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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 3: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-23} \lor \neg \left(x \leq 8.5 \cdot 10^{+62}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e-23) (not (<= x 8.5e+62)))
   (* (cos y) x)
   (- (* x 1.0) (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e-23) || !(x <= 8.5e+62)) {
		tmp = cos(y) * x;
	} else {
		tmp = (x * 1.0) - (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 <= (-5.8d-23)) .or. (.not. (x <= 8.5d+62))) then
        tmp = cos(y) * x
    else
        tmp = (x * 1.0d0) - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e-23) || !(x <= 8.5e+62)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = (x * 1.0) - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.8e-23) or not (x <= 8.5e+62):
		tmp = math.cos(y) * x
	else:
		tmp = (x * 1.0) - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.8e-23) || !(x <= 8.5e+62))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(Float64(x * 1.0) - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.8e-23) || ~((x <= 8.5e+62)))
		tmp = cos(y) * x;
	else
		tmp = (x * 1.0) - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.8e-23], N[Not[LessEqual[x, 8.5e+62]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000003e-23 or 8.4999999999999997e62 < x

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 85.0%

       \[\leadsto \cos y \cdot x \]

   if -5.8000000000000003e-23 < x < 8.4999999999999997e62

   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 90.4%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-23} \lor \neg \left(x \leq 8.5 \cdot 10^{+62}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-67} \lor \neg \left(x \leq 3.5 \cdot 10^{-46}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.9e-67) (not (<= x 3.5e-46)))
   (* (cos y) x)
   (* (- z) (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.9e-67) || !(x <= 3.5e-46)) {
		tmp = cos(y) * x;
	} else {
		tmp = -z * sin(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-4.9d-67)) .or. (.not. (x <= 3.5d-46))) then
        tmp = cos(y) * x
    else
        tmp = -z * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.9e-67) || !(x <= 3.5e-46)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = -z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.9e-67) or not (x <= 3.5e-46):
		tmp = math.cos(y) * x
	else:
		tmp = -z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.9e-67) || !(x <= 3.5e-46))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(Float64(-z) * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.9e-67) || ~((x <= 3.5e-46)))
		tmp = cos(y) * x;
	else
		tmp = -z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.9e-67], N[Not[LessEqual[x, 3.5e-46]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.89999999999999993e-67 or 3.5000000000000002e-46 < x

   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. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.3%

       \[\leadsto \cos y \cdot x \]

   if -4.89999999999999993e-67 < x < 3.5000000000000002e-46

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 73.7

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

   5. Applied rewrites 73.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-67} \lor \neg \left(x \leq 3.5 \cdot 10^{-46}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380 \lor \neg \left(y \leq 2.6\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -380.0) (not (<= y 2.6)))
   (* (cos y) x)
   (-
    (* x 1.0)
    (*
     (fma
      (* z (fma 0.008333333333333333 (* y y) -0.16666666666666666))
      (* y y)
      z)
     y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -380.0) || !(y <= 2.6)) {
		tmp = cos(y) * x;
	} else {
		tmp = (x * 1.0) - (fma((z * fma(0.008333333333333333, (y * y), -0.16666666666666666)), (y * y), z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -380.0) || !(y <= 2.6))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(Float64(x * 1.0) - Float64(fma(Float64(z * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666)), Float64(y * y), z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -380.0], N[Not[LessEqual[y, 2.6]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] - N[(N[(N[(z * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -380 or 2.60000000000000009 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 92.1

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

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 46.9%

       \[\leadsto \cos y \cdot x \]

   if -380 < y < 2.60000000000000009

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot 1 - \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot 1 - \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto x \cdot 1 - \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) + z\right)} \cdot y \]

      4. *-commutative N/A

         \[\leadsto x \cdot 1 - \left(\color{blue}{\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) \cdot {y}^{2}} + z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot 1 - \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right), {y}^{2}, z\right)} \cdot y \]

      6. +-commutative N/A

         \[\leadsto x \cdot 1 - \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{6} \cdot z}, {y}^{2}, z\right) \cdot y \]

      7. associate-*r* N/A

         \[\leadsto x \cdot 1 - \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot z} + \frac{-1}{6} \cdot z, {y}^{2}, z\right) \cdot y \]

      8. distribute-rgt-out N/A

         \[\leadsto x \cdot 1 - \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, {y}^{2}, z\right) \cdot y \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot 1 - \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, {y}^{2}, z\right) \cdot y \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot 1 - \mathsf{fma}\left(z \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, z\right) \cdot y \]

      11. unpow2 N/A

          \[\leadsto x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, z\right) \cdot y \]

      12. lower-*.f64 N/A

          \[\leadsto x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, z\right) \cdot y \]

      13. unpow2 N/A

          \[\leadsto x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, z\right) \cdot y \]

      14. lower-*.f64 98.7

          \[\leadsto x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, z\right) \cdot y \]

   8. Applied rewrites 98.7%

      \[\leadsto x \cdot 1 - \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380 \lor \neg \left(y \leq 2.6\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-104} \lor \neg \left(x \leq 1.15 \cdot 10^{-249}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.8e-104) (not (<= x 1.15e-249))) (* 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.8e-104) || !(x <= 1.15e-249)) {
		tmp = 1.0 * x;
	} else {
		tmp = -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 ((x <= (-2.8d-104)) .or. (.not. (x <= 1.15d-249))) then
        tmp = 1.0d0 * x
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.8e-104) || !(x <= 1.15e-249)) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.8e-104) or not (x <= 1.15e-249):
		tmp = 1.0 * x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.8e-104) || !(x <= 1.15e-249))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.8e-104) || ~((x <= 1.15e-249)))
		tmp = 1.0 * x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.8e-104], N[Not[LessEqual[x, 1.15e-249]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e-104 or 1.1499999999999999e-249 < x

   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. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 45.2%

       \[\leadsto 1 \cdot x \]

   if -2.8e-104 < x < 1.1499999999999999e-249

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   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 39.0%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-104} \lor \neg \left(x \leq 1.15 \cdot 10^{-249}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
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 52.0

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

5. Applied rewrites 52.0%

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

Alternative 8: 39.1% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(1.0 * x), $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. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 94.5

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

7. Applied rewrites 94.5%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 38.2%

    \[\leadsto 1 \cdot x \]

11. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2024324 
(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))))