Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 97.0%

Time: 9.6s

Alternatives: 14

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (/ (sin y) y)) z)))
   (if (<= t_0 -1e-216) t_0 (/ x (* z (/ y (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -1e-216) {
		tmp = t_0;
	} else {
		tmp = x / (z * (y / sin(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (sin(y) / y)) / z
    if (t_0 <= (-1d-216)) then
        tmp = t_0
    else
        tmp = x / (z * (y / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -1e-216) {
		tmp = t_0;
	} else {
		tmp = x / (z * (y / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -1e-216:
		tmp = t_0
	else:
		tmp = x / (z * (y / math.sin(y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -1e-216)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(z * Float64(y / sin(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -1e-216)
		tmp = t_0;
	else
		tmp = x / (z * (y / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-216], t$95$0, N[(x / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1e-216

   1. Initial program 99.8%

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

   if -1e-216 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.3%

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

      1. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.8%

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

      3. clear-num 98.9%

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

   5. Applied egg-rr 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternative 2: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.15e-8) (/ x z) (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e-8) {
		tmp = x / z;
	} else {
		tmp = sin(y) * (x / (y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.15d-8) then
        tmp = x / z
    else
        tmp = sin(y) * (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e-8) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.15e-8:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.15e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.15e-8)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.15e-8], N[(x / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1500000000000001e-8

   1. Initial program 97.5%

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

      1. associate-/l* 98.3%

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

      2. associate-/r/ 88.7%

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

      3. associate-/l/ 82.6%

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

      4. associate-/r/ 82.9%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 68.9%

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

   if 2.1500000000000001e-8 < y

   1. Initial program 97.4%

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

      1. associate-/l* 94.4%

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

      2. associate-/r/ 94.5%

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

      3. associate-/l/ 97.7%

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

      4. associate-/r/ 97.6%

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

      5. associate-/r* 94.4%

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

   3. Simplified 94.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e+77) (* (/ (sin y) y) (/ x z)) (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+77) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = sin(y) * (x / (y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.2d+77) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = sin(y) * (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+77) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.2e+77:
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e+77)
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e+77)
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e+77], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $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 < 9.19999999999999979e77

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*r/ 98.0%

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

   3. Simplified 98.0%

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

   if 9.19999999999999979e77 < y

   1. Initial program 97.4%

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

      1. associate-/l* 96.3%

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

      2. associate-/r/ 96.3%

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

      3. associate-/l/ 97.5%

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

      4. associate-/r/ 97.4%

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

      5. associate-/r* 96.4%

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

   3. Simplified 96.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 4: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

   1. clear-num 97.2%

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

   2. associate-/r/ 97.3%

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

   3. clear-num 97.3%

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

5. Applied egg-rr 97.3%

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

6. Final simplification 97.3%

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

Alternative 5: 60.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\left(y \cdot y\right) \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.3e+47)
   (* (/ x z) (+ 1.0 (* (* y y) -0.16666666666666666)))
   (/ 6.0 (* (* y y) (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.3e+47) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / ((y * y) * (z / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.3d+47) then
        tmp = (x / z) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = 6.0d0 / ((y * y) * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.3e+47) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = 6.0 / ((y * y) * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.3e+47:
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = 6.0 / ((y * y) * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.3e+47)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(6.0 / Float64(Float64(y * y) * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.3e+47)
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = 6.0 / ((y * y) * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.3e+47], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(y * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.3000000000000001e47

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*r/ 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in y around 0 64.9%

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

      1. unpow2 64.9%

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

   6. Simplified 64.9%

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

   if 7.3000000000000001e47 < y

   1. Initial program 97.7%

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

      1. associate-/l* 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in y around 0 33.5%

