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

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-define 99.9%

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

   7. sin-neg 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-z, \sin y, x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.4e-107)
   (fma (- z) (sin y) x)
   (if (<= z 6e-64) (* x (cos y)) (- x (* z (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.4e-107) {
		tmp = fma(-z, sin(y), x);
	} else if (z <= 6e-64) {
		tmp = x * cos(y);
	} else {
		tmp = x - (z * sin(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.4e-107)
		tmp = fma(Float64(-z), sin(y), x);
	elseif (z <= 6e-64)
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.4e-107], N[((-z) * N[Sin[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6e-64], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.40000000000000025e-107

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-define 99.9%

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

      7. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 86.4%

      \[\leadsto \mathsf{fma}\left(z, -\sin y, x \cdot \color{blue}{1}\right) \]

   6. Taylor expanded in z around 0 86.3%

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

      1. +-commutative 86.3%

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

      2. associate-*r* 86.3%

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

      3. neg-mul-1 86.3%

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

      4. fma-define 86.4%

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

   8. Simplified 86.4%

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

   if -6.40000000000000025e-107 < z < 6.0000000000000001e-64

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 93.0%

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

   if 6.0000000000000001e-64 < z

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. sin-neg 99.9%

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

      6. fma-define 99.9%

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

      7. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 88.5%

      \[\leadsto \mathsf{fma}\left(z, -\sin y, x \cdot \color{blue}{1}\right) \]

   6. Taylor expanded in z around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. associate-*r* 88.5%

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

      3. neg-mul-1 88.5%

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

   8. Simplified 88.5%

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

   9. Taylor expanded in z around 0 88.5%

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

       1. neg-mul-1 88.5%

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

       2. sub-neg 88.5%

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

   11. Simplified 88.5%

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

Alternative 3: 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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-104} \lor \neg \left(z \leq 2.9 \cdot 10^{-65}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e-104) (not (<= z 2.9e-65)))
   (- x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e-104) || !(z <= 2.9e-65)) {
		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 <= (-2.8d-104)) .or. (.not. (z <= 2.9d-65))) 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 <= -2.8e-104) || !(z <= 2.9e-65)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.8e-104) or not (z <= 2.9e-65):
		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 <= -2.8e-104) || !(z <= 2.9e-65))
		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 <= -2.8e-104) || ~((z <= 2.9e-65)))
		tmp = x - (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e-104], N[Not[LessEqual[z, 2.9e-65]], $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 < -2.8e-104 or 2.8999999999999998e-65 < z

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. sin-neg 99.9%

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

      6. fma-define 99.9%

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

      7. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 87.3%

      \[\leadsto \mathsf{fma}\left(z, -\sin y, x \cdot \color{blue}{1}\right) \]

   6. Taylor expanded in z around 0 87.2%

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

      1. +-commutative 87.2%

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

      2. associate-*r* 87.2%

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

      3. neg-mul-1 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in z around 0 87.2%

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

       1. neg-mul-1 87.2%

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

       2. sub-neg 87.2%

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

   11. Simplified 87.2%

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

   if -2.8e-104 < z < 2.8999999999999998e-65

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 93.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-104} \lor \neg \left(z \leq 2.9 \cdot 10^{-65}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-82} \lor \neg \left(x \leq 4.5 \cdot 10^{-132}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.8e-82) (not (<= x 4.5e-132)))
   (* x (cos y))
   (* z (- (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e-82) || !(x <= 4.5e-132)) {
		tmp = x * cos(y);
	} else {
		tmp = 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 ((x <= (-3.8d-82)) .or. (.not. (x <= 4.5d-132))) then
        tmp = x * cos(y)
    else
        tmp = z * -sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e-82) || !(x <= 4.5e-132)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = z * -Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.8e-82) or not (x <= 4.5e-132):
		tmp = x * math.cos(y)
	else:
		tmp = z * -math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.8e-82) || !(x <= 4.5e-132))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(z * Float64(-sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.8e-82) || ~((x <= 4.5e-132)))
		tmp = x * cos(y);
	else
		tmp = z * -sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.8e-82], N[Not[LessEqual[x, 4.5e-132]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e-82 or 4.4999999999999999e-132 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.2%

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

   if -3.8000000000000002e-82 < x < 4.4999999999999999e-132

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 82.0%

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

      1. neg-mul-1 82.0%

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

      2. *-commutative 82.0%

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

      3. distribute-rgt-neg-in 82.0%

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

   5. Simplified 82.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-82} \lor \neg \left(x \leq 4.5 \cdot 10^{-132}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00028 \lor \neg \left(y \leq 48\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + 0.16666666666666666 \cdot \left(z \cdot y\right)\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00028) (not (<= y 48.0)))
   (* x (cos y))
   (+ x (* y (- (* y (+ (* x -0.5) (* 0.16666666666666666 (* z y)))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00028) || !(y <= 48.0)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * 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 <= 48.0d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * ((y * ((x * (-0.5d0)) + (0.16666666666666666d0 * (z * 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 <= 48.0)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00028) or not (y <= 48.0):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00028) || !(y <= 48.0))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(0.16666666666666666 * Float64(z * y)))) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00028) || ~((y <= 48.0)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00028], N[Not[LessEqual[y, 48.0]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999998e-4 or 48 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 50.7%

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

   if -2.7999999999999998e-4 < y < 48

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.8%

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot \left(y \cdot z\right)\right) - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00028 \lor \neg \left(y \leq 48\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + 0.16666666666666666 \cdot \left(z \cdot y\right)\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.4% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e-94) x (if (<= x 1.2e-192) (* z (- y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-94) {
		tmp = x;
	} else if (x <= 1.2e-192) {
		tmp = z * -y;
	} 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 <= (-1.05d-94)) then
        tmp = x
    else if (x <= 1.2d-192) then
        tmp = z * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-94) {
		tmp = x;
	} else if (x <= 1.2e-192) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.05e-94:
		tmp = x
	elif x <= 1.2e-192:
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e-94)
		tmp = x;
	elseif (x <= 1.2e-192)
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.05e-94)
		tmp = x;
	elseif (x <= 1.2e-192)
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.05e-94], x, If[LessEqual[x, 1.2e-192], N[(z * (-y)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-94 or 1.2e-192 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 49.7%

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

   if -1.05e-94 < x < 1.2e-192

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 54.8%

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

      1. mul-1-neg 54.8%

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

      2. unsub-neg 54.8%

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

      3. *-commutative 54.8%

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

   5. Simplified 54.8%

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

   6. Taylor expanded in x around 0 45.7%

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

      1. neg-mul-1 45.7%

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

      2. distribute-rgt-neg-in 45.7%

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

   8. Simplified 45.7%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.3% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - N[(z * y), $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 55.6%

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

   1. mul-1-neg 55.6%

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

   2. unsub-neg 55.6%

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

   3. *-commutative 55.6%

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

5. Simplified 55.6%

   \[\leadsto \color{blue}{x - z \cdot y} \]
6. Add Preprocessing

Alternative 9: 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. Taylor expanded in y around 0 39.5%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))))