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

Percentage Accurate: 99.8% → 99.8%

Time: 10.6s

Alternatives: 9

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \sin y, x \cdot \cos y\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 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. Final simplification 99.8%

   \[\leadsto x \cdot \cos y + z \cdot \sin y \]

Alternative 3: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+255}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+105}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.00185 \lor \neg \left(y \leq 0.0195\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -2.6e+255)
     t_0
     (if (<= y -1.8e+105)
       (* z (sin y))
       (if (or (<= y -0.00185) (not (<= y 0.0195)))
         t_0
         (+ x (* y (+ z (* y (* x -0.5))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -2.6e+255) {
		tmp = t_0;
	} else if (y <= -1.8e+105) {
		tmp = z * sin(y);
	} else if ((y <= -0.00185) || !(y <= 0.0195)) {
		tmp = t_0;
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	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 <= (-2.6d+255)) then
        tmp = t_0
    else if (y <= (-1.8d+105)) then
        tmp = z * sin(y)
    else if ((y <= (-0.00185d0)) .or. (.not. (y <= 0.0195d0))) then
        tmp = t_0
    else
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    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 <= -2.6e+255) {
		tmp = t_0;
	} else if (y <= -1.8e+105) {
		tmp = z * Math.sin(y);
	} else if ((y <= -0.00185) || !(y <= 0.0195)) {
		tmp = t_0;
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -2.6e+255:
		tmp = t_0
	elif y <= -1.8e+105:
		tmp = z * math.sin(y)
	elif (y <= -0.00185) or not (y <= 0.0195):
		tmp = t_0
	else:
		tmp = x + (y * (z + (y * (x * -0.5))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -2.6e+255)
		tmp = t_0;
	elseif (y <= -1.8e+105)
		tmp = Float64(z * sin(y));
	elseif ((y <= -0.00185) || !(y <= 0.0195))
		tmp = t_0;
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -2.6e+255)
		tmp = t_0;
	elseif (y <= -1.8e+105)
		tmp = z * sin(y);
	elseif ((y <= -0.00185) || ~((y <= 0.0195)))
		tmp = t_0;
	else
		tmp = x + (y * (z + (y * (x * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+255], t$95$0, If[LessEqual[y, -1.8e+105], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -0.00185], N[Not[LessEqual[y, 0.0195]], $MachinePrecision]], t$95$0, N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6000000000000001e255 or -1.7999999999999999e105 < y < -0.0018500000000000001 or 0.0195 < y

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf 60.4%

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

   if -2.6000000000000001e255 < y < -1.7999999999999999e105

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around 0 65.5%

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

   if -0.0018500000000000001 < y < 0.0195

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-out 99.6%

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

      6. *-commutative 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+105}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.00185 \lor \neg \left(y \leq 0.0195\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+255}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-8} \lor \neg \left(y \leq 0.0039\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -1.8e+255)
     t_0
     (if (<= y -1.25e+105)
       (* z (sin y))
       (if (or (<= y -2.6e-8) (not (<= y 0.0039))) t_0 (fma y z x))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -1.8e+255) {
		tmp = t_0;
	} else if (y <= -1.25e+105) {
		tmp = z * sin(y);
	} else if ((y <= -2.6e-8) || !(y <= 0.0039)) {
		tmp = t_0;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.8e+255)
		tmp = t_0;
	elseif (y <= -1.25e+105)
		tmp = Float64(z * sin(y));
	elseif ((y <= -2.6e-8) || !(y <= 0.0039))
		tmp = t_0;
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+255], t$95$0, If[LessEqual[y, -1.25e+105], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.6e-8], N[Not[LessEqual[y, 0.0039]], $MachinePrecision]], t$95$0, N[(y * z + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7999999999999999e255 or -1.25000000000000011e105 < y < -2.6000000000000001e-8 or 0.0038999999999999998 < y

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf 61.2%

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

   if -1.7999999999999999e255 < y < -1.25000000000000011e105

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around 0 65.5%

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

   if -2.6000000000000001e-8 < y < 0.0038999999999999998

   1. Initial program 100.0%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, x\right)} \]

   4. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-8} \lor \neg \left(y \leq 0.0039\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]

