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

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(0 - z\right)\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.175:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- 0.0 z))))
   (if (<= y -4.3e+119)
     t_0
     (if (<= y -0.175)
       (* x (cos y))
       (if (<= y 0.82)
         (+ x (* y (- (* y (+ (* x -0.5) (* z (* y 0.16666666666666666)))) z)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * (0.0 - z);
	double tmp;
	if (y <= -4.3e+119) {
		tmp = t_0;
	} else if (y <= -0.175) {
		tmp = x * cos(y);
	} else if (y <= 0.82) {
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z));
	} 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 = sin(y) * (0.0d0 - z)
    if (y <= (-4.3d+119)) then
        tmp = t_0
    else if (y <= (-0.175d0)) then
        tmp = x * cos(y)
    else if (y <= 0.82d0) then
        tmp = x + (y * ((y * ((x * (-0.5d0)) + (z * (y * 0.16666666666666666d0)))) - z))
    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.sin(y) * (0.0 - z);
	double tmp;
	if (y <= -4.3e+119) {
		tmp = t_0;
	} else if (y <= -0.175) {
		tmp = x * Math.cos(y);
	} else if (y <= 0.82) {
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * (0.0 - z)
	tmp = 0
	if y <= -4.3e+119:
		tmp = t_0
	elif y <= -0.175:
		tmp = x * math.cos(y)
	elif y <= 0.82:
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(0.0 - z))
	tmp = 0.0
	if (y <= -4.3e+119)
		tmp = t_0;
	elseif (y <= -0.175)
		tmp = Float64(x * cos(y));
	elseif (y <= 0.82)
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(z * Float64(y * 0.16666666666666666)))) - z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * (0.0 - z);
	tmp = 0.0;
	if (y <= -4.3e+119)
		tmp = t_0;
	elseif (y <= -0.175)
		tmp = x * cos(y);
	elseif (y <= 0.82)
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.3e+119], t$95$0, If[LessEqual[y, -0.175], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.82], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.30000000000000032e119 or 0.819999999999999951 < y

   1. Initial program 99.7%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. fmm-def N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \cos y + z \cdot \sin y\right), \color{blue}{\left(\mathsf{fma}\left(x, \cos y, \mathsf{neg}\left(\sin y \cdot z\right)\right) \cdot \frac{1}{x \cdot \cos y + z \cdot \sin y}\right)}\right) \]

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z \cdot \sin y} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z \cdot \sin y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \color{blue}{\sin y}\right)\right) \]

      5. sin-lowering-sin.f64 58.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   7. Simplified 58.7%

      \[\leadsto \color{blue}{0 - z \cdot \sin y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(z \cdot \sin y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(z, \sin y\right)\right) \]

      4. sin-lowering-sin.f64 58.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   9. Applied egg-rr 58.7%

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

   if -4.30000000000000032e119 < y < -0.17499999999999999

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 61.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 61.9%

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

   if -0.17499999999999999 < y < 0.819999999999999951

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right), z\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot x\right), \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)\right), z\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{2}\right), \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)\right), z\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)\right), z\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \left(\left(\frac{1}{6} \cdot y\right) \cdot z\right)\right)\right), z\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \left(z \cdot \left(\frac{1}{6} \cdot y\right)\right)\right)\right), z\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \left(\frac{1}{6} \cdot y\right)\right)\right)\right), z\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), z\right)\right)\right) \]

      12. *-lowering-*.f64 99.4%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), z\right)\right)\right) \]

   5. Simplified 99.4%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+119}:\\ \;\;\;\;\sin y \cdot \left(0 - z\right)\\ \mathbf{elif}\;y \leq -0.175:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+97}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -5.8e+66) t_0 (if (<= x 9e+97) (- x (* z (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -5.8e+66) {
		tmp = t_0;
	} else if (x <= 9e+97) {
		tmp = x - (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 = x * cos(y)
    if (x <= (-5.8d+66)) then
        tmp = t_0
    else if (x <= 9d+97) then
        tmp = x - (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 = x * Math.cos(y);
	double tmp;
	if (x <= -5.8e+66) {
		tmp = t_0;
	} else if (x <= 9e+97) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -5.8e+66:
		tmp = t_0
	elif x <= 9e+97:
		tmp = x - (z * math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -5.8e+66)
		tmp = t_0;
	elseif (x <= 9e+97)
		tmp = Float64(x - Float64(z * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -5.8e+66)
		tmp = t_0;
	elseif (x <= 9e+97)
		tmp = x - (z * sin(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e+66], t$95$0, If[LessEqual[x, 9e+97], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.79999999999999972e66 or 8.99999999999999952e97 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 87.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 87.9%

