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

Percentage Accurate: 99.8% → 99.8%

Time: 12.9s

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: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+74}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -9.5e+122) t_0 (if (<= x 1.9e+74) (- 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 <= -9.5e+122) {
		tmp = t_0;
	} else if (x <= 1.9e+74) {
		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 <= (-9.5d+122)) then
        tmp = t_0
    else if (x <= 1.9d+74) 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 <= -9.5e+122) {
		tmp = t_0;
	} else if (x <= 1.9e+74) {
		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 <= -9.5e+122:
		tmp = t_0
	elif x <= 1.9e+74:
		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 <= -9.5e+122)
		tmp = t_0;
	elseif (x <= 1.9e+74)
		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 <= -9.5e+122)
		tmp = t_0;
	elseif (x <= 1.9e+74)
		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, -9.5e+122], t$95$0, If[LessEqual[x, 1.9e+74], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999986e122 or 1.8999999999999999e74 < 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 90.4%

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

   5. Simplified 90.4%

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

   if -9.49999999999999986e122 < x < 1.8999999999999999e74

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 86.9%

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

Alternative 3: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.86 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \left(0 - \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.86e-115) t_0 (if (<= x 8.2e-80) (* z (- 0.0 (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.86e-115) {
		tmp = t_0;
	} else if (x <= 8.2e-80) {
		tmp = z * (0.0 - 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 <= (-1.86d-115)) then
        tmp = t_0
    else if (x <= 8.2d-80) then
        tmp = z * (0.0d0 - 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 <= -1.86e-115) {
		tmp = t_0;
	} else if (x <= 8.2e-80) {
		tmp = z * (0.0 - 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 <= -1.86e-115:
		tmp = t_0
	elif x <= 8.2e-80:
		tmp = z * (0.0 - 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 <= -1.86e-115)
		tmp = t_0;
	elseif (x <= 8.2e-80)
		tmp = Float64(z * Float64(0.0 - 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 <= -1.86e-115)
		tmp = t_0;
	elseif (x <= 8.2e-80)
		tmp = z * (0.0 - 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, -1.86e-115], t$95$0, If[LessEqual[x, 8.2e-80], N[(z * N[(0.0 - N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.86e-115 or 8.1999999999999999e-80 < 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 76.5%

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

   5. Simplified 76.5%

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

   if -1.86e-115 < x < 8.1999999999999999e-80

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. 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 80.7%

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

   5. Simplified 80.7%

      \[\leadsto \color{blue}{0 - z \cdot \sin y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \left(0 - \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.038:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.34:\\ \;\;\;\;x - y \cdot \left(z + \left(z \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.038)
     t_0
     (if (<= y 0.34)
       (-
        x
        (*
         y
         (+
          z
          (*
           (* z (* y y))
           (+
            -0.16666666666666666
            (*
             (* y y)
             (+ 0.008333333333333333 (* (* y y) -0.0001984126984126984))))))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.038) {
		tmp = t_0;
	} else if (y <= 0.34) {
		tmp = x - (y * (z + ((z * (y * y)) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))));
	} 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 <= (-0.038d0)) then
        tmp = t_0
    else if (y <= 0.34d0) then
        tmp = x - (y * (z + ((z * (y * y)) * ((-0.16666666666666666d0) + ((y * y) * (0.008333333333333333d0 + ((y * y) * (-0.0001984126984126984d0))))))))
    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 <= -0.038) {
		tmp = t_0;
	} else if (y <= 0.34) {
		tmp = x - (y * (z + ((z * (y * y)) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -0.038:
		tmp = t_0
	elif y <= 0.34:
		tmp = x - (y * (z + ((z * (y * y)) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.038)
		tmp = t_0;
	elseif (y <= 0.34)
		tmp = Float64(x - Float64(y * Float64(z + Float64(Float64(z * Float64(y * y)) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * -0.0001984126984126984))))))));
	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 <= -0.038)
		tmp = t_0;
	elseif (y <= 0.34)
		tmp = x - (y * (z + ((z * (y * y)) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))));
	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, -0.038], t$95$0, If[LessEqual[y, 0.34], N[(x - N[(y * N[(z + N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0379999999999999991 or 0.340000000000000024 < 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 49.8%

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

   5. Simplified 49.8%

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

   if -0.0379999999999999991 < y < 0.340000000000000024

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 99.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{-1}{6} \cdot z} + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{-1}{6} \cdot z} + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot z\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)}\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\left(z \cdot \frac{-1}{6}\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right)} + \frac{1}{120} \cdot z\right)\right)\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{-1}{5040} \cdot {y}^{2}\right) \cdot z + \color{blue}{\frac{1}{120}} \cdot z\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(z \cdot \left(\frac{1}{120} + \color{blue}{\frac{-1}{5040} \cdot {y}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto x - \color{blue}{y \cdot \left(z + \left(y \cdot y\right) \cdot \left(z \cdot -0.16666666666666666 + y \cdot \left(y \cdot \left(z \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right) + \color{blue}{\frac{-1}{6} \cdot z}\right)\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot z + \frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right)\right) + \frac{-1}{6} \cdot z\right)\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot z + \left(\frac{-1}{5040} \cdot {y}^{2}\right) \cdot z\right) + \frac{-1}{6} \cdot z\right)\right)\right)\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left({y}^{2} \cdot \left(z \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot z\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot z\right) + \frac{-1}{6} \cdot z\right)\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right) \cdot z + \color{blue}{\frac{-1}{6}} \cdot z\right)\right)\right)\right)\right) \]

       7. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left(z \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)}\right)\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left(z \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       9. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left({y}^{2} \cdot \left(z \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \left(\left({y}^{2} \cdot z\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\left({y}^{2} \cdot z\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

   11. Simplified 99.8%

       \[\leadsto x - y \cdot \left(z + \color{blue}{\left(z \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 41.2% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-138}:\\ \;\;\;\;z \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-123) x (if (<= x 1.2e-138) (* z (- 0.0 y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-123) {
		tmp = x;
	} else if (x <= 1.2e-138) {
		tmp = z * (0.0 - y);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.45d-123)) then
        tmp = x
    else if (x <= 1.2d-138) then
        tmp = z * (0.0d0 - y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-123) {
		tmp = x;
	} else if (x <= 1.2e-138) {
		tmp = z * (0.0 - y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-123:
		tmp = x
	elif x <= 1.2e-138:
		tmp = z * (0.0 - y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-123)
		tmp = x;
	elseif (x <= 1.2e-138)
		tmp = Float64(z * Float64(0.0 - y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-123)
		tmp = x;
	elseif (x <= 1.2e-138)
		tmp = z * (0.0 - y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-123], x, If[LessEqual[x, 1.2e-138], N[(z * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000002e-123 or 1.2e-138 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 45.1%

      \[\leadsto \color{blue}{x} \]

   if -1.45000000000000002e-123 < x < 1.2e-138

   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

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

      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 56.7%

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

   5. Simplified 56.7%

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

   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 43.8%

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

   8. Simplified 43.8%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 43.8%

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

   10. Applied egg-rr 43.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-138}:\\ \;\;\;\;z \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% 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 52.4%

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

5. Simplified 52.4%

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

6. Final simplification 52.4%

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

Alternative 7: 38.3% 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 2024155 
(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))))