Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.4%

Time: 7.6s

Alternatives: 11

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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+82} \lor \neg \left(z \leq 10^{-109}\right):\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= z -3e+82) (not (<= z 1e-109)))
     (* t_0 (/ x z))
     (/ x (/ z t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((z <= -3e+82) || !(z <= 1e-109)) {
		tmp = t_0 * (x / z);
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if ((z <= (-3d+82)) .or. (.not. (z <= 1d-109))) then
        tmp = t_0 * (x / z)
    else
        tmp = x / (z / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if ((z <= -3e+82) || !(z <= 1e-109)) {
		tmp = t_0 * (x / z);
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if (z <= -3e+82) or not (z <= 1e-109):
		tmp = t_0 * (x / z)
	else:
		tmp = x / (z / t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((z <= -3e+82) || !(z <= 1e-109))
		tmp = Float64(t_0 * Float64(x / z));
	else
		tmp = Float64(x / Float64(z / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if ((z <= -3e+82) || ~((z <= 1e-109)))
		tmp = t_0 * (x / z);
	else
		tmp = x / (z / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[z, -3e+82], N[Not[LessEqual[z, 1e-109]], $MachinePrecision]], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999989e82 or 9.9999999999999999e-110 < z

   1. Initial program 98.6%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. associate-/l* 88.5%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.99999999999999989e82 < z < 9.9999999999999999e-110

   1. Initial program 95.6%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+82} \lor \neg \left(z \leq 10^{-109}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+16} \lor \neg \left(z \leq 5.2 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e+16) (not (<= z 5.2e-107)))
   (* (/ (sin y) y) (/ x z))
   (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+16) || !(z <= 5.2e-107)) {
		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 <= (-2d+16)) .or. (.not. (z <= 5.2d-107))) 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 <= -2e+16) || !(z <= 5.2e-107)) {
		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 <= -2e+16) or not (z <= 5.2e-107):
		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 <= -2e+16) || !(z <= 5.2e-107))
		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 <= -2e+16) || ~((z <= 5.2e-107)))
		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, -2e+16], N[Not[LessEqual[z, 5.2e-107]], $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 < -2e16 or 5.2000000000000001e-107 < z

   1. Initial program 98.8%

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

      1. clear-num 98.8%

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

      2. associate-/r/ 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. associate-/l* 90.3%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2e16 < z < 5.2000000000000001e-107

   1. Initial program 94.6%

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

      1. associate-*r/ 99.6%

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

      2. associate-/l/ 80.6%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+16} \lor \neg \left(z \leq 5.2 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 3: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e-20) (/ x z) (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-20) {
		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 <= 4.5d-20) 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 <= 4.5e-20) {
		tmp = x / z;
	} else {
		tmp = x * ((Math.sin(y) / z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e-20:
		tmp = x / z
	else:
		tmp = x * ((math.sin(y) / z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e-20)
		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 <= 4.5e-20)
		tmp = x / z;
	else
		tmp = x * ((sin(y) / z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e-20], 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 < 4.5000000000000001e-20

   1. Initial program 98.5%

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

      1. associate-*r/ 94.7%

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

      2. associate-/l/ 84.1%

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

      3. associate-/r* 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in y around 0 69.8%

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

   if 4.5000000000000001e-20 < y

   1. Initial program 93.6%

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

      1. associate-*r/ 92.2%

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

      2. associate-/l/ 91.8%

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

      3. associate-/r* 92.1%

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

   3. Simplified 92.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

2. Final simplification 97.1%

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

Alternative 5: 62.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{\frac{x}{y \cdot z}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-8) (/ x z) (/ y (/ 1.0 (/ x (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-8) {
		tmp = x / z;
	} else {
		tmp = y / (1.0 / (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 <= 2d-8) then
        tmp = x / z
    else
        tmp = y / (1.0d0 / (x / (y * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-8) {
		tmp = x / z;
	} else {
		tmp = y / (1.0 / (x / (y * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-8:
		tmp = x / z
	else:
		tmp = y / (1.0 / (x / (y * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(1.0 / Float64(x / Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-8)
		tmp = x / z;
	else
		tmp = y / (1.0 / (x / (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-8], N[(x / z), $MachinePrecision], N[(y / N[(1.0 / N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-8

   1. Initial program 98.5%

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

      1. associate-*r/ 94.8%

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

      2. associate-/l/ 84.3%

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

      3. associate-/r* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in y around 0 70.1%

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

   if 2e-8 < y

   1. Initial program 93.4%

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

      1. associate-*r/ 92.0%

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

      2. associate-/l/ 91.6%

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

      3. associate-/r* 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around 0 19.8%

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

      1. *-un-lft-identity 19.8%

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

      2. div-inv 19.8%

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

      3. *-inverses 19.8%

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

      4. times-frac 22.9%

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

      5. associate-/l* 33.8%

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

      6. *-commutative 33.8%

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

      7. *-un-lft-identity 33.8%

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

      8. times-frac 33.7%

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

      9. /-rgt-identity 33.7%

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

   6. Applied egg-rr 33.7%

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

      1. associate-*r/ 33.8%

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

      2. clear-num 33.8%

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

      3. *-commutative 33.8%

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

   8. Applied egg-rr 33.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{\frac{x}{y \cdot z}}}\\ \end{array} \]

Alternative 6: 60.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 100000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 100000.0) (/ x z) (* x (/ y (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 100000.0) {
		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 <= 100000.0d0) 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 <= 100000.0) {
		tmp = x / z;
	} else {
		tmp = x * (y / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 100000.0:
		tmp = x / z
	else:
		tmp = x * (y / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 100000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(x * Float64(y / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 100000.0)
		tmp = x / z;
	else
		tmp = x * (y / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 100000.0], N[(x / z), $MachinePrecision], N[(x * N[(y / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e5

