Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 99.7%

Time: 9.3s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -10000 \lor \neg \left(t_1 \leq 10^{-201}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (/ (* x t_0) z)))
   (if (or (<= t_1 -10000.0) (not (<= t_1 1e-201))) t_1 (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double t_1 = (x * t_0) / z;
	double tmp;
	if ((t_1 <= -10000.0) || !(t_1 <= 1e-201)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) / y
    t_1 = (x * t_0) / z
    if ((t_1 <= (-10000.0d0)) .or. (.not. (t_1 <= 1d-201))) then
        tmp = t_1
    else
        tmp = t_0 * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double t_1 = (x * t_0) / z;
	double tmp;
	if ((t_1 <= -10000.0) || !(t_1 <= 1e-201)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	t_1 = (x * t_0) / z
	tmp = 0
	if (t_1 <= -10000.0) or not (t_1 <= 1e-201):
		tmp = t_1
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if ((t_1 <= -10000.0) || !(t_1 <= 1e-201))
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	t_1 = (x * t_0) / z;
	tmp = 0.0;
	if ((t_1 <= -10000.0) || ~((t_1 <= 1e-201)))
		tmp = t_1;
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10000.0], N[Not[LessEqual[t$95$1, 1e-201]], $MachinePrecision]], t$95$1, N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1e4 or 9.99999999999999946e-202 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.8%

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

   if -1e4 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99999999999999946e-202

   1. Initial program 90.1%

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

      1. associate-/l* 92.7%

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

      2. associate-/r/ 83.8%

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

   3. Simplified 83.8%

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

      1. *-un-lft-identity 83.8%

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

      2. frac-times 81.2%

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

      3. clear-num 82.1%

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

      4. frac-times 88.2%

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

      5. *-commutative 88.2%

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

      6. times-frac 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -10000 \lor \neg \left(\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{-201}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-82} \lor \neg \left(z \leq 4.1 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.25e-82) (not (<= z 4.1e-51)))
   (* (/ (sin y) y) (/ x z))
   (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e-82) || !(z <= 4.1e-51)) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x * ((sin(y) / z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.25d-82)) .or. (.not. (z <= 4.1d-51))) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = x * ((sin(y) / z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.25e-82) || !(z <= 4.1e-51)) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = x * ((Math.sin(y) / z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.25e-82) or not (z <= 4.1e-51):
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = x * ((math.sin(y) / z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.25e-82) || !(z <= 4.1e-51))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(x * Float64(Float64(sin(y) / z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.25e-82) || ~((z <= 4.1e-51)))
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = x * ((sin(y) / z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.25e-82], N[Not[LessEqual[z, 4.1e-51]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2499999999999999e-82 or 4.09999999999999973e-51 < z

   1. Initial program 99.8%

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

      1. associate-/l* 91.8%

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

      2. associate-/r/ 81.3%

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

   3. Simplified 81.3%

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

      1. *-un-lft-identity 81.3%

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

      2. frac-times 83.3%

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

      3. clear-num 83.9%

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

      4. frac-times 87.3%

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

      5. *-commutative 87.3%

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

      6. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.2499999999999999e-82 < z < 4.09999999999999973e-51

   1. Initial program 87.6%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. associate-/r* 99.5%

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

      4. clear-num 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-82} \lor \neg \left(z \leq 4.1 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -500000000000 \lor \neg \left(z \leq 1.5 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -500000000000.0) (not (<= z 1.5e+21)))
   (* (/ (sin y) y) (/ x z))
   (/ x (* y (/ z (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -500000000000.0) || !(z <= 1.5e+21)) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x / (y * (z / sin(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-500000000000.0d0)) .or. (.not. (z <= 1.5d+21))) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = x / (y * (z / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -500000000000.0) || !(z <= 1.5e+21)) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = x / (y * (z / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -500000000000.0) or not (z <= 1.5e+21):
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = x / (y * (z / math.sin(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -500000000000.0) || !(z <= 1.5e+21))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(x / Float64(y * Float64(z / sin(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -500000000000.0) || ~((z <= 1.5e+21)))
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = x / (y * (z / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -500000000000.0], N[Not[LessEqual[z, 1.5e+21]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5e11 or 1.5e21 < z

