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

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

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

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

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 2: 86.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+119} \lor \neg \left(x \leq 3.8 \cdot 10^{+33}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e+119) (not (<= x 3.8e+33)))
   (* (cos y) x)
   (fma (sin y) (- z) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e+119) || !(x <= 3.8e+33)) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(sin(y), -z, (1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.8e+119) || !(x <= 3.8e+33))
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(sin(y), Float64(-z), Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.8e+119], N[Not[LessEqual[x, 3.8e+33]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.80000000000000014e119 or 3.80000000000000002e33 < 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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 52.7

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if -5.80000000000000014e119 < x < 3.80000000000000002e33

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

         \[\leadsto x \cdot \color{blue}{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)} - z \cdot \sin y \]

      4. flip3-+ N/A

         \[\leadsto x \cdot \color{blue}{\frac{{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}} - z \cdot \sin y \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}} - z \cdot \sin y \]

   4. Applied rewrites 99.8%

      \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left({\sin y}^{3}, 0, 1 \cdot {\cos y}^{3}\right)}{{\left(0 \cdot \sin y\right)}^{2} + \left({\left(1 \cdot \cos y\right)}^{2} - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} - z \cdot \sin y \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 88.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

   9. Applied rewrites 88.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+119} \lor \neg \left(x \leq 3.8 \cdot 10^{+33}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+119} \lor \neg \left(x \leq 3.8 \cdot 10^{+33}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e+119) (not (<= x 3.8e+33)))
   (* (cos y) x)
   (- (* x 1.0) (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e+119) || !(x <= 3.8e+33)) {
		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+119)) .or. (.not. (x <= 3.8d+33))) 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+119) || !(x <= 3.8e+33)) {
		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+119) or not (x <= 3.8e+33):
		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+119) || !(x <= 3.8e+33))
		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+119) || ~((x <= 3.8e+33)))
		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+119], N[Not[LessEqual[x, 3.8e+33]], $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 < -5.80000000000000014e119 or 3.80000000000000002e33 < 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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 52.7

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if -5.80000000000000014e119 < x < 3.80000000000000002e33

   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 88.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+119} \lor \neg \left(x \leq 3.8 \cdot 10^{+33}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -2.45e+234)
     t_0
     (if (<= y -1.15e-7)
       (* (- z) (sin y))
       (if (<= y 8.5e-17) (fma (- z) y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -2.45e+234) {
		tmp = t_0;
	} else if (y <= -1.15e-7) {
		tmp = -z * sin(y);
	} else if (y <= 8.5e-17) {
		tmp = fma(-z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -2.45e+234)
		tmp = t_0;
	elseif (y <= -1.15e-7)
		tmp = Float64(Float64(-z) * sin(y));
	elseif (y <= 8.5e-17)
		tmp = fma(Float64(-z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -2.45e+234], t$95$0, If[LessEqual[y, -1.15e-7], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-17], N[((-z) * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.44999999999999995e234 or 8.5e-17 < y

   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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 68.5

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

   7. Applied rewrites 68.5%

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

   if -2.44999999999999995e234 < y < -1.14999999999999997e-7

   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. 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 64.7

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

   5. Applied rewrites 64.7%

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

   if -1.14999999999999997e-7 < y < 8.5e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+234}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -540 \lor \neg \left(y \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -540.0) (not (<= y 8.5e-17)))
   (* (cos y) x)
   (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -540.0) || !(y <= 8.5e-17)) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -540.0) || !(y <= 8.5e-17))
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -540.0], N[Not[LessEqual[y, 8.5e-17]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -540 or 8.5e-17 < y

   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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -540 < y < 8.5e-17

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -540 \lor \neg \left(y \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-167} \lor \neg \left(x \leq 2.5 \cdot 10^{-24}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9e-167) (not (<= x 2.5e-24))) (* 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9e-167) || !(x <= 2.5e-24)) {
		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 <= (-9d-167)) .or. (.not. (x <= 2.5d-24))) 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 <= -9e-167) || !(x <= 2.5e-24)) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9e-167) or not (x <= 2.5e-24):
		tmp = 1.0 * x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9e-167) || !(x <= 2.5e-24))
		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 <= -9e-167) || ~((x <= 2.5e-24)))
		tmp = 1.0 * x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9e-167], N[Not[LessEqual[x, 2.5e-24]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000002e-167 or 2.4999999999999999e-24 < x

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 44.7%

       \[\leadsto 1 \cdot x \]

   if -9.0000000000000002e-167 < x < 2.4999999999999999e-24

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 53.6

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

   5. Applied rewrites 53.6%

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

   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 38.3%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-167} \lor \neg \left(x \leq 2.5 \cdot 10^{-24}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% 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. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 52.1

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

5. Applied rewrites 52.1%

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

6. Final simplification 52.1%

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

Alternative 8: 40.0% 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. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 52.1

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

5. Applied rewrites 52.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 49.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 35.6%

    \[\leadsto 1 \cdot x \]

12. Final simplification 35.6%

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

Reproduce

herbie shell --seed 2024339 
(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))))