Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 99.6%

Time: 11.4s

Alternatives: 15

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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+80} \lor \neg \left(z \leq 2.1 \cdot 10^{-41}\right):\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= z -2e+80) (not (<= z 2.1e-41)))
     (* 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 <= -2e+80) || !(z <= 2.1e-41)) {
		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 <= (-2d+80)) .or. (.not. (z <= 2.1d-41))) 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 <= -2e+80) || !(z <= 2.1e-41)) {
		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 <= -2e+80) or not (z <= 2.1e-41):
		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 <= -2e+80) || !(z <= 2.1e-41))
		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 <= -2e+80) || ~((z <= 2.1e-41)))
		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[Or[LessEqual[z, -2e+80], N[Not[LessEqual[z, 2.1e-41]], $MachinePrecision]], 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 < -2e80 or 2.10000000000000013e-41 < z

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   if -2e80 < z < 2.10000000000000013e-41

   1. Initial program 94.6%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+80} \lor \neg \left(z \leq 2.1 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 2: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-5)
   (* (/ x z) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-5) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} 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 <= 1d-5) then
        tmp = (x / z) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    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 <= 1e-5) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e-5:
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-5)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	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 <= 1e-5)
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e-5], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000008e-5

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in y around 0 64.5%

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

      1. unpow2 64.5%

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

   6. Simplified 64.5%

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

   if 1.00000000000000008e-5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

      2. associate-/r/ 88.5%

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

      3. associate-/l/ 92.9%

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

      4. associate-/r/ 92.9%

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

      5. associate-/r* 88.3%

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

   3. Simplified 88.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 3: 96.2% 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 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*r/ 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Alternative 4: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

2. Final simplification 97.2%

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

Alternative 5: 62.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (* 6.0 (* (/ 1.0 y) (/ (/ x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((1.0 / 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 <= 2.5d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * ((1.0d0 / 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 <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(1.0 / y) * Float64(Float64(x / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(1.0 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. unpow2 36.2%

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

      2. *-commutative 36.2%

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

   9. Simplified 36.2%

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

       1. associate-/r* 36.1%

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

       2. *-un-lft-identity 36.1%

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

       3. associate-*l/ 36.1%

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

       4. *-un-lft-identity 36.1%

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

       5. times-frac 36.4%

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

       6. associate-*l/ 36.4%

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

       7. *-un-lft-identity 36.4%

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

   11. Applied egg-rr 36.4%

       \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \]

Alternative 6: 60.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e+65)
   (* (/ x z) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (* 6.0 (* (/ 1.0 y) (/ (/ x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e+65) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = 6.0 * ((1.0 / 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 <= 1.8d+65) then
        tmp = (x / z) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = 6.0d0 * ((1.0d0 / 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 <= 1.8e+65) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.8e+65:
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.8e+65)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(6.0 * Float64(Float64(1.0 / y) * Float64(Float64(x / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.8e+65)
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e+65], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(1.0 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999989e65

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r/ 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in y around 0 63.7%

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

      1. unpow2 63.7%

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

   6. Simplified 63.7%

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

   if 1.79999999999999989e65 < y

   1. Initial program 92.1%

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

      1. associate-/l* 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in y around 0 38.1%

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

      1. *-commutative 38.1%

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

      2. unpow2 38.1%

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

   6. Simplified 38.1%

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

   7. Taylor expanded in y around inf 38.1%

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

      1. unpow2 38.1%

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

      2. *-commutative 38.1%

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

   9. Simplified 38.1%

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

       1. associate-/r* 38.0%

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

       2. *-un-lft-identity 38.0%

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

       3. associate-*l/ 38.0%

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

       4. *-un-lft-identity 38.0%

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

       5. times-frac 38.3%

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

       6. associate-*l/ 38.3%

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

       7. *-un-lft-identity 38.3%

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

   11. Applied egg-rr 38.3%

       \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \]

Alternative 7: 62.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{y} \cdot \frac{x}{y \cdot \left(-z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (* (/ -6.0 y) (/ x (* y (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (-6.0 / y) * (x / (y * -z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d0) then
        tmp = x / z
    else
        tmp = ((-6.0d0) / y) * (x / (y * -z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (-6.0 / y) * (x / (y * -z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = (-6.0 / y) * (x / (y * -z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(-6.0 / y) * Float64(x / Float64(y * Float64(-z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = (-6.0 / y) * (x / (y * -z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(-6.0 / y), $MachinePrecision] * N[(x / N[(y * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. unpow2 36.2%

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

      2. *-commutative 36.2%

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

   9. Simplified 36.2%

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

       1. associate-*r/ 36.2%

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

       2. frac-2neg 36.2%

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

       3. *-commutative 36.2%

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

       4. associate-*r* 36.3%

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

       5. distribute-rgt-neg-in 36.3%

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

   11. Applied egg-rr 36.3%

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

       1. distribute-lft-neg-in 36.3%

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

       2. times-frac 36.3%

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

       3. metadata-eval 36.3%

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

       4. *-commutative 36.3%

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

       5. distribute-rgt-neg-in 36.3%

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

   13. Simplified 36.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{y} \cdot \frac{x}{y \cdot \left(-z\right)}\\ \end{array} \]

