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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 3: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+52} \lor \neg \left(x \leq 1.35 \cdot 10^{+69}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e+52) (not (<= x 1.35e+69)))
   (* (cos y) x)
   (- (* x 1.0) (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e+52) || !(x <= 1.35e+69)) {
		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 <= (-1.1d+52)) .or. (.not. (x <= 1.35d+69))) 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 <= -1.1e+52) || !(x <= 1.35e+69)) {
		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 <= -1.1e+52) or not (x <= 1.35e+69):
		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 <= -1.1e+52) || !(x <= 1.35e+69))
		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 <= -1.1e+52) || ~((x <= 1.35e+69)))
		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, -1.1e+52], N[Not[LessEqual[x, 1.35e+69]], $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 < -1.1e52 or 1.3499999999999999e69 < x

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 90.9%

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

   if -1.1e52 < x < 1.3499999999999999e69

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+52} \lor \neg \left(x \leq 1.35 \cdot 10^{+69}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.46 \lor \neg \left(x \leq 1.52 \cdot 10^{-47}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.46) (not (<= x 1.52e-47))) (* (cos y) x) (* (- z) (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.46) || !(x <= 1.52e-47)) {
		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 <= (-0.46d0)) .or. (.not. (x <= 1.52d-47))) 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 <= -0.46) || !(x <= 1.52e-47)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = -z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.46) or not (x <= 1.52e-47):
		tmp = math.cos(y) * x
	else:
		tmp = -z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.46) || !(x <= 1.52e-47))
		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 <= -0.46) || ~((x <= 1.52e-47)))
		tmp = cos(y) * x;
	else
		tmp = -z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.46], N[Not[LessEqual[x, 1.52e-47]], $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 < -0.46000000000000002 or 1.52000000000000001e-47 < x

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 85.4%

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

   if -0.46000000000000002 < x < 1.52000000000000001e-47

   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 74.8

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

   5. Applied rewrites 74.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.46 \lor \neg \left(x \leq 1.52 \cdot 10^{-47}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.26 \lor \neg \left(y \leq 1.12 \cdot 10^{-7}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.26) (not (<= y 1.12e-7)))
   (* (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 <= -0.26) || !(y <= 1.12e-7)) {
		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 <= -0.26) || !(y <= 1.12e-7))
		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, -0.26], N[Not[LessEqual[y, 1.12e-7]], $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 < -0.26000000000000001 or 1.12e-7 < y

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 47.0%

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

   if -0.26000000000000001 < y < 1.12e-7

   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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.26 \lor \neg \left(y \leq 1.12 \cdot 10^{-7}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-165} \lor \neg \left(x \leq 3.4 \cdot 10^{-160}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.2e-165) (not (<= x 3.4e-160))) (* 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.2e-165) || !(x <= 3.4e-160)) {
		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 <= (-7.2d-165)) .or. (.not. (x <= 3.4d-160))) 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 <= -7.2e-165) || !(x <= 3.4e-160)) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.2e-165) or not (x <= 3.4e-160):
		tmp = 1.0 * x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.2e-165) || !(x <= 3.4e-160))
		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 <= -7.2e-165) || ~((x <= 3.4e-160)))
		tmp = 1.0 * x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.2e-165], N[Not[LessEqual[x, 3.4e-160]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999969e-165 or 3.40000000000000021e-160 < x

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 47.0%

       \[\leadsto 1 \cdot x \]

   if -7.19999999999999969e-165 < x < 3.40000000000000021e-160

   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 42.6

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

   5. Applied rewrites 42.6%

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

   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 33.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-165} \lor \neg \left(x \leq 3.4 \cdot 10^{-160}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% 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 48.9

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

5. Applied rewrites 48.9%

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

Alternative 8: 39.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-in N/A

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

   6. neg-mul-1 N/A

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

   7. lower-*.f64 N/A

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

7. Applied rewrites 90.6%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 38.8%

    \[\leadsto 1 \cdot x \]

11. Final simplification 38.8%

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

Reproduce

herbie shell --seed 2024313 
(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))))