Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.2%

Time: 10.0s

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+45} \lor \neg \left(z \leq 5 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.9e+45) (not (<= z 5e-102)))
   (* (/ (sin y) y) (/ x z))
   (/ x (* y (/ z (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.9e+45) || !(z <= 5e-102)) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x / (y * (z / sin(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-4.9d+45)) .or. (.not. (z <= 5d-102))) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = x / (y * (z / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.9e+45) || !(z <= 5e-102)) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = x / (y * (z / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.9e+45) or not (z <= 5e-102):
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = x / (y * (z / math.sin(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.9e+45) || !(z <= 5e-102))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(x / Float64(y * Float64(z / sin(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.9e+45) || ~((z <= 5e-102)))
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = x / (y * (z / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.9e+45], N[Not[LessEqual[z, 5e-102]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9000000000000002e45 or 5.00000000000000026e-102 < z

   1. Initial program 99.2%

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

      1. associate-*l/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -4.9000000000000002e45 < z < 5.00000000000000026e-102

   1. Initial program 90.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/r/ 91.3%

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

      2. *-commutative 91.3%

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

      3. div-inv 91.2%

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

      4. associate-*l* 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. *-commutative 82.7%

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

      2. *-commutative 82.7%

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

      3. associate-*l/ 81.4%

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

      4. div-inv 81.5%

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

      5. associate-/r/ 87.7%

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

      6. associate-/l/ 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+45} \lor \neg \left(z \leq 5 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+45} \lor \neg \left(z \leq 1.2 \cdot 10^{+20}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.2e+45) (not (<= z 1.2e+20)))
   (* (sin y) (/ (/ x z) y))
   (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2e+45) || !(z <= 1.2e+20)) {
		tmp = sin(y) * ((x / z) / y);
	} 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 <= (-6.2d+45)) .or. (.not. (z <= 1.2d+20))) then
        tmp = sin(y) * ((x / z) / y)
    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 <= -6.2e+45) || !(z <= 1.2e+20)) {
		tmp = Math.sin(y) * ((x / z) / y);
	} else {
		tmp = x * ((Math.sin(y) / z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.2e+45) or not (z <= 1.2e+20):
		tmp = math.sin(y) * ((x / z) / y)
	else:
		tmp = x * ((math.sin(y) / z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.2e+45) || !(z <= 1.2e+20))
		tmp = Float64(sin(y) * Float64(Float64(x / z) / y));
	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 <= -6.2e+45) || ~((z <= 1.2e+20)))
		tmp = sin(y) * ((x / z) / y);
	else
		tmp = x * ((sin(y) / z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.2e+45], N[Not[LessEqual[z, 1.2e+20]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $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 < -6.19999999999999975e45 or 1.2e20 < z

   1. Initial program 99.9%

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

      1. associate-*r/ 92.7%

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

      2. associate-/l/ 92.7%

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

      3. associate-/r* 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in x around 0 90.0%

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

      1. associate-*l/ 89.9%

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

      2. associate-/l/ 97.1%

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

      3. *-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -6.19999999999999975e45 < z < 1.2e20

   1. Initial program 91.9%

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

      1. associate-*r/ 99.6%

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

      2. associate-/l/ 91.5%

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

      3. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+45} \lor \neg \left(z \leq 1.2 \cdot 10^{+20}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-144} \lor \neg \left(z \leq 2.6 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e-144) (not (<= z 2.6e-102)))
   (* (/ (sin y) y) (/ x z))
   (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e-144) || !(z <= 2.6e-102)) {
		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 <= (-1.5d-144)) .or. (.not. (z <= 2.6d-102))) 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 <= -1.5e-144) || !(z <= 2.6e-102)) {
		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 <= -1.5e-144) or not (z <= 2.6e-102):
		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 <= -1.5e-144) || !(z <= 2.6e-102))
		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 <= -1.5e-144) || ~((z <= 2.6e-102)))
		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, -1.5e-144], N[Not[LessEqual[z, 2.6e-102]], $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 < -1.4999999999999999e-144 or 2.59999999999999986e-102 < z

   1. Initial program 98.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 -1.4999999999999999e-144 < z < 2.59999999999999986e-102

   1. Initial program 87.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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* 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-144} \lor \neg \left(z \leq 2.6 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e-11) (/ x z) (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-11) {
		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 <= 2.5d-11) 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 <= 2.5e-11) {
		tmp = x / z;
	} else {
		tmp = x * ((Math.sin(y) / z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5e-11:
		tmp = x / z
	else:
		tmp = x * ((math.sin(y) / z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e-11)
		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 <= 2.5e-11)
		tmp = x / z;
	else
		tmp = x * ((sin(y) / z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e-11], 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 < 2.50000000000000009e-11

   1. Initial program 95.8%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 91.8%

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

      3. associate-/r* 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 2.50000000000000009e-11 < y

   1. Initial program 93.9%

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

      1. associate-*r/ 92.5%

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

      2. associate-/l/ 92.5%

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

      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 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 5.5e-71) (/ x (/ z t_0)) (/ (* x t_0) z))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 5.5e-71) {
		tmp = x / (z / t_0);
	} 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 <= 5.5d-71) then
        tmp = x / (z / t_0)
    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 <= 5.5e-71) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 5.5e-71:
		tmp = x / (z / t_0)
	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 <= 5.5e-71)
		tmp = Float64(x / Float64(z / t_0));
	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 <= 5.5e-71)
		tmp = x / (z / t_0);
	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, 5.5e-71], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999997e-71

