Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

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 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos y \cdot \left(x + z \cdot \tan y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (cos y) (+ x (* z (tan y)))))

C

double code(double x, double y, double z) {
	return cos(y) * (x + (z * tan(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 = cos(y) * (x + (z * tan(y)))
end function

Java

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

Python

def code(x, y, z):
	return math.cos(y) * (x + (z * math.tan(y)))

Julia

function code(x, y, z)
	return Float64(cos(y) * Float64(x + Float64(z * tan(y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = cos(y) * (x + (z * tan(y)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. *-commutative N/A

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

   4. div-sub N/A

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

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

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

4. Applied egg-rr 41.3%

   \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot y\right)\right) \cdot \frac{x \cdot x}{x \cdot \cos y - z \cdot \sin y} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) \cdot \frac{z \cdot z}{x \cdot \cos y - z \cdot \sin y}} \]

5. Taylor expanded in z around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.5 + 0.5 \cdot \cos \left(y \cdot 2\right)\right)}{\cos y} + \frac{\left(z \cdot \sin y\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(y \cdot 2\right)\right)}{{\cos y}^{2}}} \]
8. Step-by-step derivation

   1. +-commutative N/A

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

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot \sin y}{\cos y}, \cos y, x \cdot \cos y\right)} \]
10. Step-by-step derivation

    1. distribute-rgt-out N/A

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

    2. *-commutative N/A

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

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

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

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

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

    5. associate-/l* N/A

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

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

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

    7. quot-tan N/A

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

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

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

    9. cos-lowering-cos.f64 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \cos y \cdot \left(x + z \cdot \tan y\right) \]
13. Add Preprocessing

Alternative 3: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+178}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.23:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 435:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + z \cdot \left(y \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -1.45e+178)
     (* x (cos y))
     (if (<= y -0.23)
       t_0
       (if (<= y 435.0)
         (+
          x
          (* y (+ z (* y (+ (* x -0.5) (* z (* y -0.16666666666666666)))))))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -1.45e+178) {
		tmp = x * cos(y);
	} else if (y <= -0.23) {
		tmp = t_0;
	} else if (y <= 435.0) {
		tmp = x + (y * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))));
	} 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 = z * sin(y)
    if (y <= (-1.45d+178)) then
        tmp = x * cos(y)
    else if (y <= (-0.23d0)) then
        tmp = t_0
    else if (y <= 435.0d0) then
        tmp = x + (y * (z + (y * ((x * (-0.5d0)) + (z * (y * (-0.16666666666666666d0)))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double tmp;
	if (y <= -1.45e+178) {
		tmp = x * Math.cos(y);
	} else if (y <= -0.23) {
		tmp = t_0;
	} else if (y <= 435.0) {
		tmp = x + (y * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	tmp = 0
	if y <= -1.45e+178:
		tmp = x * math.cos(y)
	elif y <= -0.23:
		tmp = t_0
	elif y <= 435.0:
		tmp = x + (y * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -1.45e+178)
		tmp = Float64(x * cos(y));
	elseif (y <= -0.23)
		tmp = t_0;
	elseif (y <= 435.0)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(Float64(x * -0.5) + Float64(z * Float64(y * -0.16666666666666666)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	tmp = 0.0;
	if (y <= -1.45e+178)
		tmp = x * cos(y);
	elseif (y <= -0.23)
		tmp = t_0;
	elseif (y <= 435.0)
		tmp = x + (y * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e+178], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -0.23], t$95$0, If[LessEqual[y, 435.0], N[(x + N[(y * N[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(z * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e178

   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 66.1%

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

   5. Simplified 66.1%

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

   if -1.45e178 < y < -0.23000000000000001 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

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

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

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

      2. sin-lowering-sin.f64 59.4%

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

   5. Simplified 59.4%

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

   if -0.23000000000000001 < 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

      \[\leadsto \color{blue}{x + y \cdot \left(z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 98.6%

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

   5. Simplified 98.6%

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

Alternative 4: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -8 \cdot 10^{+127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+86}:\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -8e+127) t_0 (if (<= x 1.12e+86) (+ 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 <= 1.12e+86) {
		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 <= 1.12d+86) 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 <= 1.12e+86) {
		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 <= 1.12e+86:
		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 <= 1.12e+86)
		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 <= 1.12e+86)
		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, 1.12e+86], 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 1.12e86 < 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 92.2%

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

   5. Simplified 92.2%

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

   if -7.99999999999999964e127 < x < 1.12e86

   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 5: 74.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.02)
     t_0
     (if (<= y 0.205)
       (+ x (* y (+ z (* y (+ (* x -0.5) (* z (* y -0.16666666666666666)))))))
       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 * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))));
	} 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 * (z + (y * ((x * (-0.5d0)) + (z * (y * (-0.16666666666666666d0)))))))
    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 * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))));
	} 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 * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))))
	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(z + Float64(y * Float64(Float64(x * -0.5) + Float64(z * Float64(y * -0.16666666666666666)))))));
	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 * (z + (y * ((x * -0.5) + (z * (y * -0.16666666666666666))))));
	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[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(z * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 47.9%

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

   5. Simplified 47.9%

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

   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

      \[\leadsto \color{blue}{x + y \cdot \left(z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

Alternative 6: 39.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+54}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.6e+54) (* y z) (if (<= z 3e+168) x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e+54) {
		tmp = y * z;
	} else if (z <= 3e+168) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.6d+54)) then
        tmp = y * z
    else if (z <= 3d+168) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e+54) {
		tmp = y * z;
	} else if (z <= 3e+168) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.6e+54:
		tmp = y * z
	elif z <= 3e+168:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.6e+54)
		tmp = Float64(y * z);
	elseif (z <= 3e+168)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.6e+54)
		tmp = y * z;
	elseif (z <= 3e+168)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.6e+54], N[(y * z), $MachinePrecision], If[LessEqual[z, 3e+168], x, N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.59999999999999988e54 or 2.9999999999999998e168 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 36.1%

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

   8. Simplified 36.1%

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

   if -4.59999999999999988e54 < 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 48.9%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+54}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.1% accurate, 41.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (y * z);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 52.7%

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

5. Simplified 52.7%

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

6. Final simplification 52.7%

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

Alternative 8: 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 37.3%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))