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

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 82.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+40} \lor \neg \left(x \leq 6 \cdot 10^{+111}\right):\\ \;\;\;\;x \cdot \cos y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.6e+40) (not (<= x 6e+111)))
   (- (* x (cos y)) (* y z))
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e+40) || !(x <= 6e+111)) {
		tmp = (x * cos(y)) - (y * z);
	} else {
		tmp = x - (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 <= (-2.6d+40)) .or. (.not. (x <= 6d+111))) then
        tmp = (x * cos(y)) - (y * z)
    else
        tmp = x - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e+40) || !(x <= 6e+111)) {
		tmp = (x * Math.cos(y)) - (y * z);
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.6e+40) or not (x <= 6e+111):
		tmp = (x * math.cos(y)) - (y * z)
	else:
		tmp = x - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.6e+40) || !(x <= 6e+111))
		tmp = Float64(Float64(x * cos(y)) - Float64(y * z));
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.6e+40) || ~((x <= 6e+111)))
		tmp = (x * cos(y)) - (y * z);
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.6e+40], N[Not[LessEqual[x, 6e+111]], $MachinePrecision]], N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001e40 or 6e111 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 81.2%

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

   if -2.6000000000000001e40 < x < 6e111

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 88.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+40} \lor \neg \left(x \leq 6 \cdot 10^{+111}\right):\\ \;\;\;\;x \cdot \cos y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 3: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+16} \lor \neg \left(x \leq 4.8 \cdot 10^{+53}\right):\\ \;\;\;\;x \cdot \left(\cos y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+16) (not (<= x 4.8e+53)))
   (- (* x (+ (cos y) 1.0)) x)
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+16) || !(x <= 4.8e+53)) {
		tmp = (x * (cos(y) + 1.0)) - x;
	} else {
		tmp = x - (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 <= (-2.4d+16)) .or. (.not. (x <= 4.8d+53))) then
        tmp = (x * (cos(y) + 1.0d0)) - x
    else
        tmp = x - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+16) || !(x <= 4.8e+53)) {
		tmp = (x * (Math.cos(y) + 1.0)) - x;
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.4e+16) or not (x <= 4.8e+53):
		tmp = (x * (math.cos(y) + 1.0)) - x
	else:
		tmp = x - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+16) || !(x <= 4.8e+53))
		tmp = Float64(Float64(x * Float64(cos(y) + 1.0)) - x);
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.4e+16) || ~((x <= 4.8e+53)))
		tmp = (x * (cos(y) + 1.0)) - x;
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+16], N[Not[LessEqual[x, 4.8e+53]], $MachinePrecision]], N[(N[(x * N[(N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e16 or 4.8e53 < x

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)} - z \cdot \sin y \]

      2. expm1-udef 99.7%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)} - z \cdot \sin y \]

      3. log1p-udef 99.6%

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

      4. add-exp-log 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 78.4%

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

      1. sub-neg 78.4%

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

      2. distribute-rgt-in 78.3%

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

      3. metadata-eval 78.3%

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in z around 0 89.9%

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

   if -2.4e16 < x < 4.8e53

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+16} \lor \neg \left(x \leq 4.8 \cdot 10^{+53}\right):\\ \;\;\;\;x \cdot \left(\cos y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -450 \lor \neg \left(y \leq 0.0034\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -450.0) (not (<= y 0.0034)))
   (* z (- (sin y)))
   (- (* x (+ 1.0 (* -0.5 (* y y)))) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -450.0) || !(y <= 0.0034)) {
		tmp = z * -sin(y);
	} else {
		tmp = (x * (1.0 + (-0.5 * (y * y)))) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-450.0d0)) .or. (.not. (y <= 0.0034d0))) then
        tmp = z * -sin(y)
    else
        tmp = (x * (1.0d0 + ((-0.5d0) * (y * y)))) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -450.0) || !(y <= 0.0034)) {
		tmp = z * -Math.sin(y);
	} else {
		tmp = (x * (1.0 + (-0.5 * (y * y)))) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -450.0) or not (y <= 0.0034):
		tmp = z * -math.sin(y)
	else:
		tmp = (x * (1.0 + (-0.5 * (y * y)))) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -450.0) || !(y <= 0.0034))
		tmp = Float64(z * Float64(-sin(y)));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.5 * Float64(y * y)))) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -450.0) || ~((y <= 0.0034)))
		tmp = z * -sin(y);
	else
		tmp = (x * (1.0 + (-0.5 * (y * y)))) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -450.0], N[Not[LessEqual[y, 0.0034]], $MachinePrecision]], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], N[(N[(x * N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -450 or 0.00339999999999999981 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 17.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - z \cdot \sin y \]
   3. Step-by-step derivation

      1. unpow2 17.9%

         \[\leadsto x \cdot \left(1 + -0.5 \cdot \color{blue}{\left(y \cdot y\right)}\right) - z \cdot \sin y \]

   4. Simplified 17.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot \left(y \cdot y\right)\right)} - z \cdot \sin y \]

   5. Taylor expanded in x around 0 41.8%

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

      1. mul-1-neg 41.8%

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

      2. distribute-rgt-neg-out 41.8%

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

   7. Simplified 41.8%

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

   if -450 < y < 0.00339999999999999981

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.4%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - z \cdot \sin y \]
   3. Step-by-step derivation

      1. unpow2 99.4%

         \[\leadsto x \cdot \left(1 + -0.5 \cdot \color{blue}{\left(y \cdot y\right)}\right) - z \cdot \sin y \]

   4. Simplified 99.4%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot \left(y \cdot y\right)\right)} - z \cdot \sin y \]

   5. Taylor expanded in y around 0 99.3%

      \[\leadsto x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) - \color{blue}{y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -450 \lor \neg \left(y \leq 0.0034\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) - y \cdot z\\ \end{array} \]

Alternative 5: 76.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 75.6%

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

3. Final simplification 75.6%

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

Alternative 6: 52.3% 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 99.9%

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

2. Taylor expanded in y around 0 75.6%

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

3. Taylor expanded in y around 0 57.5%

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

4. Final simplification 57.5%

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

Alternative 7: 16.4% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(-z\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 70.8%

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

3. Taylor expanded in x around 0 17.7%

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

   1. associate-*r* 17.7%

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

   2. neg-mul-1 17.7%

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

   3. *-commutative 17.7%

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

5. Simplified 17.7%

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

6. Final simplification 17.7%

   \[\leadsto y \cdot \left(-z\right) \]

Reproduce

herbie shell --seed 2023238 
(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))))