Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.5% → 99.5%

Time: 9.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-53} \lor \neg \left(z \leq 3.4 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e-53) (not (<= z 3.4e-73)))
   (* (/ (sin y) y) (/ x z))
   (/ x (* z (/ y (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e-53) || !(z <= 3.4e-73)) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x / (z * (y / sin(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2d-53)) .or. (.not. (z <= 3.4d-73))) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = x / (z * (y / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e-53) || !(z <= 3.4e-73)) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = x / (z * (y / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2e-53) or not (z <= 3.4e-73):
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = x / (z * (y / math.sin(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e-53) || !(z <= 3.4e-73))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(x / Float64(z * Float64(y / sin(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2e-53) || ~((z <= 3.4e-73)))
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = x / (z * (y / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e-53], N[Not[LessEqual[z, 3.4e-73]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000006e-53 or 3.40000000000000021e-73 < z

   1. Initial program 99.0%

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

      1. associate-*l/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   if -2.00000000000000006e-53 < z < 3.40000000000000021e-73

   1. Initial program 87.0%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.6%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-53} \lor \neg \left(z \leq 3.4 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-37} \lor \neg \left(z \leq 6.5 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-37) (not (<= z 6.5e-66)))
   (* (/ (sin y) y) (/ x z))
   (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-37) || !(z <= 6.5e-66)) {
		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 <= (-1d-37)) .or. (.not. (z <= 6.5d-66))) 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 <= -1e-37) || !(z <= 6.5e-66)) {
		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 <= -1e-37) or not (z <= 6.5e-66):
		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 <= -1e-37) || !(z <= 6.5e-66))
		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 <= -1e-37) || ~((z <= 6.5e-66)))
		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, -1e-37], N[Not[LessEqual[z, 6.5e-66]], $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 < -1.00000000000000007e-37 or 6.50000000000000024e-66 < z

   1. Initial program 99.0%

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

      1. associate-*l/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   if -1.00000000000000007e-37 < z < 6.50000000000000024e-66

   1. Initial program 87.6%

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

      1. associate-*r/ 99.5%

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

      2. associate-/l/ 86.4%

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

      3. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-37} \lor \neg \left(z \leq 6.5 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 3: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e-8) (/ x z) (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e-8) {
		tmp = 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 (y <= 3.2d-8) then
        tmp = 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 (y <= 3.2e-8) {
		tmp = x / z;
	} else {
		tmp = x * ((Math.sin(y) / z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.2e-8:
		tmp = x / z
	else:
		tmp = x * ((math.sin(y) / z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e-8)
		tmp = 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 (y <= 3.2e-8)
		tmp = x / z;
	else
		tmp = x * ((sin(y) / z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.2e-8], N[(x / z), $MachinePrecision], N[(x * N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000002e-8

   1. Initial program 95.5%

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

      1. associate-*r/ 97.8%

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

      2. associate-/l/ 88.7%

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

      3. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around 0 66.0%

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

   if 3.2000000000000002e-8 < y

   1. Initial program 90.1%

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

      1. associate-*r/ 92.6%

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

      2. associate-/l/ 91.1%

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

      3. associate-/r* 92.5%

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

   3. Simplified 92.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 10^{-8}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))) (if (<= x 1e-8) (* t_0 (/ x z)) (/ (* x t_0) z))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 1e-8) {
		tmp = t_0 * (x / z);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 1e-8) {
		tmp = t_0 * (x / z);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 1e-8:
		tmp = t_0 * (x / z)
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 1e-8)
		tmp = Float64(t_0 * Float64(x / z));
	else
		tmp = Float64(Float64(x * t_0) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 1e-8)
		tmp = t_0 * (x / z);
	else
		tmp = (x * t_0) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 1e-8], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-8

   1. Initial program 92.1%

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

      1. associate-*l/ 98.3%

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

      2. *-commutative 98.3%

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

   3. Simplified 98.3%

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

   if 1e-8 < x

   1. Initial program 99.8%

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

4. Final simplification 98.7%

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

Alternative 5: 63.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4) (/ x z) (* (/ x (* z (* y 0.16666666666666666))) (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / z;
	} else {
		tmp = (x / (z * (y * 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 <= 2.4d0) then
        tmp = x / z
    else
        tmp = (x / (z * (y * 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 <= 2.4) {
		tmp = x / z;
	} else {
		tmp = (x / (z * (y * 0.16666666666666666))) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4:
		tmp = x / z
	else:
		tmp = (x / (z * (y * 0.16666666666666666))) * (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / Float64(z * Float64(y * 0.16666666666666666))) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x / z;
	else
		tmp = (x / (z * (y * 0.16666666666666666))) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4], N[(x / z), $MachinePrecision], N[(N[(x / N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   1. Initial program 95.5%

