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

Percentage Accurate: 99.8% → 99.8%

Time: 13.0s

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, 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: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot x\right), \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), x\right) - t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)))
   (if (<= z -1.6e-60)
     (- (fma (* y (* y x)) (fma y (* y 0.041666666666666664) -0.5) x) t_0)
     (if (<= z 1.35e+37) (* x (cos y)) (- t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double tmp;
	if (z <= -1.6e-60) {
		tmp = fma((y * (y * x)), fma(y, (y * 0.041666666666666664), -0.5), x) - t_0;
	} else if (z <= 1.35e+37) {
		tmp = x * cos(y);
	} else {
		tmp = -t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	tmp = 0.0
	if (z <= -1.6e-60)
		tmp = Float64(fma(Float64(y * Float64(y * x)), fma(y, Float64(y * 0.041666666666666664), -0.5), x) - t_0);
	elseif (z <= 1.35e+37)
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(-t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.6e-60], N[(N[(N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + x), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[z, 1.35e+37], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], (-t$95$0)]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6000000000000001e-60

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + {y}^{2} \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} - z \cdot \sin y \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right) + x\right)} - z \cdot \sin y \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot x\right) \cdot {y}^{2} + \left(\frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} + x\right) - z \cdot \sin y \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\frac{-1}{2} \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right) + x\right) - z \cdot \sin y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(x \cdot {y}^{2}\right) \cdot \frac{-1}{2}} + \left(\frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right) + x\right) - z \cdot \sin y \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot {y}^{2}\right) \cdot \frac{-1}{2} + \color{blue}{\left(\left(x \cdot {y}^{2}\right) \cdot \frac{1}{24}\right)} \cdot {y}^{2}\right) + x\right) - z \cdot \sin y \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\left(x \cdot {y}^{2}\right) \cdot \frac{-1}{2} + \color{blue}{\left(x \cdot {y}^{2}\right) \cdot \left(\frac{1}{24} \cdot {y}^{2}\right)}\right) + x\right) - z \cdot \sin y \]

      7. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot {y}^{2}\right) \cdot \left(\frac{-1}{2} + \frac{1}{24} \cdot {y}^{2}\right)} + x\right) - z \cdot \sin y \]

      8. +-commutative N/A

         \[\leadsto \left(\left(x \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}\right)} + x\right) - z \cdot \sin y \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot {y}^{2}\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) + x\right) - z \cdot \sin y \]

      10. sub-neg N/A

          \[\leadsto \left(\left(x \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)} + x\right) - z \cdot \sin y \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {y}^{2}, \frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, x\right)} - z \cdot \sin y \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot x\right), \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), x\right)} - z \cdot \sin y \]

   if -1.6000000000000001e-60 < z < 1.34999999999999993e37

   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 88.9

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

   5. Applied rewrites 88.9%

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

   if 1.34999999999999993e37 < z

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 70.4

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

   5. Applied rewrites 70.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot x\right), \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), x\right) - \sin y \cdot z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin y \cdot z\\ \mathbf{if}\;z \leq -3 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (sin y) z))))
   (if (<= z -3e+66) t_0 (if (<= z 1.35e+37) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(sin(y) * z);
	double tmp;
	if (z <= -3e+66) {
		tmp = t_0;
	} else if (z <= 1.35e+37) {
		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 = -(sin(y) * z)
    if (z <= (-3d+66)) then
        tmp = t_0
    else if (z <= 1.35d+37) 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 = -(Math.sin(y) * z);
	double tmp;
	if (z <= -3e+66) {
		tmp = t_0;
	} else if (z <= 1.35e+37) {
		tmp = x * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(math.sin(y) * z)
	tmp = 0
	if z <= -3e+66:
		tmp = t_0
	elif z <= 1.35e+37:
		tmp = x * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -3e+66)
		tmp = t_0;
	elseif (z <= 1.35e+37)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(sin(y) * z);
	tmp = 0.0;
	if (z <= -3e+66)
		tmp = t_0;
	elseif (z <= 1.35e+37)
		tmp = x * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision])}, If[LessEqual[z, -3e+66], t$95$0, If[LessEqual[z, 1.35e+37], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.00000000000000002e66 or 1.34999999999999993e37 < z

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

   if -3.00000000000000002e66 < z < 1.34999999999999993e37

   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 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+66}:\\ \;\;\;\;-\sin y \cdot z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.118)
     t_0
     (if (<= y 3.6e-22)
       (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 <= -0.118) {
		tmp = t_0;
	} else if (y <= 3.6e-22) {
		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 <= -0.118)
		tmp = t_0;
	elseif (y <= 3.6e-22)
		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, -0.118], t$95$0, If[LessEqual[y, 3.6e-22], 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 < -0.11799999999999999 or 3.5999999999999998e-22 < 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 53.3

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

   5. Applied rewrites 53.3%

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

   if -0.11799999999999999 < y < 3.5999999999999998e-22

   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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 7: 41.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{+223}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z))))
   (if (<= z -2.25e+223) t_0 (if (<= z 3.5e+202) (* x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if (z <= -2.25e+223) {
		tmp = t_0;
	} else if (z <= 3.5e+202) {
		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 <= (-2.25d+223)) then
        tmp = t_0
    else if (z <= 3.5d+202) 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 <= -2.25e+223) {
		tmp = t_0;
	} else if (z <= 3.5e+202) {
		tmp = x * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	tmp = 0
	if z <= -2.25e+223:
		tmp = t_0
	elif z <= 3.5e+202:
		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 <= -2.25e+223)
		tmp = t_0;
	elseif (z <= 3.5e+202)
		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 <= -2.25e+223)
		tmp = t_0;
	elseif (z <= 3.5e+202)
		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, -2.25e+223], t$95$0, If[LessEqual[z, 3.5e+202], N[(x * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e223 or 3.49999999999999987e202 < z

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.1%

      \[\leadsto -y \cdot z \]

   if -2.25e223 < z < 3.49999999999999987e202

   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. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-cos.f64 71.0

          \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(y \cdot -2\right)}{\cos y}, \frac{0.5}{\color{blue}{\cos y}}\right) \]

   7. Applied rewrites 71.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(y \cdot -2\right)}{\cos y}, \frac{0.5}{\cos y}\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 45.3%

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

Alternative 8: 52.3% 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 52.7

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

5. Applied rewrites 52.7%

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

6. Final simplification 52.7%

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

Alternative 9: 38.7% 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 65.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-neg-in N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 61.5

       \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(y \cdot -2\right)}{\cos y}, \frac{0.5}{\color{blue}{\cos y}}\right) \]

7. Applied rewrites 61.5%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(y \cdot -2\right)}{\cos y}, \frac{0.5}{\cos y}\right)} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 39.6%

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

Reproduce

herbie shell --seed 2024221 
(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))))