59999999999999997e-148 < x

   1. Initial program 99.1%

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

      1. associate-*l/ 94.6%

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

      2. times-frac 84.5%

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

      3. *-commutative 84.5%

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

      4. associate-*r/ 81.2%

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

      5. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in y around 0 57.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{\frac{y \cdot z}{y}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 15: 65.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x / (1.0 + ((y * y) * 0.16666666666666666))) / 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 / (1.0d0 + ((y * y) * 0.16666666666666666d0))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

   1. clear-num 97.4%

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

   2. un-div-inv 97.4%

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

3. Applied egg-rr 97.4%

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

4. Taylor expanded in y around 0 63.5%

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

   1. *-commutative 63.5%

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

   2. unpow2 63.5%

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

6. Simplified 63.5%

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

7. Final simplification 63.5%

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

Alternative 16: 58.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+113) (* (/ x y) (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+113) {
		tmp = (x / y) * (y / z);
	} else {
		tmp = 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 <= (-3d+113)) then
        tmp = (x / y) * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+113) {
		tmp = (x / y) * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3e+113:
		tmp = (x / y) * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+113)
		tmp = Float64(Float64(x / y) * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e+113)
		tmp = (x / y) * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e+113], N[(N[(x / y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e113

   1. Initial program 94.0%

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

      1. associate-*r/ 94.0%

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

      2. associate-/l/ 90.0%

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

      3. *-commutative 90.0%

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

      4. times-frac 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in y around 0 32.1%

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

   if -3e113 < y

   1. Initial program 98.3%

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

      1. associate-*l/ 98.0%

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

      2. times-frac 85.0%

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

      3. *-commutative 85.0%

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

      4. associate-*r/ 83.6%

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

      5. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in y around 0 64.5%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 17: 57.5% 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 97.4%

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

   1. associate-*l/ 96.1%

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

   2. times-frac 86.0%

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

   3. *-commutative 86.0%

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

   4. associate-*r/ 84.9%

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

   5. *-commutative 84.9%

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

3. Simplified 84.9%

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

4. Taylor expanded in y around 0 55.4%

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

5. Final simplification 55.4%

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

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 2023196 
(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))