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

Percentage Accurate: 99.8% → 99.8%

Time: 10.9s

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: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(0 - z\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+178}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.12:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 435:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- 0.0 z))))
   (if (<= y -1.6e+178)
     (* x (cos y))
     (if (<= y -0.12)
       t_0
       (if (<= y 435.0)
         (+ 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 <= -1.6e+178) {
		tmp = x * cos(y);
	} else if (y <= -0.12) {
		tmp = t_0;
	} else if (y <= 435.0) {
		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 <= (-1.6d+178)) then
        tmp = x * cos(y)
    else if (y <= (-0.12d0)) then
        tmp = t_0
    else if (y <= 435.0d0) 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 <= -1.6e+178) {
		tmp = x * Math.cos(y);
	} else if (y <= -0.12) {
		tmp = t_0;
	} else if (y <= 435.0) {
		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 <= -1.6e+178:
		tmp = x * math.cos(y)
	elif y <= -0.12:
		tmp = t_0
	elif y <= 435.0:
		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 <= -1.6e+178)
		tmp = Float64(x * cos(y));
	elseif (y <= -0.12)
		tmp = t_0;
	elseif (y <= 435.0)
		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 <= -1.6e+178)
		tmp = x * cos(y);
	elseif (y <= -0.12)
		tmp = t_0;
	elseif (y <= 435.0)
		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, -1.6e+178], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -0.12], t$95$0, If[LessEqual[y, 435.0], 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 < -1.6e178

   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 65.9%

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

   5. Simplified 65.9%

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

   if -1.6e178 < y < -0.12 or 435 < y

   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

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 59.3%

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

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\sin y \cdot \left(0 - z\right)} \]

   if -0.12 < y < 435

   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 98.6%

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

   5. Simplified 98.6%

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

Alternative 3: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -8 \cdot 10^{+127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+85}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -8e+127) t_0 (if (<= x 2.55e+85) (- 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 <= -8e+127) {
		tmp = t_0;
	} else if (x <= 2.55e+85) {
		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 <= (-8d+127)) then
        tmp = t_0
    else if (x <= 2.55d+85) 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 <= -8e+127) {
		tmp = t_0;
	} else if (x <= 2.55e+85) {
		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 <= -8e+127:
		tmp = t_0
	elif x <= 2.55e+85:
		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 <= -8e+127)
		tmp = t_0;
	elseif (x <= 2.55e+85)
		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 <= -8e+127)
		tmp = t_0;
	elseif (x <= 2.55e+85)
		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, -8e+127], t$95$0, If[LessEqual[x, 2.55e+85], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e127 or 2.5499999999999999e85 < 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 92.1%

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

   5. Simplified 92.1%

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

   if -7.99999999999999964e127 < x < 2.5499999999999999e85

   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 88.9%

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

Alternative 4: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.205:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.02)
     t_0
     (if (<= y 0.205)
       (+ 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 <= -0.02) {
		tmp = t_0;
	} else if (y <= 0.205) {
		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 <= (-0.02d0)) then
        tmp = t_0
    else if (y <= 0.205d0) 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 <= -0.02) {
		tmp = t_0;
	} else if (y <= 0.205) {
		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 <= -0.02:
		tmp = t_0
	elif y <= 0.205:
		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 <= -0.02)
		tmp = t_0;
	elseif (y <= 0.205)
		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 <= -0.02)
		tmp = t_0;
	elseif (y <= 0.205)
		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, -0.02], t$95$0, If[LessEqual[y, 0.205], 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 < -0.0200000000000000004 or 0.204999999999999988 < y

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 47.7%

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

   5. Simplified 47.7%

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

   if -0.0200000000000000004 < y < 0.204999999999999988

   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: 39.2% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot z\\ \mathbf{if}\;z \leq -52000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y z))))
   (if (<= z -52000000000000.0) t_0 (if (<= z 3e+168) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.0 - (y * z);
	double tmp;
	if (z <= -52000000000000.0) {
		tmp = t_0;
	} else if (z <= 3e+168) {
		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 <= (-52000000000000.0d0)) then
        tmp = t_0
    else if (z <= 3d+168) 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 <= -52000000000000.0) {
		tmp = t_0;
	} else if (z <= 3e+168) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.0 - (y * z)
	tmp = 0
	if z <= -52000000000000.0:
		tmp = t_0
	elif z <= 3e+168:
		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 <= -52000000000000.0)
		tmp = t_0;
	elseif (z <= 3e+168)
		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 <= -52000000000000.0)
		tmp = t_0;
	elseif (z <= 3e+168)
		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, -52000000000000.0], t$95$0, If[LessEqual[z, 3e+168], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e13 or 2.9999999999999998e168 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      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 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 35.4%

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

   8. Simplified 35.4%

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

      1. sub0-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. neg-lowering-neg.f64 35.4%

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

   10. Applied egg-rr 35.4%

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

   if -5.2e13 < z < 2.9999999999999998e168

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 50.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -52000000000000:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% 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 52.6%

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

5. Simplified 52.6%

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

6. Final simplification 52.6%

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

Alternative 7: 37.8% 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 37.5%

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

Reproduce

herbie shell --seed 2024161 
(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))))