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

Percentage Accurate: 99.8% → 99.8%

Time: 11.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 99.9%

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

Alternative 2: 86.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+91}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -5.2e+94) t_0 (if (<= x 2.2e+91) (- 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 <= -5.2e+94) {
		tmp = t_0;
	} else if (x <= 2.2e+91) {
		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 <= (-5.2d+94)) then
        tmp = t_0
    else if (x <= 2.2d+91) 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 <= -5.2e+94) {
		tmp = t_0;
	} else if (x <= 2.2e+91) {
		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 <= -5.2e+94:
		tmp = t_0
	elif x <= 2.2e+91:
		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 <= -5.2e+94)
		tmp = t_0;
	elseif (x <= 2.2e+91)
		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 <= -5.2e+94)
		tmp = t_0;
	elseif (x <= 2.2e+91)
		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, -5.2e+94], t$95$0, If[LessEqual[x, 2.2e+91], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1999999999999998e94 or 2.19999999999999999e91 < x

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 92.4%

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

   5. Simplified 92.4%

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

   if -5.1999999999999998e94 < x < 2.19999999999999999e91

   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 89.9%

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

Alternative 3: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(0 - z\right)\\ \mathbf{if}\;y \leq -0.0058:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0011:\\ \;\;\;\;x + y \cdot \left(x \cdot \left(y \cdot -0.5\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- 0.0 z))))
   (if (<= y -0.0058)
     t_0
     (if (<= y 0.0011) (+ x (* y (- (* x (* y -0.5)) z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * (0.0 - z);
	double tmp;
	if (y <= -0.0058) {
		tmp = t_0;
	} else if (y <= 0.0011) {
		tmp = x + (y * ((x * (y * -0.5)) - 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 <= (-0.0058d0)) then
        tmp = t_0
    else if (y <= 0.0011d0) then
        tmp = x + (y * ((x * (y * (-0.5d0))) - 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 <= -0.0058) {
		tmp = t_0;
	} else if (y <= 0.0011) {
		tmp = x + (y * ((x * (y * -0.5)) - 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 <= -0.0058:
		tmp = t_0
	elif y <= 0.0011:
		tmp = x + (y * ((x * (y * -0.5)) - 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 <= -0.0058)
		tmp = t_0;
	elseif (y <= 0.0011)
		tmp = Float64(x + Float64(y * Float64(Float64(x * Float64(y * -0.5)) - 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 <= -0.0058)
		tmp = t_0;
	elseif (y <= 0.0011)
		tmp = x + (y * ((x * (y * -0.5)) - 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, -0.0058], t$95$0, If[LessEqual[y, 0.0011], N[(x + N[(y * N[(N[(x * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0058 or 0.00110000000000000007 < 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. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

      6. fmm-def N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\mathsf{fma}\left(x, \cos y, \mathsf{neg}\left(z \cdot \sin y\right)\right)\right)\right)\right) \]

      10. fmm-def N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(x \cdot \cos y - \color{blue}{z \cdot \sin y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(x \cdot \cos y\right), \color{blue}{\left(z \cdot \sin y\right)}\right)\right)\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot \cos y - z \cdot \sin y}}} \]

   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 59.3%

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

   7. Simplified 59.3%

      \[\leadsto \color{blue}{0 - z \cdot \sin y} \]
   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 59.3%

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

   9. Applied egg-rr 59.3%

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

   if -0.0058 < y < 0.00110000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\frac{-1}{2} \cdot \left(x \cdot y\right) - z\right)} \]
   4. Step-by-step derivation

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0058:\\ \;\;\;\;\sin y \cdot \left(0 - z\right)\\ \mathbf{elif}\;y \leq 0.0011:\\ \;\;\;\;x + y \cdot \left(x \cdot \left(y \cdot -0.5\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(-1 - \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -3.9e-8)
     t_0
     (if (<= y 3.3)
       (+
        x
        (*
         z
         (*
          y
          (-
           -1.0
           (*
            (* 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 <= -3.9e-8) {
		tmp = t_0;
	} else if (y <= 3.3) {
		tmp = x + (z * (y * (-1.0 - ((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 <= (-3.9d-8)) then
        tmp = t_0
    else if (y <= 3.3d0) then
        tmp = x + (z * (y * ((-1.0d0) - ((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 <= -3.9e-8) {
		tmp = t_0;
	} else if (y <= 3.3) {
		tmp = x + (z * (y * (-1.0 - ((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 <= -3.9e-8:
		tmp = t_0
	elif y <= 3.3:
		tmp = x + (z * (y * (-1.0 - ((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 <= -3.9e-8)
		tmp = t_0;
	elseif (y <= 3.3)
		tmp = Float64(x + Float64(z * Float64(y * Float64(-1.0 - Float64(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 <= -3.9e-8)
		tmp = t_0;
	elseif (y <= 3.3)
		tmp = x + (z * (y * (-1.0 - ((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, -3.9e-8], t$95$0, If[LessEqual[y, 3.3], N[(x + N[(z * N[(y * N[(-1.0 - N[(N[(y * y), $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]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.89999999999999985e-8 or 3.2999999999999998 < 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 43.3%