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

      1. *-commutative 33.5%

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

      2. unpow2 33.5%

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

   6. Simplified 33.5%

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

   7. Taylor expanded in y around inf 33.5%

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

      1. unpow2 33.5%

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

      2. associate-*r* 33.6%

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

   9. Simplified 33.6%

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

       1. clear-num 33.6%

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

       2. un-div-inv 33.6%

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

   11. Applied egg-rr 33.6%

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

       1. div-inv 33.6%

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

       2. associate-*r* 33.5%

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

       3. associate-*l* 35.3%

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

       4. div-inv 35.3%

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

   13. Applied egg-rr 35.3%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\left(y \cdot y\right) \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 6: 62.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\left(y \cdot y\right) \cdot \frac{-z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 17000.0) (/ x z) (/ 6.0 (* (* y y) (/ (- z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 17000.0) {
		tmp = x / z;
	} else {
		tmp = 6.0 / ((y * y) * (-z / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 17000.0d0) then
        tmp = x / z
    else
        tmp = 6.0d0 / ((y * y) * (-z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 17000.0) {
		tmp = x / z;
	} else {
		tmp = 6.0 / ((y * y) * (-z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 17000.0:
		tmp = x / z
	else:
		tmp = 6.0 / ((y * y) * (-z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 17000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 / Float64(Float64(y * y) * Float64(Float64(-z) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 17000.0)
		tmp = x / z;
	else
		tmp = 6.0 / ((y * y) * (-z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 17000.0], N[(x / z), $MachinePrecision], N[(6.0 / N[(N[(y * y), $MachinePrecision] * N[((-z) / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 17000

   1. Initial program 97.5%

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

      1. associate-/l* 98.3%

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

      2. associate-/r/ 88.7%

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

      3. associate-/l/ 82.6%

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

      4. associate-/r/ 82.9%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 68.9%

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

   if 17000 < y

   1. Initial program 97.4%

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

      1. associate-/l* 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 28.4%

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

      1. *-commutative 28.4%

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

      2. unpow2 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in y around inf 28.4%

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

      1. unpow2 28.4%

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

      2. associate-*r* 28.5%

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

   9. Simplified 28.5%

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

       1. clear-num 28.5%

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

       2. un-div-inv 28.5%

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

   11. Applied egg-rr 28.5%

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

       1. add-sqr-sqrt 14.0%

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

       2. sqrt-unprod 29.8%

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

       3. sqr-neg 29.8%

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

       4. sqrt-unprod 14.8%

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

       5. add-sqr-sqrt 28.6%

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

       6. distribute-rgt-neg-in 28.6%

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

       7. distribute-rgt-neg-in 28.6%

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

       8. distribute-neg-frac 28.6%

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

       9. div-inv 28.6%

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

       10. associate-*r* 28.6%

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

       11. associate-*l* 29.8%

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

       12. div-inv 29.8%

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

   13. Applied egg-rr 29.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\left(y \cdot y\right) \cdot \frac{-z}{x}}\\ \end{array} \]

Alternative 7: 61.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+62) (/ x z) (* 6.0 (/ x (* y (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+62) {
		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 <= 6d+62) 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 <= 6e+62) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (y * (y * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+62:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (y * (y * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+62)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(x / Float64(y * Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+62)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (y * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+62], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6e62

   1. Initial program 97.5%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 89.2%

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

      3. associate-/l/ 83.9%

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

      4. associate-/r/ 84.2%

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

      5. associate-/r* 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in y around 0 64.3%

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

   if 6e62 < y

   1. Initial program 97.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 35.1%

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

      1. *-commutative 35.1%

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

      2. unpow2 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in y around inf 35.1%

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

      1. unpow2 35.1%

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

      2. associate-*r* 35.2%

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

   9. Simplified 35.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 8: 61.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+62) (/ x z) (* 6.0 (/ (/ x y) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+62) {
		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 <= 6d+62) 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 <= 6e+62) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((x / y) / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+62:
		tmp = x / z
	else:
		tmp = 6.0 * ((x / y) / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+62)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(x / y) / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+62)
		tmp = x / z;
	else
		tmp = 6.0 * ((x / y) / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+62], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(x / y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6e62

   1. Initial program 97.5%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 89.2%

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

      3. associate-/l/ 83.9%

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

      4. associate-/r/ 84.2%

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

      5. associate-/r* 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in y around 0 64.3%

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

   if 6e62 < y

   1. Initial program 97.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 35.1%

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

      1. *-commutative 35.1%

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

      2. unpow2 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in y around inf 35.1%

