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

Percentage Accurate: 99.8% → 99.8%

Time: 13.9s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] * (-z) + N[(x * N[Cos[y], $MachinePrecision]), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

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(-Float64(sin(y) * 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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - 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. Final simplification 99.8%

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

Alternative 4: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := -\sin y \cdot z\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (- (* (sin y) z))))
   (if (<= y -3.6e+217)
     t_0
     (if (<= y -0.1)
       t_1
       (if (<= y 0.00055)
         (fma y (fma y (fma z (* y 0.16666666666666666) (* x -0.5)) (- z)) x)
         (if (<= y 6.8e+86) t_1 (if (<= y 7.5e+245) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = -(sin(y) * z);
	double tmp;
	if (y <= -3.6e+217) {
		tmp = t_0;
	} else if (y <= -0.1) {
		tmp = t_1;
	} else if (y <= 0.00055) {
		tmp = fma(y, fma(y, fma(z, (y * 0.16666666666666666), (x * -0.5)), -z), x);
	} else if (y <= 6.8e+86) {
		tmp = t_1;
	} else if (y <= 7.5e+245) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(-Float64(sin(y) * z))
	tmp = 0.0
	if (y <= -3.6e+217)
		tmp = t_0;
	elseif (y <= -0.1)
		tmp = t_1;
	elseif (y <= 0.00055)
		tmp = fma(y, fma(y, fma(z, Float64(y * 0.16666666666666666), Float64(x * -0.5)), Float64(-z)), x);
	elseif (y <= 6.8e+86)
		tmp = t_1;
	elseif (y <= 7.5e+245)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision])}, If[LessEqual[y, -3.6e+217], t$95$0, If[LessEqual[y, -0.1], t$95$1, If[LessEqual[y, 0.00055], N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 6.8e+86], t$95$1, If[LessEqual[y, 7.5e+245], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6000000000000002e217 or 6.7999999999999995e86 < y < 7.5e245

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -3.6000000000000002e217 < y < -0.10000000000000001 or 5.50000000000000033e-4 < y < 6.7999999999999995e86 or 7.5e245 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if -0.10000000000000001 < y < 5.50000000000000033e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, x\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}, x\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right), \mathsf{neg}\left(z\right)\right)}, x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{-1}{2} \cdot x}, \mathsf{neg}\left(z\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot z} + \frac{-1}{2} \cdot x, \mathsf{neg}\left(z\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{z \cdot \left(\frac{1}{6} \cdot y\right)} + \frac{-1}{2} \cdot x, \mathsf{neg}\left(z\right)\right), x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \frac{1}{6} \cdot y, \frac{-1}{2} \cdot x\right)}, \mathsf{neg}\left(z\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{6}}, \frac{-1}{2} \cdot x\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{6}}, \frac{-1}{2} \cdot x\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \frac{1}{6}, \color{blue}{x \cdot \frac{-1}{2}}\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \frac{1}{6}, \color{blue}{x \cdot \frac{-1}{2}}\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      13. lower-neg.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), \color{blue}{-z}\right), x\right) \]

   5. Applied rewrites 99.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.1:\\ \;\;\;\;-\sin y \cdot z\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;-\sin y \cdot z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+245}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 1 - \sin y \cdot z\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 650000000:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x 1.0) (* (sin y) z))))
   (if (<= z -5.9e-27) t_0 (if (<= z 650000000.0) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 1.0) - (sin(y) * z);
	double tmp;
	if (z <= -5.9e-27) {
		tmp = t_0;
	} else if (z <= 650000000.0) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	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 = (x * 1.0d0) - (sin(y) * z)
    if (z <= (-5.9d-27)) then
        tmp = t_0
    else if (z <= 650000000.0d0) then
        tmp = x * cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 1.0) - (Math.sin(y) * z);
	double tmp;
	if (z <= -5.9e-27) {
		tmp = t_0;
	} else if (z <= 650000000.0) {
		tmp = x * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 1.0) - (math.sin(y) * z)
	tmp = 0
	if z <= -5.9e-27:
		tmp = t_0
	elif z <= 650000000.0:
		tmp = x * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 1.0) - Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -5.9e-27)
		tmp = t_0;
	elseif (z <= 650000000.0)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 1.0) - (sin(y) * z);
	tmp = 0.0;
	if (z <= -5.9e-27)
		tmp = t_0;
	elseif (z <= 650000000.0)
		tmp = x * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 1.0), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.9e-27], t$95$0, If[LessEqual[z, 650000000.0], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.8999999999999998e-27 or 6.5e8 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 92.3%

