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

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

Alternatives: 6

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: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -54 \lor \neg \left(y \leq 4 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + x \cdot \left(y \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -54.0) (not (<= y 4e-23)))
   (* x (cos y))
   (+ x (* y (+ z (* x (* y -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -54.0) || !(y <= 4e-23)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (x * (y * -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 <= (-54.0d0)) .or. (.not. (y <= 4d-23))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (x * (y * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -54.0) || !(y <= 4e-23)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (x * (y * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -54.0) or not (y <= 4e-23):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (x * (y * -0.5))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -54.0) || !(y <= 4e-23))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(x * Float64(y * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -54.0) || ~((y <= 4e-23)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (x * (y * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -54.0], N[Not[LessEqual[y, 4e-23]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(x * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -54 or 3.99999999999999984e-23 < 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 -54 < y < 3.99999999999999984e-23

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.4%

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

      2. pow3 99.5%

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

   3. Applied egg-rr 99.5%

      \[\leadsto x \cdot \cos y + \color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{3}} \]
   4. Step-by-step derivation

      1. rem-cube-cbrt 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 52.6%

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

      4. associate-*r* 52.6%

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

   5. Applied egg-rr 52.6%

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

      1. add-sqr-sqrt 51.1%

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

      2. associate-*r* 51.1%

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

      3. associate-*l* 51.1%

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

      4. add-sqr-sqrt 98.5%

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

      5. fma-def 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. unpow2 98.6%

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

      5. associate-*r* 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-out 98.6%

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

      8. associate-*l* 98.6%

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

      9. *-commutative 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -54 \lor \neg \left(y \leq 4 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + x \cdot \left(y \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+137} \lor \neg \left(z \leq 1.55 \cdot 10^{+122}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e+137) (not (<= z 1.55e+122))) (* z (sin y)) (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e+137) || !(z <= 1.55e+122)) {
		tmp = 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 <= (-8d+137)) .or. (.not. (z <= 1.55d+122))) then
        tmp = 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 <= -8e+137) || !(z <= 1.55e+122)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e+137) or not (z <= 1.55e+122):
		tmp = z * math.sin(y)
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e+137) || !(z <= 1.55e+122))
		tmp = 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 <= -8e+137) || ~((z <= 1.55e+122)))
		tmp = z * sin(y);
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e+137], N[Not[LessEqual[z, 1.55e+122]], $MachinePrecision]], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.0000000000000003e137 or 1.54999999999999999e122 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 70.6%

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

   if -8.0000000000000003e137 < z < 1.54999999999999999e122

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 84.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+137} \lor \neg \left(z \leq 1.55 \cdot 10^{+122}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 5: 52.1% 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 54.5%

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

   1. +-commutative 54.5%

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

4. Simplified 54.5%

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

5. Final simplification 54.5%

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

Alternative 6: 38.8% 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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. add-sqr-sqrt 47.3%

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

   3. associate-*r* 47.3%

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

   4. fma-def 47.3%

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

3. Applied egg-rr 47.3%

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

4. Taylor expanded in y around 0 43.5%

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

5. Final simplification 43.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023280 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))