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

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-def 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 75.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+269}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;t_1 + y \cdot z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= y -3.7e+269)
     t_0
     (if (<= y -7.2e+182)
       t_1
       (if (<= y -5.1e+23)
         t_0
         (if (<= y 2.2e+68) (+ t_1 (* y z)) (if (<= y 2e+193) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -3.7e+269) {
		tmp = t_0;
	} else if (y <= -7.2e+182) {
		tmp = t_1;
	} else if (y <= -5.1e+23) {
		tmp = t_0;
	} else if (y <= 2.2e+68) {
		tmp = t_1 + (y * z);
	} else if (y <= 2e+193) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = x * cos(y)
    if (y <= (-3.7d+269)) then
        tmp = t_0
    else if (y <= (-7.2d+182)) then
        tmp = t_1
    else if (y <= (-5.1d+23)) then
        tmp = t_0
    else if (y <= 2.2d+68) then
        tmp = t_1 + (y * z)
    else if (y <= 2d+193) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -3.7e+269) {
		tmp = t_0;
	} else if (y <= -7.2e+182) {
		tmp = t_1;
	} else if (y <= -5.1e+23) {
		tmp = t_0;
	} else if (y <= 2.2e+68) {
		tmp = t_1 + (y * z);
	} else if (y <= 2e+193) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -3.7e+269:
		tmp = t_0
	elif y <= -7.2e+182:
		tmp = t_1
	elif y <= -5.1e+23:
		tmp = t_0
	elif y <= 2.2e+68:
		tmp = t_1 + (y * z)
	elif y <= 2e+193:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -3.7e+269)
		tmp = t_0;
	elseif (y <= -7.2e+182)
		tmp = t_1;
	elseif (y <= -5.1e+23)
		tmp = t_0;
	elseif (y <= 2.2e+68)
		tmp = Float64(t_1 + Float64(y * z));
	elseif (y <= 2e+193)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -3.7e+269)
		tmp = t_0;
	elseif (y <= -7.2e+182)
		tmp = t_1;
	elseif (y <= -5.1e+23)
		tmp = t_0;
	elseif (y <= 2.2e+68)
		tmp = t_1 + (y * z);
	elseif (y <= 2e+193)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.7e+269], t$95$0, If[LessEqual[y, -7.2e+182], t$95$1, If[LessEqual[y, -5.1e+23], t$95$0, If[LessEqual[y, 2.2e+68], N[(t$95$1 + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+193], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6999999999999999e269 or -7.2e182 < y < -5.10000000000000021e23 or 2.19999999999999987e68 < y < 2.00000000000000013e193

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 70.0%

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

   if -3.6999999999999999e269 < y < -7.2e182 or 2.00000000000000013e193 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 69.7%

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

   if -5.10000000000000021e23 < y < 2.19999999999999987e68

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 96.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+269}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{+23}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;x \cdot \cos y + y \cdot z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+193}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0028 \lor \neg \left(y \leq 0.052\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0028) (not (<= y 0.052)))
   (* x (cos y))
   (+ x (* y (+ z (* y (* x -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0028) || !(y <= 0.052)) {
		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 <= (-0.0028d0)) .or. (.not. (y <= 0.052d0))) 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 <= -0.0028) || !(y <= 0.052)) {
		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 <= -0.0028) or not (y <= 0.052):
		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 <= -0.0028) || !(y <= 0.052))
		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 <= -0.0028) || ~((y <= 0.052)))
		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, -0.0028], N[Not[LessEqual[y, 0.052]], $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 < -0.00279999999999999997 or 0.0519999999999999976 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 50.1%

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

   if -0.00279999999999999997 < y < 0.0519999999999999976

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. unpow2 99.9%

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

      5. associate-*r* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0028 \lor \neg \left(y \leq 0.052\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-45} \lor \neg \left(x \leq 1.95 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6e-45) (not (<= x 1.95e-28))) (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6e-45) || !(x <= 1.95e-28)) {
		tmp = x * cos(y);
	} else {
		tmp = z * sin(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 ((x <= (-6d-45)) .or. (.not. (x <= 1.95d-28))) then
        tmp = x * cos(y)
    else
        tmp = z * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6e-45) || !(x <= 1.95e-28)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6e-45) or not (x <= 1.95e-28):
		tmp = x * math.cos(y)
	else:
		tmp = z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6e-45) || !(x <= 1.95e-28))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(z * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6e-45) || ~((x <= 1.95e-28)))
		tmp = x * cos(y);
	else
		tmp = z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6e-45], N[Not[LessEqual[x, 1.95e-28]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000022e-45 or 1.94999999999999999e-28 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 86.8%

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

   if -6.00000000000000022e-45 < x < 1.94999999999999999e-28

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 69.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-45} \lor \neg \left(x \leq 1.95 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0 55.8%

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

   1. +-commutative 55.8%

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

5. Simplified 55.8%

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

6. Final simplification 55.8%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))