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

   if -5.79999999999999972e66 < x < 8.99999999999999952e97

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 89.1%

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

Alternative 4: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.13:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.13)
     t_0
     (if (<= y 0.00095)
       (+ x (* y (- (* y (+ (* x -0.5) (* z (* y 0.16666666666666666)))) z)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.13) {
		tmp = t_0;
	} else if (y <= 0.00095) {
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z));
	} 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 = x * cos(y)
    if (y <= (-0.13d0)) then
        tmp = t_0
    else if (y <= 0.00095d0) then
        tmp = x + (y * ((y * ((x * (-0.5d0)) + (z * (y * 0.16666666666666666d0)))) - z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (y <= -0.13) {
		tmp = t_0;
	} else if (y <= 0.00095) {
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -0.13:
		tmp = t_0
	elif y <= 0.00095:
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.13)
		tmp = t_0;
	elseif (y <= 0.00095)
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(z * Float64(y * 0.16666666666666666)))) - z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -0.13)
		tmp = t_0;
	elseif (y <= 0.00095)
		tmp = x + (y * ((y * ((x * -0.5) + (z * (y * 0.16666666666666666)))) - z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.13], t$95$0, If[LessEqual[y, 0.00095], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.13 or 9.49999999999999998e-4 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 47.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 47.8%

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

   if -0.13 < y < 9.49999999999999998e-4

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right), \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right), z\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot x\right), \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)\right), z\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{2}\right), \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)\right), z\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)\right), z\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \left(\left(\frac{1}{6} \cdot y\right) \cdot z\right)\right)\right), z\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \left(z \cdot \left(\frac{1}{6} \cdot y\right)\right)\right)\right), z\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \left(\frac{1}{6} \cdot y\right)\right)\right)\right), z\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), z\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), z\right)\right)\right) \]

   5. Simplified 100.0%

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

Alternative 5: 42.0% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot z\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y z))))
   (if (<= z -1.9e+171) t_0 (if (<= z 1.08e+96) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.0 - (y * z);
	double tmp;
	if (z <= -1.9e+171) {
		tmp = t_0;
	} else if (z <= 1.08e+96) {
		tmp = x;
	} 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 = 0.0d0 - (y * z)
    if (z <= (-1.9d+171)) then
        tmp = t_0
    else if (z <= 1.08d+96) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.0 - (y * z);
	double tmp;
	if (z <= -1.9e+171) {
		tmp = t_0;
	} else if (z <= 1.08e+96) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.0 - (y * z)
	tmp = 0
	if z <= -1.9e+171:
		tmp = t_0
	elif z <= 1.08e+96:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.0 - Float64(y * z))
	tmp = 0.0
	if (z <= -1.9e+171)
		tmp = t_0;
	elseif (z <= 1.08e+96)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.0 - (y * z);
	tmp = 0.0;
	if (z <= -1.9e+171)
		tmp = t_0;
	elseif (z <= 1.08e+96)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+171], t$95$0, If[LessEqual[z, 1.08e+96], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000001e171 or 1.08e96 < z

   1. Initial program 99.8%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. fmm-def N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \cos y + z \cdot \sin y\right), \color{blue}{\left(\mathsf{fma}\left(x, \cos y, \mathsf{neg}\left(\sin y \cdot z\right)\right) \cdot \frac{1}{x \cdot \cos y + z \cdot \sin y}\right)}\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z \cdot \sin y} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z \cdot \sin y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \color{blue}{\sin y}\right)\right) \]

      5. sin-lowering-sin.f64 81.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   7. Simplified 81.4%

      \[\leadsto \color{blue}{0 - z \cdot \sin y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(z \cdot \sin y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(z, \sin y\right)\right) \]

      4. sin-lowering-sin.f64 81.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   9. Applied egg-rr 81.4%

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

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{neg.f64}\left(\color{blue}{\left(y \cdot z\right)}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(z \cdot y\right)\right) \]

       2. *-lowering-*.f64 33.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(z, y\right)\right) \]

   12. Simplified 33.2%

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

   if -1.9000000000000001e171 < z < 1.08e96

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 47.4%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+171}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.1% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(z \cdot \color{blue}{y}\right)\right) \]

   5. *-lowering-*.f64 53.5%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

5. Simplified 53.5%

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

6. Final simplification 53.5%

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

Alternative 7: 38.4% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 37.9%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))