Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 97.3%

Time: 8.8s

Alternatives: 12

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: 96.3% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-194}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (sin y) y))))
   (if (<= t_0 5e-194) (/ x (* (/ y (sin y)) z)) (/ t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * (sin(y) / y);
	double tmp;
	if (t_0 <= 5e-194) {
		tmp = x / ((y / sin(y)) * z);
	} else {
		tmp = t_0 / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (sin(y) / y)
    if (t_0 <= 5d-194) then
        tmp = x / ((y / sin(y)) * z)
    else
        tmp = t_0 / z
    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);
	double tmp;
	if (t_0 <= 5e-194) {
		tmp = x / ((y / Math.sin(y)) * z);
	} else {
		tmp = t_0 / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (math.sin(y) / y)
	tmp = 0
	if t_0 <= 5e-194:
		tmp = x / ((y / math.sin(y)) * z)
	else:
		tmp = t_0 / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= 5e-194)
		tmp = Float64(x / Float64(Float64(y / sin(y)) * z));
	else
		tmp = Float64(t_0 / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= 5e-194)
		tmp = x / ((y / sin(y)) * z);
	else
		tmp = t_0 / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-194], N[(x / N[(N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 5.0000000000000002e-194

   1. Initial program 95.3%

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

      1. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 98.1%

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

      3. clear-num 98.2%

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

   5. Applied egg-rr 98.2%

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

   if 5.0000000000000002e-194 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 5 \cdot 10^{-194}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 2: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-13) (/ x z) (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-13) {
		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 <= 2.4d-13) 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 <= 2.4e-13) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-13:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-13)
		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 <= 2.4e-13)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-13], 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 < 2.3999999999999999e-13

   1. Initial program 97.8%

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

   2. Taylor expanded in y around 0 75.7%

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

   if 2.3999999999999999e-13 < y

   1. Initial program 94.1%

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

      1. associate-*l/ 92.5%

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

      2. times-frac 91.2%

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

      3. *-commutative 91.2%

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

      4. associate-*r/ 91.2%

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

      5. *-commutative 91.2%

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

   3. Simplified 91.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.7e-47) (/ x z) (* (/ (sin y) z) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e-47) {
		tmp = x / z;
	} else {
		tmp = (sin(y) / z) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.7d-47) then
        tmp = x / z
    else
        tmp = (sin(y) / z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e-47) {
		tmp = x / z;
	} else {
		tmp = (Math.sin(y) / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.7e-47:
		tmp = x / z
	else:
		tmp = (math.sin(y) / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.7e-47)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(sin(y) / z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.7e-47)
		tmp = x / z;
	else
		tmp = (sin(y) / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.7e-47], N[(x / z), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7000000000000001e-47

   1. Initial program 97.8%

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

   2. Taylor expanded in y around 0 75.1%

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

   if 1.7000000000000001e-47 < y

   1. Initial program 94.5%

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

      1. associate-*r/ 93.8%

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

      2. associate-/l/ 91.0%

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

      3. *-commutative 91.0%

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

      4. times-frac 94.8%

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

   3. Simplified 94.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. associate-/l* 96.2%

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

3. Simplified 96.2%

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

   1. clear-num 96.1%

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

   2. associate-/r/ 96.2%

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

   3. clear-num 96.3%

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

5. Applied egg-rr 96.3%

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

6. Final simplification 96.3%

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

Alternative 5: 63.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (* (/ (/ x y) (* z 0.16666666666666666)) (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d0) then
        tmp = x / z
    else
        tmp = ((x / y) / (z * 0.16666666666666666d0)) * (1.0d0 / 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 = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(z * 0.16666666666666666)) * Float64(1.0 / 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 = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 76.0%

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

   if 2.5 < y

   1. Initial program 93.9%

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

      1. associate-/l* 90.9%

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

      2. associate-/r/ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around 0 28.5%

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

   5. Taylor expanded in y around inf 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-*r* 28.5%

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

   7. Simplified 28.5%

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

      1. associate-/r* 28.6%

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

      2. div-inv 28.6%

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

      3. associate-/r* 28.6%

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

   9. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 6: 60.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 62:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 62.0)
   (/ x (/ z (+ 1.0 (* -0.16666666666666666 (* y y)))))
   (* (/ (/ x y) (* z 0.16666666666666666)) (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 62.0) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 62.0d0) then
        tmp = x / (z / (1.0d0 + ((-0.16666666666666666d0) * (y * y))))
    else
        tmp = ((x / y) / (z * 0.16666666666666666d0)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 62.0) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 62.0:
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))))
	else:
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 62.0)
		tmp = Float64(x / Float64(z / Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))));
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(z * 0.16666666666666666)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 62.0)
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	else
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 62.0], N[(x / N[(z / N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 62

   1. Initial program 97.9%

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

      1. associate-/l* 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in y around 0 74.0%

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

      1. unpow2 74.0%

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

   6. Simplified 74.0%

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

   if 62 < y

   1. Initial program 93.8%

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

      1. associate-/l* 90.8%

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

      2. associate-/r/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in y around 0 29.0%

