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

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision] - 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. Final simplification 99.8%

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

Alternative 2: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= x -1.42e-83) t_0 (if (<= x 0.036) (* (- z) (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (x <= -1.42e-83) {
		tmp = t_0;
	} else if (x <= 0.036) {
		tmp = -z * sin(y);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * x
    if (x <= (-1.42d-83)) then
        tmp = t_0
    else if (x <= 0.036d0) then
        tmp = -z * sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * x;
	double tmp;
	if (x <= -1.42e-83) {
		tmp = t_0;
	} else if (x <= 0.036) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * x
	tmp = 0
	if x <= -1.42e-83:
		tmp = t_0
	elif x <= 0.036:
		tmp = -z * math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (x <= -1.42e-83)
		tmp = t_0;
	elseif (x <= 0.036)
		tmp = Float64(Float64(-z) * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * x;
	tmp = 0.0;
	if (x <= -1.42e-83)
		tmp = t_0;
	elseif (x <= 0.036)
		tmp = -z * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.42e-83], t$95$0, If[LessEqual[x, 0.036], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999e-83 or 0.0359999999999999973 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -1.4199999999999999e-83 < x < 0.0359999999999999973

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   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 71.4

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

   5. Applied rewrites 71.4%

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

Alternative 3: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.165)
     t_0
     (if (<= y 0.08)
       (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.165) {
		tmp = t_0;
	} else if (y <= 0.08) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.165)
		tmp = t_0;
	elseif (y <= 0.08)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.165], t$95$0, If[LessEqual[y, 0.08], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.165000000000000008 or 0.0800000000000000017 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 51.0

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

   5. Applied rewrites 51.0%

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

   if -0.165000000000000008 < y < 0.0800000000000000017

   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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 41.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+103}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.85e+103) (* 1.0 x) (* (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.85e+103) {
		tmp = 1.0 * x;
	} else {
		tmp = -z * 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 (z <= 1.85d+103) then
        tmp = 1.0d0 * x
    else
        tmp = -z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.85e+103) {
		tmp = 1.0 * x;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.85e+103:
		tmp = 1.0 * x
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.85e+103)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.85e+103)
		tmp = 1.0 * x;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.85e+103], N[(1.0 * x), $MachinePrecision], N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.85000000000000016e103

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. flip-- N/A

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

      6. clear-num N/A

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 92.9

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 34.8%

       \[\leadsto 1 \cdot x \]

   if 1.85000000000000016e103 < z

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 43.1%

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

Alternative 5: 53.0% 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. flip-- N/A

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

   6. clear-num N/A

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

4. Applied rewrites 56.7%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 46.6

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

7. Applied rewrites 46.6%

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

Alternative 6: 53.0% 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 46.6

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

5. Applied rewrites 46.6%

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

Alternative 7: 39.6% 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. flip-- N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. flip-- N/A

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

   6. clear-num N/A

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

4. Applied rewrites 56.7%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 90.4

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

7. Applied rewrites 90.4%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 32.2%

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

Reproduce

herbie shell --seed 2024268 
(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))))