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

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 6

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y, x, \sin y \cdot \left(-z\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(\cos y, x, \sin y \cdot \left(-z\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (cos(y) * x) - (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 = (cos(y) * x) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \cos y \cdot x - z \cdot \sin y \]
4. Add Preprocessing

Alternative 3: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-7} \lor \neg \left(z \leq 6.8 \cdot 10^{+155}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.2e-7) (not (<= z 6.8e+155)))
   (* (sin y) (- z))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.2e-7) || !(z <= 6.8e+155)) {
		tmp = sin(y) * -z;
	} else {
		tmp = cos(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 ((z <= (-4.2d-7)) .or. (.not. (z <= 6.8d+155))) then
        tmp = sin(y) * -z
    else
        tmp = cos(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.2e-7) || !(z <= 6.8e+155)) {
		tmp = Math.sin(y) * -z;
	} else {
		tmp = Math.cos(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.2e-7) or not (z <= 6.8e+155):
		tmp = math.sin(y) * -z
	else:
		tmp = math.cos(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.2e-7) || !(z <= 6.8e+155))
		tmp = Float64(sin(y) * Float64(-z));
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.2e-7) || ~((z <= 6.8e+155)))
		tmp = sin(y) * -z;
	else
		tmp = cos(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.2e-7], N[Not[LessEqual[z, 6.8e+155]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2e-7 or 6.8000000000000002e155 < z

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 78.5%

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

      1. mul-1-neg 78.5%

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

      2. *-commutative 78.5%

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

      3. distribute-rgt-neg-in 78.5%

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

   5. Simplified 78.5%

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

   if -4.2e-7 < z < 6.8000000000000002e155

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-7} \lor \neg \left(z \leq 6.8 \cdot 10^{+155}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00028 \lor \neg \left(y \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00028) (not (<= y 5e-9))) (* (cos y) x) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00028) || !(y <= 5e-9)) {
		tmp = cos(y) * x;
	} 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 <= (-0.00028d0)) .or. (.not. (y <= 5d-9))) then
        tmp = cos(y) * x
    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 <= -0.00028) || !(y <= 5e-9)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00028) or not (y <= 5e-9):
		tmp = math.cos(y) * x
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00028) || !(y <= 5e-9))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(x - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00028) || ~((y <= 5e-9)))
		tmp = cos(y) * x;
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00028], N[Not[LessEqual[y, 5e-9]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999998e-4 or 5.0000000000000001e-9 < y

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 53.7%

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

   if -2.7999999999999998e-4 < y < 5.0000000000000001e-9

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00028 \lor \neg \left(y \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.7% 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. Add Preprocessing

3. Taylor expanded in y around 0 56.0%

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

   1. mul-1-neg 56.0%

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

   2. unsub-neg 56.0%

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

5. Simplified 56.0%

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

6. Final simplification 56.0%

   \[\leadsto x - y \cdot z \]
7. Add Preprocessing

Alternative 6: 39.2% 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. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0 40.8%

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

6. Final simplification 40.8%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))