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

   5. Taylor expanded in y around inf 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*r* 29.0%

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

   7. Simplified 29.0%

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

      1. associate-/r* 29.1%

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

      2. div-inv 29.1%

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

      3. associate-/r* 29.1%

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

   9. Applied egg-rr 29.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 62:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 7: 60.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 62:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 62.0)
   (/ (* x (+ 1.0 (* -0.16666666666666666 (* y y)))) z)
   (* (/ (/ x y) (* z 0.16666666666666666)) (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 62.0) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 62.0d0) then
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))) / z
    else
        tmp = ((x / y) / (z * 0.16666666666666666d0)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 62.0) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 62.0:
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z
	else:
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 62.0)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))) / z);
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(z * 0.16666666666666666)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 62.0)
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	else
		tmp = ((x / y) / (z * 0.16666666666666666)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 62.0], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 62

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 74.0%

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

      1. unpow2 74.0%

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

   4. Simplified 74.0%

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

   if 62 < y

   1. Initial program 93.8%

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

      1. associate-/l* 90.8%

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

      2. associate-/r/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in y around 0 29.0%

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

   5. Taylor expanded in y around inf 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*r* 29.0%

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

   7. Simplified 29.0%

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

      1. associate-/r* 29.1%

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

      2. div-inv 29.1%

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

      3. associate-/r* 29.1%

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

   9. Applied egg-rr 29.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 62:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.16666666666666666} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 8: 63.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (/ (/ x (* z (* y 0.16666666666666666))) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = (x / (z * (y * 0.16666666666666666))) / 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 = (x / (z * (y * 0.16666666666666666d0))) / 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 = (x / (z * (y * 0.16666666666666666))) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = (x / (z * (y * 0.16666666666666666))) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / Float64(z * Float64(y * 0.16666666666666666))) / 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 = (x / (z * (y * 0.16666666666666666))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(x / N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 76.0%

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

   if 2.5 < y

   1. Initial program 93.9%

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

      1. associate-/l* 90.9%

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

      2. associate-/r/ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around 0 28.5%

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

   5. Taylor expanded in y around inf 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-*r* 28.5%

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

   7. Simplified 28.5%

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

      1. clear-num 28.5%

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

      2. associate-/r/ 28.5%

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

      3. associate-*r* 28.5%

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

      4. associate-*l* 28.5%

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

      5. *-commutative 28.5%

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

   9. Applied egg-rr 28.5%

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

       1. associate-*l/ 28.5%

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

       2. *-un-lft-identity 28.5%

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

       3. associate-*r* 28.5%

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

       4. associate-/r* 28.6%

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

       5. associate-*l* 28.6%

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

   11. Applied egg-rr 28.6%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}}{y}\\ \end{array} \]

Alternative 9: 62.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 200000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 200000000000.0) (/ x z) (* y (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 200000000000.0) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 200000000000.0d0) then
        tmp = x / z
    else
        tmp = y * (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 200000000000.0) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 200000000000.0:
		tmp = x / z
	else:
		tmp = y * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 200000000000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 200000000000.0)
		tmp = x / z;
	else
		tmp = y * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 200000000000.0], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e11

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 74.6%

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

   if 2e11 < y

   1. Initial program 93.5%

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

      1. associate-*l/ 91.8%

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

      2. times-frac 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-*r/ 90.3%

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

      5. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around 0 28.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 200000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 10: 62.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-6) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-6) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4d-6) then
        tmp = x / z
    else
        tmp = y / (z * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-6) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e-6:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e-6)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(z * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e-6)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e-6], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999982e-6

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 76.0%

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

   if 3.99999999999999982e-6 < y

   1. Initial program 93.9%

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

      1. associate-*r/ 94.0%

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

      2. associate-/l/ 90.9%

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

      3. *-commutative 90.9%

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

      4. times-frac 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in y around 0 18.5%

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

      1. *-commutative 18.5%

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

      2. clear-num 18.5%

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

      3. frac-times 27.8%

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

      4. *-un-lft-identity 27.8%

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

   6. Applied egg-rr 27.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 11: 58.5% 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 96.9%

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

   1. associate-*r/ 91.6%

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

   2. associate-/l/ 86.3%

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

   3. *-commutative 86.3%

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

   4. times-frac 83.1%

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

3. Simplified 83.1%

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

4. Taylor expanded in y around 0 49.2%

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

   1. *-commutative 49.2%

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

   2. clear-num 48.3%

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

   3. un-div-inv 48.3%

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

6. Applied egg-rr 48.3%

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

   1. associate-/r/ 53.7%

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

   2. associate-/r* 56.5%

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

   3. div-inv 56.5%

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

   4. associate-*l* 56.9%

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

   5. *-commutative 56.9%

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

8. Applied egg-rr 56.9%

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

   1. *-commutative 56.9%

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

   2. associate-*l/ 57.2%

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

   3. *-un-lft-identity 57.2%

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

   4. associate-/r/ 57.4%

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

   5. *-commutative 57.4%

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

   6. associate-*l/ 54.9%

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

   7. associate-/r* 54.0%

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

   8. clear-num 54.1%

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

   9. associate-/r/ 60.9%

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

   10. *-inverses 60.9%

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

   11. *-un-lft-identity 60.9%

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

10. Applied egg-rr 60.9%

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

11. Final simplification 60.9%

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

Alternative 12: 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 96.9%

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

2. Taylor expanded in y around 0 60.7%

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

3. Final simplification 60.7%

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

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 2023221 
(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))