   1. Initial program 98.5%

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

      1. associate-*r/ 94.9%

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

      2. associate-/l/ 84.7%

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

      3. associate-/r* 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in y around 0 70.1%

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

   if 1e5 < y

   1. Initial program 92.9%

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

      1. associate-*r/ 91.4%

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

      2. associate-/l/ 91.0%

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

      3. associate-/r* 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in y around 0 15.9%

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

      1. div-inv 15.9%

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

   6. Applied egg-rr 15.9%

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

      1. un-div-inv 15.9%

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

      2. associate-/l/ 21.6%

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

   8. Applied egg-rr 21.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 100000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y \cdot z}\\ \end{array} \]

Alternative 7: 62.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+18) (/ x z) (* y (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+18) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d+18) then
        tmp = x / z
    else
        tmp = y * (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+18) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e+18:
		tmp = x / z
	else:
		tmp = y * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+18)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+18)
		tmp = x / z;
	else
		tmp = y * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+18], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e18

   1. Initial program 98.5%

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

      1. associate-*r/ 94.9%

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

      2. associate-/l/ 84.7%

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

      3. associate-/r* 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in y around 0 70.1%

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

   if 1e18 < y

   1. Initial program 92.9%

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

      1. associate-*r/ 91.4%

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

      2. associate-/l/ 91.0%

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

      3. associate-/r* 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in y around 0 16.0%

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

      1. div-inv 16.0%

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

      2. clear-num 17.1%

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

   6. Applied egg-rr 17.1%

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

      1. clear-num 17.1%

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

      2. associate-/r/ 17.1%

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

      3. clear-num 16.0%

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

      4. rgt-mult-inverse 16.0%

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

      5. un-div-inv 16.0%

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

      6. times-frac 19.4%

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

      7. div-inv 19.4%

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

      8. *-commutative 19.4%

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

      9. associate-*r* 30.0%

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

      10. associate-*l/ 30.0%

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

      11. *-un-lft-identity 30.0%

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

      12. *-commutative 30.0%

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

   8. Applied egg-rr 30.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 8: 62.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e-8) (/ x z) (/ y (* y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-8) {
		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 <= 3.5d-8) 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 <= 3.5e-8) {
		tmp = x / z;
	} else {
		tmp = y / (y * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e-8:
		tmp = x / z
	else:
		tmp = y / (y * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(y * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e-8)
		tmp = x / z;
	else
		tmp = y / (y * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e-8], N[(x / z), $MachinePrecision], N[(y / N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.50000000000000024e-8

   1. Initial program 98.5%

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

      1. associate-*r/ 94.8%

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

      2. associate-/l/ 84.3%

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

      3. associate-/r* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in y around 0 70.1%

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

   if 3.50000000000000024e-8 < y

   1. Initial program 93.4%

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

      1. associate-*r/ 92.0%

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

      2. associate-/l/ 91.6%

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

      3. associate-/r* 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around 0 19.8%

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

      1. *-un-lft-identity 19.8%

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

      2. *-inverses 19.8%

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

      3. div-inv 19.8%

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

      4. clear-num 21.1%

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

      5. frac-times 33.5%

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

      6. *-rgt-identity 33.5%

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

   6. Applied egg-rr 33.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 9: 62.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-8) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-8) {
		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 <= 2d-8) 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 <= 2e-8) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-8:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-8)
		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 <= 2e-8)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-8], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e-8

   1. Initial program 98.5%

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

      1. associate-*r/ 94.8%

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

      2. associate-/l/ 84.3%

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

      3. associate-/r* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in y around 0 70.1%

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

   if 2e-8 < y

   1. Initial program 93.4%

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

      1. associate-*r/ 92.0%

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

      2. associate-/l/ 91.6%

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

      3. associate-/r* 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around 0 19.8%

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

      1. *-un-lft-identity 19.8%

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

      2. div-inv 19.8%

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

      3. *-inverses 19.8%

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

      4. times-frac 22.9%

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

      5. associate-/l* 33.8%

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

      6. *-commutative 33.8%

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

      7. *-un-lft-identity 33.8%

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

      8. times-frac 33.7%

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

      9. /-rgt-identity 33.7%

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

   6. Applied egg-rr 33.7%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 10: 58.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. associate-*r/ 94.0%

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

   2. associate-/l/ 86.3%

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

   3. associate-/r* 87.5%

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

3. Simplified 87.5%

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

4. Taylor expanded in y around 0 56.0%

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

   1. div-inv 56.1%

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

   2. clear-num 56.6%

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

6. Applied egg-rr 56.6%

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

7. Final simplification 56.6%

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

Alternative 11: 58.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. associate-*r/ 94.0%

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

   2. associate-/l/ 86.3%

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

   3. associate-/r* 87.5%

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

3. Simplified 87.5%

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

4. Taylor expanded in y around 0 56.1%

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

5. Final simplification 56.1%

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

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 2023336 
(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))