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

      1. unpow2 35.1%

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

      2. associate-*r* 35.2%

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

   9. Simplified 35.2%

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

   10. Taylor expanded in x around 0 35.1%

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

       1. unpow2 35.1%

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

       2. associate-*r* 35.2%

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

       3. associate-/r* 35.2%

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

       4. *-commutative 35.2%

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

   12. Simplified 35.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 9: 61.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 17000.0) (/ x z) (* -6.0 (/ x (* y (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 17000.0) {
		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 <= 17000.0d0) 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 <= 17000.0) {
		tmp = x / z;
	} else {
		tmp = -6.0 * (x / (y * (y * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 17000.0:
		tmp = x / z
	else:
		tmp = -6.0 * (x / (y * (y * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 17000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(-6.0 * Float64(x / Float64(y * Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 17000.0)
		tmp = x / z;
	else
		tmp = -6.0 * (x / (y * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 17000.0], N[(x / z), $MachinePrecision], N[(-6.0 * N[(x / N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 17000

   1. Initial program 97.5%

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

      1. associate-/l* 98.3%

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

      2. associate-/r/ 88.7%

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

      3. associate-/l/ 82.6%

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

      4. associate-/r/ 82.9%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 68.9%

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

   if 17000 < y

   1. Initial program 97.4%

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

      1. associate-/l* 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 28.4%

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

      1. *-commutative 28.4%

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

      2. unpow2 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in y around inf 28.4%

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

      1. unpow2 28.4%

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

      2. associate-*r* 28.5%

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

   9. Simplified 28.5%

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

       1. clear-num 28.5%

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

       2. un-div-inv 28.5%

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

   11. Applied egg-rr 28.5%

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

       1. frac-2neg 28.5%

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

       2. div-inv 28.5%

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

       3. metadata-eval 28.5%

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

       4. distribute-neg-frac 28.5%

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

       5. distribute-rgt-neg-in 28.5%

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

       6. distribute-rgt-neg-in 28.5%

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

       7. add-sqr-sqrt 14.5%

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

       8. sqrt-unprod 27.9%

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

       9. sqr-neg 27.9%

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

       10. sqrt-unprod 13.8%

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

       11. add-sqr-sqrt 28.6%

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

       12. clear-num 28.6%

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

       13. *-commutative 28.6%

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

       14. *-commutative 28.6%

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

       15. associate-*l* 28.6%

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

   13. Applied egg-rr 28.6%

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

       1. *-commutative 28.6%

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

       2. associate-*r* 28.6%

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

       3. *-commutative 28.6%

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

       4. /-rgt-identity 28.6%

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

       5. associate-/l* 28.6%

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

       6. associate-*l/ 28.6%

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

       7. associate-*r/ 28.6%

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

       8. associate-/l* 28.6%

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

       9. /-rgt-identity 28.6%

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

       10. *-commutative 28.6%

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

   15. Simplified 28.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 10: 61.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(y \cdot y\right)} \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 17000.0) (/ x z) (* (/ x (* z (* y y))) -6.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 17000.0) {
		tmp = x / z;
	} else {
		tmp = (x / (z * (y * y))) * -6.0;
	}
	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) :: tmp
    if (y <= 17000.0d0) then
        tmp = x / z
    else
        tmp = (x / (z * (y * y))) * (-6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 17000.0) {
		tmp = x / z;
	} else {
		tmp = (x / (z * (y * y))) * -6.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 17000.0:
		tmp = x / z
	else:
		tmp = (x / (z * (y * y))) * -6.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 17000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / Float64(z * Float64(y * y))) * -6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 17000.0)
		tmp = x / z;
	else
		tmp = (x / (z * (y * y))) * -6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 17000.0], N[(x / z), $MachinePrecision], N[(N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 17000

   1. Initial program 97.5%

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

      1. associate-/l* 98.3%

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

      2. associate-/r/ 88.7%

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

      3. associate-/l/ 82.6%

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

      4. associate-/r/ 82.9%

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

      5. associate-/r* 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in y around 0 68.9%

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

   if 17000 < y

   1. Initial program 97.4%

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

      1. associate-/l* 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 28.4%

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

      1. *-commutative 28.4%

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

      2. unpow2 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in y around inf 28.4%

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

      1. unpow2 28.4%

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

      2. associate-*r* 28.5%

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

   9. Simplified 28.5%

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

       1. clear-num 28.5%

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

       2. un-div-inv 28.5%

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

   11. Applied egg-rr 28.5%

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

       1. frac-2neg 28.5%

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

       2. div-inv 28.5%

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

       3. metadata-eval 28.5%

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

       4. distribute-neg-frac 28.5%

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

       5. distribute-rgt-neg-in 28.5%

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

       6. distribute-rgt-neg-in 28.5%

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

       7. add-sqr-sqrt 14.5%

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

       8. sqrt-unprod 27.9%

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

       9. sqr-neg 27.9%

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

       10. sqrt-unprod 13.8%

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

       11. add-sqr-sqrt 28.6%

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

       12. clear-num 28.6%

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

       13. *-commutative 28.6%

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

       14. *-commutative 28.6%

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

       15. associate-*l* 28.6%

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

   13. Applied egg-rr 28.6%

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

       1. *-commutative 28.6%

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

   15. Simplified 28.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 17000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(y \cdot y\right)} \cdot -6\\ \end{array} \]

