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

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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), Float64(-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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.8

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. 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

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

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: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-56} \lor \neg \left(x \leq 4.5 \cdot 10^{+48}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.3e-56) (not (<= x 4.5e+48)))
   (* (cos y) x)
   (fma (sin y) (- z) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.3e-56) || !(x <= 4.5e+48)) {
		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 <= -2.3e-56) || !(x <= 4.5e+48))
		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, -2.3e-56], N[Not[LessEqual[x, 4.5e+48]], $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 < -2.30000000000000002e-56 or 4.49999999999999995e48 < 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. 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 49.8

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

   5. Applied rewrites 49.8%

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

   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 5.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\cos y + -1 \cdot \frac{z \cdot \sin y}{x}\right)} \]
   10. 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. *-commutative N/A

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

       5. associate-*l/ N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-cos.f64 99.7

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

   11. Applied rewrites 99.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 87.5%

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

   if -2.30000000000000002e-56 < x < 4.49999999999999995e48

   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 91.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 91.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-56} \lor \neg \left(x \leq 4.5 \cdot 10^{+48}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-56} \lor \neg \left(x \leq 4.5 \cdot 10^{+48}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.3e-56) (not (<= x 4.5e+48)))
   (* (cos y) x)
   (- (* x 1.0) (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.3e-56) || !(x <= 4.5e+48)) {
		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 <= (-2.3d-56)) .or. (.not. (x <= 4.5d+48))) 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 <= -2.3e-56) || !(x <= 4.5e+48)) {
		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 <= -2.3e-56) or not (x <= 4.5e+48):
		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 <= -2.3e-56) || !(x <= 4.5e+48))
		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 <= -2.3e-56) || ~((x <= 4.5e+48)))
		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, -2.3e-56], N[Not[LessEqual[x, 4.5e+48]], $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 < -2.30000000000000002e-56 or 4.49999999999999995e48 < 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. 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 49.8

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

   5. Applied rewrites 49.8%

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

   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 5.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\cos y + -1 \cdot \frac{z \cdot \sin y}{x}\right)} \]
   10. 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. *-commutative N/A

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

       5. associate-*l/ N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-cos.f64 99.7

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

   11. Applied rewrites 99.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 87.5%

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

   if -2.30000000000000002e-56 < x < 4.49999999999999995e48

   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 91.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-56} \lor \neg \left(x \leq 4.5 \cdot 10^{+48}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-59} \lor \neg \left(x \leq 1.05 \cdot 10^{-182}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55e-59) (not (<= x 1.05e-182)))
   (* (cos y) x)
   (* (- z) (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55e-59) || !(x <= 1.05e-182)) {
		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 <= (-1.55d-59)) .or. (.not. (x <= 1.05d-182))) 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 <= -1.55e-59) || !(x <= 1.05e-182)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = -z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.55e-59) or not (x <= 1.05e-182):
		tmp = math.cos(y) * x
	else:
		tmp = -z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.55e-59) || !(x <= 1.05e-182))
		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 <= -1.55e-59) || ~((x <= 1.05e-182)))
		tmp = cos(y) * x;
	else
		tmp = -z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.55e-59], N[Not[LessEqual[x, 1.05e-182]], $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 < -1.55e-59 or 1.05e-182 < 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. 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 50.6

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

   5. Applied rewrites 50.6%

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

   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 8.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\cos y + -1 \cdot \frac{z \cdot \sin y}{x}\right)} \]
   10. 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. *-commutative N/A

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

       5. associate-*l/ N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-cos.f64 98.6

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 79.4%

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

   if -1.55e-59 < x < 1.05e-182

   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 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-59} \lor \neg \left(x \leq 1.05 \cdot 10^{-182}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.055 \lor \neg \left(y \leq 6300\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.055) (not (<= y 6300.0)))
   (* (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 <= -0.055) || !(y <= 6300.0)) {
		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 <= -0.055) || !(y <= 6300.0))
		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, -0.055], N[Not[LessEqual[y, 6300.0]], $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 < -0.0550000000000000003 or 6300 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 7.2

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

   5. Applied rewrites 7.2%

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

   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 3.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\cos y + -1 \cdot \frac{z \cdot \sin y}{x}\right)} \]
   10. 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. *-commutative N/A

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

       5. associate-*l/ N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-cos.f64 88.3

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

   11. Applied rewrites 88.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 52.1%

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

   if -0.0550000000000000003 < y < 6300

   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 98.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.055 \lor \neg \left(y \leq 6300\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+201} \lor \neg \left(z \leq 1.95 \cdot 10^{+86}\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e+201) (not (<= z 1.95e+86))) (* (- y) z) (* (- x) -1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+201) || !(z <= 1.95e+86)) {
		tmp = -y * z;
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.3d+201)) .or. (.not. (z <= 1.95d+86))) then
        tmp = -y * z
    else
        tmp = -x * (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+201) || !(z <= 1.95e+86)) {
		tmp = -y * z;
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.3e+201) or not (z <= 1.95e+86):
		tmp = -y * z
	else:
		tmp = -x * -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e+201) || !(z <= 1.95e+86))
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(Float64(-x) * -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.3e+201) || ~((z <= 1.95e+86)))
		tmp = -y * z;
	else
		tmp = -x * -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e+201], N[Not[LessEqual[z, 1.95e+86]], $MachinePrecision]], N[((-y) * z), $MachinePrecision], N[((-x) * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e201 or 1.9500000000000001e86 < z

   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 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   if -2.3000000000000001e201 < z < 1.9500000000000001e86

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. 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 x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-cos.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto \left(-x\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 44.5%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+201} \lor \neg \left(z \leq 1.95 \cdot 10^{+86}\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.2% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 51.1

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

5. Applied rewrites 51.1%

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

7. Applied rewrites 51.1%

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

8. Final simplification 51.1%

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

Alternative 9: 53.2% 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 51.1

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

5. Applied rewrites 51.1%

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

6. Final simplification 51.1%

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

Alternative 10: 38.7% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.8

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. 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 x around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-cos.f64 88.3

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

7. Applied rewrites 88.3%

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

8. Taylor expanded in y around 0

   \[\leadsto \left(-x\right) \cdot -1 \]
9. Step-by-step derivation

10. Applied rewrites 37.9%

    \[\leadsto \left(-x\right) \cdot -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024311 
(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))))