   1. Initial program 99.8%

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

      1. associate-/l* 89.0%

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

      2. associate-/r/ 74.9%

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

   3. Simplified 74.9%

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

      1. *-un-lft-identity 74.9%

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

      2. frac-times 81.5%

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

      3. clear-num 82.3%

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

      4. frac-times 86.3%

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

      5. *-commutative 86.3%

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

      6. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -5e11 < z < 1.5e21

   1. Initial program 91.4%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -500000000000 \lor \neg \left(z \leq 1.5 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e+81)
   (/ (sin y) (/ y (/ x z)))
   (if (<= z 1.15e+22) (/ x (* y (/ z (sin y)))) (* (/ (sin y) y) (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+81) {
		tmp = sin(y) / (y / (x / z));
	} else if (z <= 1.15e+22) {
		tmp = x / (y * (z / sin(y)));
	} else {
		tmp = (sin(y) / y) * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.3d+81)) then
        tmp = sin(y) / (y / (x / z))
    else if (z <= 1.15d+22) then
        tmp = x / (y * (z / sin(y)))
    else
        tmp = (sin(y) / y) * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+81) {
		tmp = Math.sin(y) / (y / (x / z));
	} else if (z <= 1.15e+22) {
		tmp = x / (y * (z / Math.sin(y)));
	} else {
		tmp = (Math.sin(y) / y) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.3e+81:
		tmp = math.sin(y) / (y / (x / z))
	elif z <= 1.15e+22:
		tmp = x / (y * (z / math.sin(y)))
	else:
		tmp = (math.sin(y) / y) * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e+81)
		tmp = Float64(sin(y) / Float64(y / Float64(x / z)));
	elseif (z <= 1.15e+22)
		tmp = Float64(x / Float64(y * Float64(z / sin(y))));
	else
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.3e+81)
		tmp = sin(y) / (y / (x / z));
	elseif (z <= 1.15e+22)
		tmp = x / (y * (z / sin(y)));
	else
		tmp = (sin(y) / y) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e+81], N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+22], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e81

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

      2. times-frac 91.1%

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

      3. *-commutative 91.1%

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

      4. associate-*r/ 91.7%

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

      5. *-commutative 91.7%

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

   3. Simplified 91.7%

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

      1. clear-num 91.7%

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

      2. un-div-inv 91.7%

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

      3. associate-/l* 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -3.3e81 < z < 1.1500000000000001e22

   1. Initial program 92.0%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

   if 1.1500000000000001e22 < z

   1. Initial program 99.8%

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

      1. associate-/l* 85.1%

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

      2. associate-/r/ 64.2%

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

   3. Simplified 64.2%

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

      1. *-un-lft-identity 64.2%

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

      2. frac-times 72.5%

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

      3. clear-num 73.3%

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

      4. frac-times 82.0%

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

      5. *-commutative 82.0%

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

      6. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.42e-19) (/ x z) (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e-19) {
		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 <= 1.42d-19) 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 <= 1.42e-19) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.42e-19:
		tmp = x / z
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.42e-19)
		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 <= 1.42e-19)
		tmp = x / z;
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.42e-19], 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 < 1.42e-19

   1. Initial program 97.3%

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

      1. associate-*l/ 96.3%

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

      2. times-frac 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*r/ 79.5%

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

      5. *-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in y around 0 71.1%

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

   if 1.42e-19 < y

   1. Initial program 90.1%

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

      1. associate-*l/ 94.3%

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

      2. times-frac 91.5%

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

      3. *-commutative 91.5%

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

      4. associate-*r/ 91.5%

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

      5. *-commutative 91.5%

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

   3. Simplified 91.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 6: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-/l* 94.6%

