Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 97.5%

Time: 12.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 1.95e+56) (/ x (/ z t_0)) (/ (* x t_0) z))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 1.95e+56) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= 1.95d+56) then
        tmp = x / (z / t_0)
    else
        tmp = (x * t_0) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 1.95e+56) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 1.95e+56:
		tmp = x / (z / t_0)
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 1.95e+56)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x * t_0) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 1.95e+56)
		tmp = x / (z / t_0);
	else
		tmp = (x * t_0) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 1.95e+56], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999997e56

   1. Initial program 94.2%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   if 1.94999999999999997e56 < x

   1. Initial program 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 2: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.6e-12) (/ x z) (* (sin y) (/ x (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e-12) {
		tmp = x / z;
	} else {
		tmp = sin(y) * (x / (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.6d-12) then
        tmp = x / z
    else
        tmp = sin(y) * (x / (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e-12) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.6e-12:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.6e-12)
		tmp = Float64(x / z);
	else
		tmp = Float64(sin(y) * Float64(x / Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.6e-12)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.6e-12], N[(x / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999983e-12

   1. Initial program 95.6%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 90.8%

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

      3. associate-/l/ 82.3%

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

      4. associate-/r/ 83.0%

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

      5. associate-/r* 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in y around 0 70.9%

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

   if 2.59999999999999983e-12 < y

   1. Initial program 94.6%

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

      1. associate-/l* 94.0%

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

      2. associate-/r/ 94.1%

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

      3. associate-/l/ 94.9%

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

      4. associate-/r/ 94.9%

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

      5. associate-/r* 94.0%

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

   3. Simplified 94.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= z -5e+33) (* t_0 (/ x z)) (/ x (/ z t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z <= -5e+33) {
		tmp = t_0 * (x / 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 (z <= (-5d+33)) then
        tmp = t_0 * (x / 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 (z <= -5e+33) {
		tmp = t_0 * (x / z);
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= -5e+33:
		tmp = t_0 * (x / 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 (z <= -5e+33)
		tmp = Float64(t_0 * Float64(x / 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 (z <= -5e+33)
		tmp = t_0 * (x / 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[LessEqual[z, -5e+33], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.99999999999999973e33

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   if -4.99999999999999973e33 < z

   1. Initial program 93.9%

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

      1. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 4: 95.7% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 95.4%

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

   1. *-commutative 95.4%

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

   2. associate-*r/ 96.1%

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

3. Simplified 96.1%

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

4. Final simplification 96.1%

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

Alternative 5: 61.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 350:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{6}{y}}{\frac{y}{\frac{x}{z}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 350.0)
   (* (/ x z) (+ 1.0 (* (* y y) -0.16666666666666666)))
   (/ (/ 6.0 y) (/ y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 350.0) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = (6.0 / y) / (y / (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 350.0d0) then
        tmp = (x / z) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = (6.0d0 / y) / (y / (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 350.0) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = (6.0 / y) / (y / (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 350.0:
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = (6.0 / y) / (y / (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 350.0)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(6.0 / y) / Float64(y / Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 350.0)
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = (6.0 / y) / (y / (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 350.0], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 / y), $MachinePrecision] / N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 350

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*r/ 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in y around 0 69.4%

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

      1. unpow2 69.4%

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

   6. Simplified 69.4%

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

   if 350 < y

   1. Initial program 94.1%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

   7. Taylor expanded in y around inf 30.3%

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

      1. unpow2 30.3%

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

      2. associate-*r* 30.3%

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

   9. Simplified 30.3%

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

       1. associate-*r/ 30.3%

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

       2. frac-times 30.3%

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

       3. clear-num 30.3%

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

       4. un-div-inv 30.3%

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

       5. *-commutative 30.3%

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

   11. Applied egg-rr 30.3%

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

   12. Taylor expanded in z around 0 30.3%

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

       1. associate-/l* 30.4%

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

   14. Simplified 30.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 350:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{6}{y}}{\frac{y}{\frac{x}{z}}}\\ \end{array} \]