Alternative 11: 65.9% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z * (1.0d0 + (y * (y * 0.16666666666666666d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

   1. clear-num 97.2%

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

   2. associate-/r/ 97.3%

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

   3. clear-num 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in y around 0 62.3%

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

   1. unpow2 62.3%

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

   2. *-commutative 62.3%

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

   3. associate-*l* 62.3%

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

8. Simplified 62.3%

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

9. Final simplification 62.3%

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

Alternative 12: 65.9% 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 97.5%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in y around 0 62.3%

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

   1. *-commutative 62.3%

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

   2. unpow2 62.3%

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

6. Simplified 62.3%

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

7. Taylor expanded in z around 0 62.3%

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

   1. unpow2 62.3%

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

   2. associate-*r* 62.4%

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

9. Simplified 62.4%

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

    1. *-commutative 62.4%

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

    2. add-sqr-sqrt 29.3%

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

    3. sqrt-unprod 57.9%

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

    4. sqr-neg 57.9%

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

    5. sqrt-unprod 33.6%

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

    6. add-sqr-sqrt 62.7%

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

    7. distribute-rgt-neg-in 62.7%

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

    8. distribute-rgt-neg-in 62.7%

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

    9. cancel-sign-sub-inv 62.7%

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

    10. associate-*r* 62.7%

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

    11. associate-*l* 62.7%

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

11. Applied egg-rr 62.7%

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

    1. associate-*r* 62.7%

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

    2. unpow2 62.7%

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

    3. *-commutative 62.7%

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

    4. unpow2 62.7%

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

    5. *-commutative 62.7%

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

    6. cancel-sign-sub-inv 62.7%

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

    7. metadata-eval 62.7%

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

    8. associate-*r* 62.7%

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

    9. *-commutative 62.7%

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

    10. /-rgt-identity 62.7%

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

    11. associate-/l* 62.7%

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

    12. associate-*l/ 62.7%

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

    13. associate-*r/ 62.7%

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

    14. associate-/l* 62.7%

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

    15. /-rgt-identity 62.7%

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

    16. *-commutative 62.7%

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

13. Simplified 62.7%

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

14. Final simplification 62.7%

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

Alternative 13: 65.9% 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 97.5%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in y around 0 62.3%

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

   1. *-commutative 62.3%

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

   2. unpow2 62.3%

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

6. Simplified 62.3%

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

7. Taylor expanded in z around 0 62.3%

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

   1. unpow2 62.3%

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

   2. associate-*r* 62.4%

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

9. Simplified 62.4%

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

    1. *-commutative 62.4%

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

    2. add-sqr-sqrt 29.3%

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

    3. sqrt-unprod 57.9%

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

    4. sqr-neg 57.9%

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

    5. sqrt-unprod 33.6%

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

    6. add-sqr-sqrt 62.7%

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

    7. distribute-rgt-neg-in 62.7%

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

    8. distribute-rgt-neg-in 62.7%

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

    9. cancel-sign-sub-inv 62.7%

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

    10. associate-*r* 62.7%

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

    11. associate-*l* 62.7%

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

11. Applied egg-rr 62.7%

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

    1. associate-*r* 62.7%

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

    2. unpow2 62.7%

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

    3. *-commutative 62.7%

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

    4. unpow2 62.7%

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

    5. *-commutative 62.7%

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

13. Simplified 62.7%

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

14. Final simplification 62.7%

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

Alternative 14: 58.4% 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 97.5%

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

   1. associate-/l* 97.3%

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

   2. associate-/r/ 90.2%

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

   3. associate-/l/ 86.5%

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

   4. associate-/r/ 86.7%

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

   5. associate-/r* 84.9%

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

3. Simplified 84.9%

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

4. Taylor expanded in y around 0 54.6%

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

5. Final simplification 54.6%

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

Developer target: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023290 
(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))