Alternative 8: 62.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (* 6.0 (/ x (* z (* y y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (z * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		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 <= 2.5)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (z * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. unpow2 36.2%

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

      2. *-commutative 36.2%

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

   9. Simplified 36.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 9: 62.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (* (/ 6.0 z) (/ x (* y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (6.0 / z) * (x / (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 <= 2.5d0) then
        tmp = x / z
    else
        tmp = (6.0d0 / z) * (x / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (6.0 / z) * (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = (6.0 / z) * (x / (y * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(6.0 / z) * Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = (6.0 / z) * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(6.0 / z), $MachinePrecision] * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. associate-*r/ 36.2%

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

      2. unpow2 36.2%

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

      3. *-commutative 36.2%

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

      4. times-frac 36.3%

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

   9. Simplified 36.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \end{array} \]

Alternative 10: 62.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z \cdot \frac{y \cdot y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (/ 6.0 (* z (/ (* y y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 / (z * ((y * y) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d0) then
        tmp = x / z
    else
        tmp = 6.0d0 / (z * ((y * y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 / (z * ((y * y) / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = 6.0 / (z * ((y * y) / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 / Float64(z * Float64(Float64(y * y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = 6.0 / (z * ((y * y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(6.0 / N[(z * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. unpow2 36.2%

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

      2. *-commutative 36.2%

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

   9. Simplified 36.2%

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

       1. clear-num 36.2%

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

       2. un-div-inv 36.2%

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

       3. *-un-lft-identity 36.2%

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

       4. times-frac 36.3%

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

       5. /-rgt-identity 36.3%

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

   11. Applied egg-rr 36.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z \cdot \frac{y \cdot y}{x}}\\ \end{array} \]

Alternative 11: 62.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (/ (* x 6.0) (* y (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (x * 6.0) / (y * (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = (x * 6.0) / (y * (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x * 6.0) / Float64(y * Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = (x * 6.0) / (y * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. unpow2 36.2%

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

      2. *-commutative 36.2%

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

   9. Simplified 36.2%

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

       1. associate-*r/ 36.2%

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

       2. *-commutative 36.2%

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

       3. associate-*r* 36.3%

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

   11. Applied egg-rr 36.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 12: 62.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (/ (* 6.0 (/ x y)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (6.0 * (x / y)) / (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = (6.0 * (x / y)) / (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(6.0 * Float64(x / y)) / Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = (6.0 * (x / y)) / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(6.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 98.5%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 88.6%

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

      3. associate-/l/ 86.9%

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

      4. associate-/r/ 82.7%

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

      5. associate-/r* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 67.4%

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

   if 2.5 < y

   1. Initial program 92.9%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in y around inf 36.2%

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

      1. associate-*r/ 36.2%

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

      2. unpow2 36.2%

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

      3. *-commutative 36.2%

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

      4. times-frac 36.3%

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

   9. Simplified 36.3%

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

       1. *-commutative 36.3%

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

       2. associate-/r* 36.4%

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

       3. frac-times 36.3%

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

       4. *-commutative 36.3%

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

   11. Applied egg-rr 36.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 13: 66.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. associate-/l* 94.6%

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

3. Simplified 94.6%

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

4. Taylor expanded in y around 0 67.8%

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

   1. *-commutative 67.8%

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

   2. unpow2 67.8%

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

6. Simplified 67.8%

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

7. Taylor expanded in z around 0 67.8%

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

   1. unpow2 67.8%

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

   2. associate-*l* 67.8%

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

9. Simplified 67.8%

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

10. Final simplification 67.8%

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

Alternative 14: 58.3% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{z}{x}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (/ z x)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (z / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. associate-/l* 94.6%

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

   2. associate-/r/ 88.6%

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

   3. associate-/l/ 88.3%

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

   4. associate-/r/ 85.1%

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

   5. associate-/r* 83.6%

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

3. Simplified 83.6%

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

   1. *-commutative 83.6%

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

   2. associate-*r/ 85.8%

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

   3. frac-times 95.4%

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

   4. clear-num 95.0%

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

   5. un-div-inv 95.4%

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

5. Applied egg-rr 95.4%

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

6. Taylor expanded in y around 0 55.5%

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

7. Final simplification 55.5%

   \[\leadsto \frac{1}{\frac{z}{x}} \]

Alternative 15: 58.6% 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 97.2%

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

   1. associate-/l* 94.6%

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

   2. associate-/r/ 88.6%

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

   3. associate-/l/ 88.3%

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

   4. associate-/r/ 85.1%

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

   5. associate-/r* 83.6%

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

3. Simplified 83.6%

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

4. Taylor expanded in y around 0 55.1%

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

5. Final simplification 55.1%

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

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 2023275 
(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))