Alternative 6: 62.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * (x / (y * (z * y)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 95.8%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 91.1%

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

      3. associate-/l/ 82.8%

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

      4. associate-/r/ 83.6%

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

      5. associate-/r* 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in y around 0 71.4%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

   7. Taylor expanded in y around inf 30.3%

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

      1. unpow2 30.3%

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

      2. associate-*r* 30.3%

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

   9. Simplified 30.3%

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

4. Final simplification 61.8%

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

Alternative 7: 62.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 95.8%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 91.1%

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

      3. associate-/l/ 82.8%

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

      4. associate-/r/ 83.6%

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

      5. associate-/r* 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in y around 0 71.4%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

   7. Taylor expanded in y around inf 30.3%

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

      1. unpow2 30.3%

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

      2. associate-*r* 30.3%

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

   9. Simplified 30.3%

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

   10. Taylor expanded in x around 0 30.3%

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

       1. unpow2 30.3%

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

       2. *-commutative 30.3%

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

       3. associate-/r* 30.1%

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

   12. Simplified 30.1%

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

4. Final simplification 61.7%

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

Alternative 8: 62.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 95.8%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 91.1%

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

      3. associate-/l/ 82.8%

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

      4. associate-/r/ 83.6%

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

      5. associate-/r* 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in y around 0 71.4%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

   7. Taylor expanded in y around inf 30.3%

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

      1. unpow2 30.3%

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

      2. associate-*r* 30.3%

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

      3. associate-*r/ 30.3%

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

      4. times-frac 30.3%

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

   9. Simplified 30.3%

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

4. Final simplification 61.8%

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

Alternative 9: 62.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x / z
    else
        tmp = (6.0d0 / y) / (y / (x / z))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 95.8%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 91.1%

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

      3. associate-/l/ 82.8%

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

      4. associate-/r/ 83.6%

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

      5. associate-/r* 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in y around 0 71.4%

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

   if 2.39999999999999991 < y

   1. Initial program 94.1%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

   7. Taylor expanded in y around inf 30.3%

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

      1. unpow2 30.3%

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

      2. associate-*r* 30.3%

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

   9. Simplified 30.3%

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

       1. associate-*r/ 30.3%

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

       2. frac-times 30.3%

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

       3. clear-num 30.3%

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

       4. un-div-inv 30.3%

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

       5. *-commutative 30.3%

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

   11. Applied egg-rr 30.3%

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

   12. Taylor expanded in z around 0 30.3%

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

       1. associate-/l* 30.4%

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

   14. Simplified 30.4%

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

4. Final simplification 61.8%

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

Alternative 10: 66.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z + (0.16666666666666666d0 * (z * (y * y))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-/l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in y around 0 68.1%

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

   1. *-commutative 68.1%

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

   2. unpow2 68.1%

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

6. Simplified 68.1%

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

7. Final simplification 68.1%

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

Alternative 11: 66.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z + (y * ((z * y) * 0.16666666666666666d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-/l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in y around 0 68.1%

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

   1. *-commutative 68.1%

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

   2. unpow2 68.1%

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

6. Simplified 68.1%

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

7. Taylor expanded in z around 0 68.1%

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

   1. unpow2 68.1%

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

   2. associate-*r* 68.1%

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

   3. *-commutative 68.1%

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

   4. associate-*l* 68.1%

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

9. Simplified 68.1%

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

10. Final simplification 68.1%

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

Alternative 12: 59.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-/l* 97.1%

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

   2. associate-/r/ 91.6%

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

   3. associate-/l/ 85.5%

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

   4. associate-/r/ 86.1%

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

   5. associate-/r* 86.6%

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

3. Simplified 86.6%

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

4. Taylor expanded in y around 0 58.2%

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

5. Final simplification 58.2%

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

Developer target: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2023285 
(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))