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

      1. associate-*r/ 97.8%

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

      2. associate-/l/ 88.7%

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

      3. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around 0 66.0%

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

   if 2.39999999999999991 < y

   1. Initial program 90.1%

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

      1. associate-/l* 91.1%

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

   3. Simplified 91.1%

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

      1. associate-/r/ 91.1%

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

   5. Applied egg-rr 91.1%

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

   6. Taylor expanded in y around 0 28.8%

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

   7. Taylor expanded in y around inf 28.8%

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

      1. *-commutative 28.8%

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

      2. *-commutative 28.8%

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

      3. *-commutative 28.8%

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

      4. associate-*r* 28.8%

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

   9. Simplified 28.8%

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

       1. associate-/r* 30.1%

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

       2. div-inv 30.1%

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

       3. associate-*r* 30.1%

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

       4. *-commutative 30.1%

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

       5. associate-*l* 30.1%

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

   11. Applied egg-rr 30.1%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 6: 62.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{\frac{1}{y}}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5000000.0) (/ x z) (* y (* x (/ (/ 1.0 y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5000000.0) {
		tmp = x / z;
	} else {
		tmp = y * (x * ((1.0 / 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 <= 5000000.0d0) then
        tmp = x / z
    else
        tmp = y * (x * ((1.0d0 / y) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5000000.0) {
		tmp = x / z;
	} else {
		tmp = y * (x * ((1.0 / y) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5000000.0:
		tmp = x / z
	else:
		tmp = y * (x * ((1.0 / y) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5000000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x * Float64(Float64(1.0 / y) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5000000.0)
		tmp = x / z;
	else
		tmp = y * (x * ((1.0 / y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5000000.0], N[(x / z), $MachinePrecision], N[(y * N[(x * N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e6

   1. Initial program 95.5%

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

      1. associate-*r/ 97.8%

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

      2. associate-/l/ 88.8%

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

      3. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around 0 65.7%

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

   if 5e6 < y

   1. Initial program 90.0%

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

      1. associate-*r/ 92.5%

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

      2. associate-/l/ 91.0%

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

      3. associate-/r* 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in y around 0 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. *-commutative 17.3%

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

      3. associate-/l/ 21.1%

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

   6. Applied egg-rr 21.1%

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

      1. *-commutative 21.1%

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

      2. *-rgt-identity 21.1%

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

      3. associate-*l/ 19.1%

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

      4. *-commutative 19.1%

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

   8. Applied egg-rr 19.1%

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

      1. div-inv 19.1%

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

      2. associate-*l* 27.7%

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

      3. *-commutative 27.7%

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

      4. associate-/r* 27.1%

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

   10. Applied egg-rr 27.1%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{\frac{1}{y}}{z}\right)\\ \end{array} \]

Alternative 7: 62.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 200:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{y \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 200.0) (/ x z) (* y (* x (/ 1.0 (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 200.0) {
		tmp = x / z;
	} else {
		tmp = y * (x * (1.0 / (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 <= 200.0d0) then
        tmp = x / z
    else
        tmp = y * (x * (1.0d0 / (y * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 200.0) {
		tmp = x / z;
	} else {
		tmp = y * (x * (1.0 / (y * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 200.0:
		tmp = x / z
	else:
		tmp = y * (x * (1.0 / (y * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 200.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x * Float64(1.0 / Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 200.0)
		tmp = x / z;
	else
		tmp = y * (x * (1.0 / (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 200.0], N[(x / z), $MachinePrecision], N[(y * N[(x * N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 200

   1. Initial program 95.5%

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

      1. associate-*r/ 97.8%

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

      2. associate-/l/ 88.8%

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

      3. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around 0 65.7%

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

   if 200 < y

   1. Initial program 90.0%

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

      1. associate-*r/ 92.5%

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

      2. associate-/l/ 91.0%

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

      3. associate-/r* 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in y around 0 17.5%

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

      1. un-div-inv 17.5%

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

      2. clear-num 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. associate-/r/ 17.5%

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

      2. *-inverses 17.5%

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

      3. associate-/r* 21.1%

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

      4. div-inv 21.1%

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

      5. associate-*l* 27.7%

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

      6. *-commutative 27.7%

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

   8. Applied egg-rr 27.7%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 200:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{y \cdot z}\right)\\ \end{array} \]

Alternative 8: 63.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e-5) (/ x z) (* y (/ 1.0 (* z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e-5) {
		tmp = x / z;
	} else {
		tmp = y * (1.0 / (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 <= 1.95d-5) then
        tmp = x / z
    else
        tmp = y * (1.0d0 / (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e-5) {
		tmp = x / z;
	} else {
		tmp = y * (1.0 / (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.95e-5:
		tmp = x / z
	else:
		tmp = y * (1.0 / (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.95e-5)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(1.0 / Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.95e-5)
		tmp = x / z;
	else
		tmp = y * (1.0 / (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.95e-5], N[(x / z), $MachinePrecision], N[(y * N[(1.0 / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.95e-5

   1. Initial program 95.5%

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

      1. associate-*r/ 97.8%

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

      2. associate-/l/ 88.7%

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

      3. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around 0 66.0%

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

   if 1.95e-5 < y

   1. Initial program 90.1%

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

      1. associate-*r/ 92.6%

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

      2. associate-/l/ 91.1%

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

      3. associate-/r* 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in y around 0 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. *-commutative 17.3%

