Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))

C

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

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 * cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))

C

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

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 * cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))

C

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

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 * cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Final simplification 99.8%

   \[\leadsto x \cdot \cos y - z \cdot \sin y \]

Alternative 2: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -122 \lor \neg \left(z \leq 8.8 \cdot 10^{-32}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -122.0) (not (<= z 8.8e-32)))
   (- x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -122.0) || !(z <= 8.8e-32)) {
		tmp = x - (z * sin(y));
	} else {
		tmp = x * cos(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 <= (-122.0d0)) .or. (.not. (z <= 8.8d-32))) then
        tmp = x - (z * sin(y))
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -122.0) || !(z <= 8.8e-32)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -122.0) or not (z <= 8.8e-32):
		tmp = x - (z * math.sin(y))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -122.0) || !(z <= 8.8e-32))
		tmp = Float64(x - Float64(z * sin(y)));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -122.0) || ~((z <= 8.8e-32)))
		tmp = x - (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -122.0], N[Not[LessEqual[z, 8.8e-32]], $MachinePrecision]], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -122 or 8.7999999999999999e-32 < z

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0 90.5%

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

   if -122 < z < 8.7999999999999999e-32

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around inf 87.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -122 \lor \neg \left(z \leq 8.8 \cdot 10^{-32}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+120} \lor \neg \left(z \leq 4.5 \cdot 10^{+23}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.5e+120) (not (<= z 4.5e+23)))
   (* (sin y) (- z))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.5e+120) || !(z <= 4.5e+23)) {
		tmp = sin(y) * -z;
	} else {
		tmp = x * cos(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 <= (-3.5d+120)) .or. (.not. (z <= 4.5d+23))) then
        tmp = sin(y) * -z
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.5e+120) || !(z <= 4.5e+23)) {
		tmp = Math.sin(y) * -z;
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.5e+120) or not (z <= 4.5e+23):
		tmp = math.sin(y) * -z
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.5e+120) || !(z <= 4.5e+23))
		tmp = Float64(sin(y) * Float64(-z));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.5e+120) || ~((z <= 4.5e+23)))
		tmp = sin(y) * -z;
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.5e+120], N[Not[LessEqual[z, 4.5e+23]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000007e120 or 4.49999999999999979e23 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around 0 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

   4. Simplified 73.9%

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

   if -3.50000000000000007e120 < z < 4.49999999999999979e23

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around inf 80.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+120} \lor \neg \left(z \leq 4.5 \cdot 10^{+23}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4: 73.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-6} \lor \neg \left(y \leq 4.1 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.5e-6) (not (<= y 4.1e+18))) (* x (cos y)) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e-6) || !(y <= 4.1e+18)) {
		tmp = x * cos(y);
	} else {
		tmp = 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 <= (-4.5d-6)) .or. (.not. (y <= 4.1d+18))) then
        tmp = x * cos(y)
    else
        tmp = x - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e-6) || !(y <= 4.1e+18)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.5e-6) or not (y <= 4.1e+18):
		tmp = x * math.cos(y)
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.5e-6) || !(y <= 4.1e+18))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.5e-6) || ~((y <= 4.1e+18)))
		tmp = x * cos(y);
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.5e-6], N[Not[LessEqual[y, 4.1e+18]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000011e-6 or 4.1e18 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around inf 50.0%

      \[\leadsto \color{blue}{x \cdot \cos y} \]

   if -4.50000000000000011e-6 < y < 4.1e18

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

   4. Simplified 98.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-6} \lor \neg \left(y \leq 4.1 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]

Alternative 5: 42.2% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-205}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8e-181) x (if (<= x 3.1e-205) (* y (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-181) {
		tmp = x;
	} else if (x <= 3.1e-205) {
		tmp = y * -z;
	} else {
		tmp = 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 (x <= (-5.8d-181)) then
        tmp = x
    else if (x <= 3.1d-205) then
        tmp = y * -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-181) {
		tmp = x;
	} else if (x <= 3.1e-205) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8e-181:
		tmp = x
	elif x <= 3.1e-205:
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.8e-181)
		tmp = x;
	elseif (x <= 3.1e-205)
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.8e-181)
		tmp = x;
	elseif (x <= 3.1e-205)
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8e-181], x, If[LessEqual[x, 3.1e-205], N[(y * (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999996e-181 or 3.09999999999999983e-205 < x

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, -z \cdot \sin y\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{z \cdot \left(-\sin y\right)}\right) \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, z \cdot \left(-\sin y\right)\right)} \]

   4. Taylor expanded in y around 0 48.2%

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

   if -5.7999999999999996e-181 < x < 3.09999999999999983e-205

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. *-commutative 84.7%

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

      3. distribute-rgt-neg-in 84.7%

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

   4. Simplified 84.7%

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

   5. Taylor expanded in y around 0 38.3%

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

      1. mul-1-neg 38.3%

         \[\leadsto \color{blue}{-y \cdot z} \]

      2. *-commutative 38.3%

         \[\leadsto -\color{blue}{z \cdot y} \]

      3. distribute-rgt-neg-in 38.3%

         \[\leadsto \color{blue}{z \cdot \left(-y\right)} \]

   7. Simplified 38.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-205}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 52.4% accurate, 41.4× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- x (* y z)))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x - (y * z)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Taylor expanded in y around 0 53.7%

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

   1. mul-1-neg 53.7%

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

   2. unsub-neg 53.7%

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

4. Simplified 53.7%

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

5. Final simplification 53.7%

   \[\leadsto x - y \cdot z \]

Alternative 7: 38.9% accurate, 207.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, -z \cdot \sin y\right)} \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{z \cdot \left(-\sin y\right)}\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, z \cdot \left(-\sin y\right)\right)} \]

4. Taylor expanded in y around 0 42.2%

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

5. Final simplification 42.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023309 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))