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

   2. associate-/r/ 88.0%

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

3. Simplified 88.0%

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

   1. *-un-lft-identity 88.0%

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

   2. frac-times 83.3%

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

   3. clear-num 83.7%

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

   4. frac-times 84.8%

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

   5. *-commutative 84.8%

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

   6. times-frac 95.8%

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

5. Applied egg-rr 95.8%

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

6. Final simplification 95.8%

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

Alternative 7: 60.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 275000000:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 275000000.0)
   (* (/ x z) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (* (/ x (* y z)) (/ 6.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 275000000.0) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 275000000.0d0) then
        tmp = (x / z) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = (x / (y * z)) * (6.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 275000000.0) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = (x / (y * z)) * (6.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 275000000.0:
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = (x / (y * z)) * (6.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 275000000.0)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(Float64(x / Float64(y * z)) * Float64(6.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 275000000.0)
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = (x / (y * z)) * (6.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 275000000.0], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(6.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.75e8

   1. Initial program 97.4%

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

      1. associate-/l* 96.0%

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

      2. associate-/r/ 87.2%

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

   3. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. frac-times 81.5%

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

      3. clear-num 82.0%

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

      4. frac-times 83.0%

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

      5. *-commutative 83.0%

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

      6. times-frac 96.5%

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

   5. Applied egg-rr 96.5%

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

   6. Taylor expanded in y around 0 70.8%

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

      1. unpow2 70.8%

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

   8. Simplified 70.8%

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

   if 2.75e8 < y

   1. Initial program 89.1%

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

      1. associate-/l* 90.4%

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

      2. associate-/r/ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 32.6%

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

   5. Taylor expanded in y around inf 32.6%

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

      1. associate-*r/ 32.6%

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

      2. *-commutative 32.6%

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

      3. unpow2 32.6%

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

   7. Simplified 32.6%

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

      1. *-commutative 32.6%

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

      2. associate-*r* 32.6%

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

      3. times-frac 32.7%

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

      4. *-commutative 32.7%

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

   9. Applied egg-rr 32.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 275000000:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \end{array} \]

Alternative 8: 60.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 275000000:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \frac{x \cdot 6}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 275000000.0)
   (* (/ x z) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (* (/ 1.0 (* y z)) (/ (* x 6.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 275000000.0) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = (1.0 / (y * z)) * ((x * 6.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 275000000.0d0) then
        tmp = (x / z) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = (1.0d0 / (y * z)) * ((x * 6.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 275000000.0) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = (1.0 / (y * z)) * ((x * 6.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 275000000.0:
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = (1.0 / (y * z)) * ((x * 6.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 275000000.0)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(Float64(x * 6.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 275000000.0)
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = (1.0 / (y * z)) * ((x * 6.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 275000000.0], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(N[(x * 6.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.75e8

   1. Initial program 97.4%

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

      1. associate-/l* 96.0%

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

      2. associate-/r/ 87.2%

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

   3. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. frac-times 81.5%

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

      3. clear-num 82.0%

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

      4. frac-times 83.0%

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

      5. *-commutative 83.0%

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

      6. times-frac 96.5%

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

   5. Applied egg-rr 96.5%

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

   6. Taylor expanded in y around 0 70.8%

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

      1. unpow2 70.8%

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

   8. Simplified 70.8%

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

   if 2.75e8 < y

   1. Initial program 89.1%

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

      1. associate-/l* 90.4%

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

      2. associate-/r/ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 32.6%

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

   5. Taylor expanded in y around inf 32.6%

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

      1. associate-*r/ 32.6%

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

      2. *-commutative 32.6%

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

      3. unpow2 32.6%

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

   7. Simplified 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. associate-*r* 32.6%

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

      3. times-frac 32.7%

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

      4. *-commutative 32.7%

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

      5. *-commutative 32.7%

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

   9. Applied egg-rr 32.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 275000000:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \frac{x \cdot 6}{y}\\ \end{array} \]

Alternative 9: 62.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.75e+31) (/ x z) (* 6.0 (/ x (* z (* y y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e+31) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.75d+31) then
        tmp = x / z
    else
        tmp = 6.0d0 * (x / (z * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e+31) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.75e+31:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (z * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.75e+31)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(x / Float64(z * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.75e+31)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (z * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.75e+31], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.75e31