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

   if -5.8999999999999998e-27 < z < 6.5e8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;x \cdot 1 - \sin y \cdot z\\ \mathbf{elif}\;z \leq 650000000:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -22000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1300000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -22000000000.0)
     t_0
     (if (<= y 1300000.0)
       (fma y (fma y (fma z (* y 0.16666666666666666) (* x -0.5)) (- z)) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -22000000000.0) {
		tmp = t_0;
	} else if (y <= 1300000.0) {
		tmp = fma(y, fma(y, fma(z, (y * 0.16666666666666666), (x * -0.5)), -z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -22000000000.0)
		tmp = t_0;
	elseif (y <= 1300000.0)
		tmp = fma(y, fma(y, fma(z, Float64(y * 0.16666666666666666), Float64(x * -0.5)), Float64(-z)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -22000000000.0], t$95$0, If[LessEqual[y, 1300000.0], N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e10 or 1.3e6 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 48.9

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

   5. Applied rewrites 48.9%

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

   if -2.2e10 < y < 1.3e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, x\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}, x\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right), \mathsf{neg}\left(z\right)\right)}, x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{-1}{2} \cdot x}, \mathsf{neg}\left(z\right)\right), x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot z} + \frac{-1}{2} \cdot x, \mathsf{neg}\left(z\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{z \cdot \left(\frac{1}{6} \cdot y\right)} + \frac{-1}{2} \cdot x, \mathsf{neg}\left(z\right)\right), x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \frac{1}{6} \cdot y, \frac{-1}{2} \cdot x\right)}, \mathsf{neg}\left(z\right)\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{6}}, \frac{-1}{2} \cdot x\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{6}}, \frac{-1}{2} \cdot x\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \frac{1}{6}, \color{blue}{x \cdot \frac{-1}{2}}\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \frac{1}{6}, \color{blue}{x \cdot \frac{-1}{2}}\right), \mathsf{neg}\left(z\right)\right), x\right) \]

      13. lower-neg.f64 95.4

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), \color{blue}{-z}\right), x\right) \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 41.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+160}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z))))
   (if (<= z -2.4e+73) t_0 (if (<= z 1.4e+160) (* x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if (z <= -2.4e+73) {
		tmp = t_0;
	} else if (z <= 1.4e+160) {
		tmp = x * 1.0;
	} else {
		tmp = t_0;
	}
	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 = -(y * z)
    if (z <= (-2.4d+73)) then
        tmp = t_0
    else if (z <= 1.4d+160) then
        tmp = x * 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if (z <= -2.4e+73) {
		tmp = t_0;
	} else if (z <= 1.4e+160) {
		tmp = x * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	tmp = 0
	if z <= -2.4e+73:
		tmp = t_0
	elif z <= 1.4e+160:
		tmp = x * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * z))
	tmp = 0.0
	if (z <= -2.4e+73)
		tmp = t_0;
	elseif (z <= 1.4e+160)
		tmp = Float64(x * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(y * z);
	tmp = 0.0;
	if (z <= -2.4e+73)
		tmp = t_0;
	elseif (z <= 1.4e+160)
		tmp = x * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * z), $MachinePrecision])}, If[LessEqual[z, -2.4e+73], t$95$0, If[LessEqual[z, 1.4e+160], N[(x * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000002e73 or 1.4e160 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.8%

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

   if -2.40000000000000002e73 < z < 1.4e160

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 48.1%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+73}:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+160}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.8

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

5. Applied rewrites 52.8%

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

7. Applied rewrites 52.8%

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

Alternative 9: 51.6% accurate, 23.8× 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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Final simplification 52.8%

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

Alternative 10: 38.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

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

C

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

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 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * 1.0), $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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sin.f64 92.7

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

8. Applied rewrites 92.7%

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

9. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
10. Step-by-step derivation

11. Applied rewrites 37.9%

    \[\leadsto x \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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))))