Alternative 5: 85.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.88 \cdot 10^{-34} \lor \neg \left(z \leq 1.85 \cdot 10^{-134}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.88e-34) (not (<= z 1.85e-134)))
   (+ x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.88e-34) || !(z <= 1.85e-134)) {
		tmp = x + (z * sin(y));
	} else {
		tmp = x * cos(y);
	}
	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 <= (-1.88d-34)) .or. (.not. (z <= 1.85d-134))) then
        tmp = x + (z * sin(y))
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.88e-34) || !(z <= 1.85e-134)) {
		tmp = x + (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.88e-34) or not (z <= 1.85e-134):
		tmp = x + (z * math.sin(y))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.88e-34) || !(z <= 1.85e-134))
		tmp = Float64(x + Float64(z * sin(y)));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.88e-34) || ~((z <= 1.85e-134)))
		tmp = x + (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.88e-34], N[Not[LessEqual[z, 1.85e-134]], $MachinePrecision]], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.88e-34 or 1.85e-134 < z

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 86.9%

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

   if -1.88e-34 < z < 1.85e-134

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf 93.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.88 \cdot 10^{-34} \lor \neg \left(z \leq 1.85 \cdot 10^{-134}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 6: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00195 \lor \neg \left(y \leq 0.0195\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00195) (not (<= y 0.0195)))
   (* x (cos y))
   (+ x (* y (+ z (* y (* x -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00195) || !(y <= 0.0195)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	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 ((y <= (-0.00195d0)) .or. (.not. (y <= 0.0195d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00195) || !(y <= 0.0195)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00195) or not (y <= 0.0195):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (y * (x * -0.5))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00195) || !(y <= 0.0195))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00195) || ~((y <= 0.0195)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (y * (x * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00195], N[Not[LessEqual[y, 0.0195]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0019499999999999999 or 0.0195 < y

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf 54.3%

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

   if -0.0019499999999999999 < y < 0.0195

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-out 99.6%

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

      6. *-commutative 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00195 \lor \neg \left(y \leq 0.0195\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7: 41.9% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+235}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e+235) (* z y) (if (<= z 1.02e+149) x (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+235) {
		tmp = z * y;
	} else if (z <= 1.02e+149) {
		tmp = x;
	} else {
		tmp = z * y;
	}
	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 <= (-5.2d+235)) then
        tmp = z * y
    else if (z <= 1.02d+149) then
        tmp = x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+235) {
		tmp = z * y;
	} else if (z <= 1.02e+149) {
		tmp = x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.2e+235:
		tmp = z * y
	elif z <= 1.02e+149:
		tmp = x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e+235)
		tmp = Float64(z * y);
	elseif (z <= 1.02e+149)
		tmp = x;
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.2e+235)
		tmp = z * y;
	elseif (z <= 1.02e+149)
		tmp = x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e+235], N[(z * y), $MachinePrecision], If[LessEqual[z, 1.02e+149], x, N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999996e235 or 1.01999999999999997e149 < z

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around 0 81.8%

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

   3. Taylor expanded in y around 0 43.2%

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

   if -5.1999999999999996e235 < z < 1.01999999999999997e149

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 73.5%

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

      1. add-cube-cbrt 73.0%

         \[\leadsto x + \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}} \]

      2. pow3 73.0%

         \[\leadsto x + \color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{3}} \]

   4. Applied egg-rr 73.0%

      \[\leadsto x + \color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{3}} \]

   5. Taylor expanded in x around inf 46.0%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+235}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 8: 52.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in y around 0 53.8%

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

3. Final simplification 53.8%

   \[\leadsto x + z \cdot y \]

Alternative 9: 38.7% 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. Taylor expanded in y around 0 76.9%

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

   1. add-cube-cbrt 76.2%

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}} \]

   2. pow3 76.2%

      \[\leadsto x + \color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{3}} \]

4. Applied egg-rr 76.2%

   \[\leadsto x + \color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{3}} \]

5. Taylor expanded in x around inf 40.9%

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

6. Final simplification 40.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))