   1. Initial program 97.4%

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

      1. associate-*l/ 96.1%

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

      2. times-frac 83.5%

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

      3. *-commutative 83.5%

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

      4. associate-*r/ 80.8%

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

      5. *-commutative 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in y around 0 70.4%

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

   if 1.75e31 < y

   1. Initial program 87.9%

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

      1. associate-/l* 89.4%

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

      2. associate-/r/ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in y around 0 32.0%

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

   5. Taylor expanded in y around inf 32.0%

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

      1. *-commutative 32.0%

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

      2. unpow2 32.0%

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

   7. Simplified 32.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 10: 62.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 97.4%

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

      1. associate-*l/ 96.4%

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

      2. times-frac 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-*r/ 80.0%

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

      5. *-commutative 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 2.39999999999999991 < y

   1. Initial program 89.4%

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

      1. associate-/l* 90.7%

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

      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 33.2%

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

   5. Taylor expanded in y around inf 33.2%

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

      1. associate-*r/ 33.2%

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

      2. *-commutative 33.2%

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

      3. unpow2 33.2%

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

   7. Simplified 33.2%

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

      1. *-commutative 33.2%

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

      2. associate-*r* 33.2%

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

      3. times-frac 33.3%

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

      4. *-commutative 33.3%

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

   9. Applied egg-rr 33.3%

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

4. Final simplification 62.1%

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

Alternative 11: 66.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-/l* 94.6%

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

   2. associate-/r/ 88.0%

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

3. Simplified 88.0%

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

4. Taylor expanded in y around 0 60.2%

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

   1. *-commutative 60.2%

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

   2. distribute-rgt-in 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-*l* 60.2%

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

   5. *-commutative 60.2%

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

6. Applied egg-rr 60.2%

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

7. Taylor expanded in y around 0 66.9%

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

8. Final simplification 66.9%

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

Alternative 12: 59.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-9) (/ x z) (* (/ x y) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-9) {
		tmp = x / z;
	} else {
		tmp = (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 <= 7d-9) then
        tmp = x / z
    else
        tmp = (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 <= 7e-9) {
		tmp = x / z;
	} else {
		tmp = (x / y) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7e-9:
		tmp = x / z
	else:
		tmp = (x / y) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e-9)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / y) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7e-9)
		tmp = x / z;
	else
		tmp = (x / y) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e-9], N[(x / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.9999999999999998e-9

   1. Initial program 97.3%

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

      1. associate-*l/ 96.4%

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

      2. times-frac 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in y around 0 71.6%

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

   if 6.9999999999999998e-9 < y

   1. Initial program 89.7%

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

      1. associate-*r/ 89.6%

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

      2. associate-/l/ 91.1%

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

      3. *-commutative 91.1%

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

      4. times-frac 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in y around 0 25.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 13: 60.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-20) (/ x z) (/ x (/ (* y z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-20) {
		tmp = x / z;
	} else {
		tmp = x / ((y * z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d-20) then
        tmp = x / z
    else
        tmp = x / ((y * z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-20) {
		tmp = x / z;
	} else {
		tmp = x / ((y * z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e-20:
		tmp = x / z
	else:
		tmp = x / ((y * z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-20)
		tmp = Float64(x / z);
	else
		tmp = Float64(x / Float64(Float64(y * z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e-20)
		tmp = x / z;
	else
		tmp = x / ((y * z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e-20], N[(x / z), $MachinePrecision], N[(x / N[(N[(y * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999945e-21

   1. Initial program 97.3%

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

      1. associate-*l/ 96.3%

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

      2. times-frac 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*r/ 79.5%

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

      5. *-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in y around 0 71.1%

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

   if 9.99999999999999945e-21 < y

   1. Initial program 90.1%

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

      1. associate-/l* 91.3%

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

      2. associate-/r/ 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in y around 0 26.1%

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

      1. associate-*l/ 33.3%

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

      2. *-commutative 33.3%

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

   6. Applied egg-rr 33.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot z}{y}}\\ \end{array} \]

Alternative 14: 62.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-9) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-9) {
		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 <= 7d-9) 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 <= 7e-9) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7e-9:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e-9)
		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 <= 7e-9)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e-9], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.9999999999999998e-9