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

   5. Simplified 43.3%

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

   if -3.89999999999999985e-8 < y < 3.2999999999999998

   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 100.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 99.4%

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

   8. Simplified 99.4%

      \[\leadsto x - z \cdot \color{blue}{\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 3.3:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(-1 - \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{-1 + \frac{-0.25 \cdot \left(x \cdot \left(x \cdot \frac{y \cdot y}{z}\right)\right) + y \cdot \left(x \cdot 0.5\right)}{z}}{z}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   y
   (/
    (+ -1.0 (/ (+ (* -0.25 (* x (* x (/ (* y y) z)))) (* y (* x 0.5))) z))
    z))))

C

double code(double x, double y, double z) {
	return x + (y / ((-1.0 + (((-0.25 * (x * (x * ((y * y) / z)))) + (y * (x * 0.5))) / z)) / 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 / (((-1.0d0) + ((((-0.25d0) * (x * (x * ((y * y) / z)))) + (y * (x * 0.5d0))) / z)) / z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((-1.0 + (((-0.25 * (x * (x * ((y * y) / z)))) + (y * (x * 0.5))) / z)) / z));
}

Python

def code(x, y, z):
	return x + (y / ((-1.0 + (((-0.25 * (x * (x * ((y * y) / z)))) + (y * (x * 0.5))) / z)) / z))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(-1.0 + Float64(Float64(Float64(-0.25 * Float64(x * Float64(x * Float64(Float64(y * y) / z)))) + Float64(y * Float64(x * 0.5))) / z)) / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((-1.0 + (((-0.25 * (x * (x * ((y * y) / z)))) + (y * (x * 0.5))) / z)) / z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(-1.0 + N[(N[(N[(-0.25 * N[(x * N[(x * N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + y \cdot \left(\frac{-1}{2} \cdot \left(x \cdot y\right) - z\right)} \]
4. Step-by-step derivation

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 55.6%

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

5. Simplified 55.6%

   \[\leadsto \color{blue}{x + y \cdot \left(x \cdot \left(y \cdot -0.5\right) - z\right)} \]
6. Step-by-step derivation

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-- N/A

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

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

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

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

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

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

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

   10. *-lowering-*.f64 55.5%

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

7. Applied egg-rr 55.5%

   \[\leadsto x + \color{blue}{\frac{y}{\frac{1}{x \cdot \left(y \cdot -0.5\right) - z}}} \]

8. Taylor expanded in z around inf

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\frac{-1}{4} \cdot \frac{{x}^{2} \cdot {y}^{2}}{{z}^{2}} - \left(1 + \frac{-1}{2} \cdot \frac{x \cdot y}{z}\right)}{z}\right)}\right)\right) \]

10. Simplified 56.5%

    \[\leadsto x + \frac{y}{\color{blue}{\frac{-1 + \frac{-0.25 \cdot \left(x \cdot \left(x \cdot \frac{y \cdot y}{z}\right)\right) + y \cdot \left(x \cdot 0.5\right)}{z}}{z}}} \]
11. Add Preprocessing

Alternative 6: 42.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot z\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y z))))
   (if (<= z -1.1e+100) t_0 (if (<= z 4.8e+95) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.0 - (y * z);
	double tmp;
	if (z <= -1.1e+100) {
		tmp = t_0;
	} else if (z <= 4.8e+95) {
		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 <= (-1.1d+100)) then
        tmp = t_0
    else if (z <= 4.8d+95) 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 <= -1.1e+100) {
		tmp = t_0;
	} else if (z <= 4.8e+95) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.0 - (y * z)
	tmp = 0
	if z <= -1.1e+100:
		tmp = t_0
	elif z <= 4.8e+95:
		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 <= -1.1e+100)
		tmp = t_0;
	elseif (z <= 4.8e+95)
		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 <= -1.1e+100)
		tmp = t_0;
	elseif (z <= 4.8e+95)
		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, -1.1e+100], t$95$0, If[LessEqual[z, 4.8e+95], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e100 or 4.8000000000000001e95 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\frac{-1}{2} \cdot \left(x \cdot y\right) - z\right)} \]
   4. Step-by-step derivation

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 51.3%

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{x + y \cdot \left(x \cdot \left(y \cdot -0.5\right) - z\right)} \]

   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 36.1%

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

   8. Simplified 36.1%

      \[\leadsto \color{blue}{0 - z \cdot y} \]
   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. *-lowering-*.f64 36.1%

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

   10. Applied egg-rr 36.1%

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

   if -1.1e100 < z < 4.8000000000000001e95

   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 54.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+100}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.0% 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 99.9%

   \[x \cdot \cos y - z \cdot \sin y \]
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 56.3%

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

5. Simplified 56.3%

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

6. Final simplification 56.3%

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

Alternative 8: 40.8% 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 99.9%

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

3. Taylor expanded in y around 0

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

5. Simplified 41.7%

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

Reproduce

herbie shell --seed 2024163 
(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))))