   1. Initial program 93.1%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   if 5.4999999999999997e-71 < x

   1. Initial program 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq 7 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= z 7e-102) (/ x (/ z t_0)) (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z <= 7e-102) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= 7d-102) then
        tmp = x / (z / t_0)
    else
        tmp = t_0 * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= 7e-102) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= 7e-102:
		tmp = x / (z / t_0)
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= 7e-102)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= 7e-102)
		tmp = x / (z / t_0);
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, 7e-102], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 6.99999999999999973e-102

   1. Initial program 93.5%

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

      1. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   if 6.99999999999999973e-102 < z

   1. Initial program 98.7%

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

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7: 62.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{z} \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e-11) (/ x z) (* y (* (/ 1.0 z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-11) {
		tmp = x / z;
	} else {
		tmp = y * ((1.0 / z) * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d-11) then
        tmp = x / z
    else
        tmp = y * ((1.0d0 / z) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-11) {
		tmp = x / z;
	} else {
		tmp = y * ((1.0 / z) * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5e-11:
		tmp = x / z
	else:
		tmp = y * ((1.0 / z) * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e-11)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(Float64(1.0 / z) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5e-11)
		tmp = x / z;
	else
		tmp = y * ((1.0 / z) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e-11], N[(x / z), $MachinePrecision], N[(y * N[(N[(1.0 / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.50000000000000009e-11

   1. Initial program 95.8%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 91.8%

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

      3. associate-/r* 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 2.50000000000000009e-11 < y

   1. Initial program 93.9%

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

      1. associate-*r/ 92.5%

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

      2. associate-/l/ 92.5%

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

      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 15.7%

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

      1. *-un-lft-identity 15.7%

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

      2. *-inverses 15.7%

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

      3. associate-*l/ 15.6%

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

      4. *-commutative 15.6%

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

      5. associate-*l* 15.4%

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

      6. div-inv 15.4%

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

      7. associate-/l* 23.6%

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

   6. Applied egg-rr 23.6%

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

      1. div-inv 23.6%

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

      2. div-inv 23.6%

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

      3. clear-num 22.5%

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

      4. associate-*l* 36.0%

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

   8. Applied egg-rr 36.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{z} \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 8: 60.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+21) (/ x z) (* x (/ y (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+21) {
		tmp = x / z;
	} else {
		tmp = x * (y / (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d+21) then
        tmp = x / z
    else
        tmp = x * (y / (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+21) {
		tmp = x / z;
	} else {
		tmp = x * (y / (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e+21:
		tmp = x / z
	else:
		tmp = x * (y / (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+21)
		tmp = Float64(x / z);
	else
		tmp = Float64(x * Float64(y / Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+21)
		tmp = x / z;
	else
		tmp = x * (y / (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+21], N[(x / z), $MachinePrecision], N[(x * N[(y / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e21

   1. Initial program 95.9%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 92.0%

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

      3. associate-/r* 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in y around 0 71.3%

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

   if 1e21 < y

   1. Initial program 93.5%

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

      1. associate-*r/ 92.1%

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

      2. associate-/l/ 92.1%

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

      3. associate-/r* 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in y around 0 12.6%

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

      1. *-commutative 12.6%

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

      2. associate-/l/ 21.8%

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

      3. associate-*l/ 21.5%

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

   6. Applied egg-rr 21.5%

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

      1. associate-/l* 34.4%

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

      2. associate-/r/ 21.8%

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

   8. Applied egg-rr 21.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \]

Alternative 9: 62.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e-11) (/ x z) (/ y (* z (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-11) {
		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 <= 2.5d-11) 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 <= 2.5e-11) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5e-11:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e-11)
		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 <= 2.5e-11)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e-11], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.50000000000000009e-11

   1. Initial program 95.8%

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

      1. associate-*r/ 98.6%

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

      2. associate-/l/ 91.8%

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

      3. associate-/r* 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in y around 0 71.7%

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

   if 2.50000000000000009e-11 < y

   1. Initial program 93.9%

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

      1. associate-*r/ 92.5%

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

      2. associate-/l/ 92.5%

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

      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 15.7%

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

      1. *-un-lft-identity 15.7%

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

      2. *-inverses 15.7%

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

      3. un-div-inv 15.7%

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

      4. times-frac 23.8%

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

      5. associate-/l* 35.9%

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

      6. *-commutative 35.9%

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

      7. *-un-lft-identity 35.9%

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

      8. times-frac 36.0%

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

      9. /-rgt-identity 36.0%

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

   6. Applied egg-rr 36.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 10: 58.3% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. associate-*r/ 96.7%

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

   2. associate-/l/ 92.0%

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

   3. associate-/r* 88.9%

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

3. Simplified 88.9%

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

   1. associate-/l/ 92.0%

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

   2. associate-*r/ 88.6%

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

   3. *-commutative 88.6%

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

   4. frac-times 95.8%

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

   5. clear-num 95.6%

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

   6. un-div-inv 95.9%

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

5. Applied egg-rr 95.9%

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

6. Taylor expanded in y around 0 54.2%

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

7. Final simplification 54.2%

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

Alternative 11: 58.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. associate-*r/ 96.7%

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

   2. associate-/l/ 92.0%

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

   3. associate-/r* 88.9%

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

3. Simplified 88.9%

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

4. Taylor expanded in y around 0 54.2%

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

5. Final simplification 54.2%

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

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 2023320 
(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))