   1. Initial program 97.3%

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

      1. associate-*l/ 96.4%

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

      2. times-frac 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in y around 0 71.6%

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

   if 6.9999999999999998e-9 < y

   1. Initial program 89.7%

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

      1. associate-*r/ 89.6%

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

      2. associate-/l/ 91.1%

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

      3. *-commutative 91.1%

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

      4. times-frac 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in y around 0 25.7%

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

      1. *-commutative 25.7%

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

      2. clear-num 27.1%

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

      3. frac-times 34.0%

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

      4. *-un-lft-identity 34.0%

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

   6. Applied egg-rr 34.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 15: 62.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-14) (/ x z) (/ y (/ z (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-14) {
		tmp = x / z;
	} else {
		tmp = 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 <= 2d-14) then
        tmp = x / z
    else
        tmp = 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 <= 2e-14) {
		tmp = x / z;
	} else {
		tmp = y / (z / (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-14:
		tmp = x / z
	else:
		tmp = y / (z / (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-14)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(z / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-14)
		tmp = x / z;
	else
		tmp = y / (z / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-14], N[(x / z), $MachinePrecision], N[(y / N[(z / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-14

   1. Initial program 97.3%

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

      1. associate-*l/ 96.3%

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

      2. times-frac 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*r/ 79.5%

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

      5. *-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in y around 0 71.1%

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

   if 2e-14 < y

   1. Initial program 90.1%

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

      1. associate-*r/ 90.1%

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

      2. associate-/l/ 91.5%

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

      3. *-commutative 91.5%

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

      4. times-frac 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 28.9%

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

      1. associate-*l/ 31.0%

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

      2. associate-/l* 36.8%

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

   6. Applied egg-rr 36.8%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{y}}}\\ \end{array} \]

Alternative 16: 62.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-14) (/ x z) (/ y (/ (* y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-14) {
		tmp = x / z;
	} else {
		tmp = 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 <= 2d-14) then
        tmp = x / z
    else
        tmp = 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 <= 2e-14) {
		tmp = x / z;
	} else {
		tmp = y / ((y * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-14:
		tmp = x / z
	else:
		tmp = y / ((y * z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-14)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(Float64(y * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-14)
		tmp = x / z;
	else
		tmp = y / ((y * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-14], N[(x / z), $MachinePrecision], N[(y / N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-14

   1. Initial program 97.3%

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

      1. associate-*l/ 96.3%

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

      2. times-frac 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*r/ 79.5%

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

      5. *-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in y around 0 71.1%

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

   if 2e-14 < y

   1. Initial program 90.1%

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

      1. associate-*r/ 90.1%

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

      2. associate-/l/ 91.5%

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

      3. *-commutative 91.5%

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

      4. times-frac 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 28.9%

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

      1. frac-times 32.9%

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

      2. associate-/l* 36.7%

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

      3. *-commutative 36.7%

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

   6. Applied egg-rr 36.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \end{array} \]

Alternative 17: 58.3% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-/l* 94.6%

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

   2. associate-/r/ 88.0%

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

3. Simplified 88.0%

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

   1. clear-num 87.8%

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

   2. associate-/r/ 87.8%

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

   3. associate-/r* 88.3%

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

   4. clear-num 88.8%

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

5. Applied egg-rr 88.8%

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

6. Taylor expanded in y around 0 58.9%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot x \]
7. Step-by-step derivation

   1. associate-*l/ 59.1%

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

   2. *-un-lft-identity 59.1%

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

   3. clear-num 59.1%

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

8. Applied egg-rr 59.1%

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

9. Final simplification 59.1%

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

Alternative 18: 58.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. associate-*l/ 95.8%

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

   2. times-frac 84.8%

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

   3. *-commutative 84.8%

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

   4. associate-*r/ 82.7%

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

   5. *-commutative 82.7%

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

3. Simplified 82.7%

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

4. Taylor expanded in y around 0 59.1%

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

5. Final simplification 59.1%

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

Developer target: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023202 
(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))