Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, z, \cos y \cdot x\right)} \]
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(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. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 3: 85.4% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e-23) (not (<= x 8.5e+62)))
   (* (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-23) || !(x <= 8.5e+62)) {
		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-23) || !(x <= 8.5e+62))
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(sin(y), z, Float64(1.0 * x));
	end
	return 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[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000003e-23 or 8.4999999999999997e62 < 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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

   4. Applied rewrites 28.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 85.2%

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

   if -5.8000000000000003e-23 < x < 8.4999999999999997e62

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.4%

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

4. Final simplification 88.0%

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

Alternative 4: 73.7% accurate, 1.8× speedup

Math

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

FPCore

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

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 = sin(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 <= (-4.9d-67)) .or. (.not. (x <= 3.5d-46))) then
        tmp = cos(y) * x
    else
        tmp = sin(y) * z
    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 = Math.sin(y) * z;
	}
	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 = math.sin(y) * z
	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(sin(y) * z);
	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 = sin(y) * z;
	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[(N[Sin[y], $MachinePrecision] * z), $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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

   4. Applied rewrites 37.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.4%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 77.7%

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

Alternative 5: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0038 \lor \neg \left(y \leq 0.0042\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0038) (not (<= y 0.0042)))
   (* (sin y) z)
   (fma 1.0 x (* (* (fma (* -0.16666666666666666 y) y 1.0) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0038) || !(y <= 0.0042)) {
		tmp = sin(y) * z;
	} else {
		tmp = fma(1.0, x, ((fma((-0.16666666666666666 * y), y, 1.0) * z) * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0038) || !(y <= 0.0042))
		tmp = Float64(sin(y) * z);
	else
		tmp = fma(1.0, x, Float64(Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0038], N[Not[LessEqual[y, 0.0042]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(1.0 * x + N[(N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00379999999999999999 or 0.00419999999999999974 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if -0.00379999999999999999 < y < 0.00419999999999999974

   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

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

   8. Applied rewrites 100.0%

      \[\leadsto x \cdot 1 + \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0038 \lor \neg \left(y \leq 0.0042\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.3% accurate, 30.6× 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(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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 52.1

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

5. Applied rewrites 52.1%

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

Alternative 7: 17.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 52.1

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

5. Applied rewrites 52.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 17.3%

   \[\leadsto z \cdot \color{blue}{y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))