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

      3. associate-/l/ 21.0%

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

   6. Applied egg-rr 21.0%

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

      1. *-commutative 21.0%

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

      2. *-rgt-identity 21.0%

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

      3. associate-*l/ 19.0%

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

      4. *-commutative 19.0%

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

   8. Applied egg-rr 19.0%

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

      1. associate-/l* 29.7%

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

      2. div-inv 29.7%

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

      3. *-un-lft-identity 29.7%

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

      4. times-frac 29.5%

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

      5. /-rgt-identity 29.5%

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

   10. Applied egg-rr 29.5%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 9: 66.9% 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 94.0%

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

   1. associate-/l* 95.5%

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

3. Simplified 95.5%

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

   1. associate-/r/ 89.5%

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

5. Applied egg-rr 89.5%

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

6. Taylor expanded in y around 0 52.6%

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

   1. *-commutative 52.6%

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

   2. +-commutative 52.6%

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

   3. distribute-rgt-in 52.6%

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

   4. *-commutative 52.6%

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

   5. associate-*r/ 52.2%

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

   6. *-commutative 52.2%

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

   7. associate-/l* 58.6%

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

   8. *-inverses 58.6%

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

   9. *-commutative 58.6%

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

   10. associate-*l* 58.6%

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

8. Applied egg-rr 58.6%

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

9. Final simplification 58.6%

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

Alternative 10: 60.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+105) (/ x z) (* (/ x y) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+105) {
		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 <= 3d+105) 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 <= 3e+105) {
		tmp = x / z;
	} else {
		tmp = (x / y) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+105:
		tmp = x / z
	else:
		tmp = (x / y) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+105)
		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 <= 3e+105)
		tmp = x / z;
	else
		tmp = (x / y) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+105], N[(x / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e105

   1. Initial program 94.8%

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

      1. associate-*r/ 98.1%

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

      2. associate-/l/ 90.3%

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

      3. associate-/r* 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in y around 0 57.5%

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

   if 3.0000000000000001e105 < y

   1. Initial program 89.4%

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

      1. associate-*r/ 86.6%

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

      2. associate-/l/ 84.1%

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

      3. associate-/r* 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in y around 0 23.4%

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

      1. *-un-lft-identity 23.4%

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

      2. *-commutative 23.4%

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

      3. associate-/l/ 29.9%

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

   6. Applied egg-rr 29.9%

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

      1. *-commutative 29.9%

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

      2. *-rgt-identity 29.9%

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

      3. associate-*l/ 26.3%

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

      4. *-commutative 26.3%

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

   8. Applied egg-rr 26.3%

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

      1. times-frac 30.0%

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

   10. Applied egg-rr 30.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 11: 60.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.75e+105) (/ x z) (/ (/ x y) (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.75e+105) {
		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 <= 3.75d+105) 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 <= 3.75e+105) {
		tmp = x / z;
	} else {
		tmp = (x / y) / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.75e+105:
		tmp = x / z
	else:
		tmp = (x / y) / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.75e+105)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / y) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.75e+105)
		tmp = x / z;
	else
		tmp = (x / y) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.75e+105], N[(x / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7500000000000001e105

   1. Initial program 94.8%

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

      1. associate-*r/ 98.1%

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

      2. associate-/l/ 90.3%

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

      3. associate-/r* 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in y around 0 57.5%

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

   if 3.7500000000000001e105 < y

   1. Initial program 89.4%

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

      1. associate-*r/ 86.6%

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

      2. associate-/l/ 84.1%

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

      3. associate-/r* 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in y around 0 23.4%

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

      1. *-un-lft-identity 23.4%

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

      2. *-commutative 23.4%

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

      3. associate-/l/ 29.9%

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

   6. Applied egg-rr 29.9%

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

      1. *-rgt-identity 29.9%

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

      2. clear-num 29.9%

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

      3. un-div-inv 29.9%

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

      4. associate-*r/ 23.4%

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

      5. associate-/r* 30.0%

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

   8. Applied egg-rr 30.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{z}{y}}\\ \end{array} \]

Alternative 12: 59.0% 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 94.0%

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

   1. associate-*r/ 96.3%

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

   2. associate-/l/ 89.4%

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

   3. associate-/r* 91.0%

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

3. Simplified 91.0%

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

4. Taylor expanded in y around 0 52.2%

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

   1. un-div-inv 52.4%

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

   2. clear-num 52.4%

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

6. Applied egg-rr 52.4%

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

7. Final simplification 52.4%

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

Alternative 13: 59.2% 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 94.0%

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

   1. associate-*r/ 96.3%

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

   2. associate-/l/ 89.4%

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

   3. associate-/r* 91.0%

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

3. Simplified 91.0%

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

4. Taylor expanded in y around 0 52.4%

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

5. Final simplification 52.4%

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

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 2023311 
(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))