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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * x + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -3 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.0102:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.36:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -3e+116)
     t_0
     (if (<= y -0.0102)
       (* z (sin y))
       (if (<= y 0.36)
         (fma y (fma y (fma x -0.5 (* (* y z) -0.16666666666666666)) z) x)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -3e+116) {
		tmp = t_0;
	} else if (y <= -0.0102) {
		tmp = z * sin(y);
	} else if (y <= 0.36) {
		tmp = fma(y, fma(y, fma(x, -0.5, ((y * z) * -0.16666666666666666)), z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -3e+116)
		tmp = t_0;
	elseif (y <= -0.0102)
		tmp = Float64(z * sin(y));
	elseif (y <= 0.36)
		tmp = fma(y, fma(y, fma(x, -0.5, Float64(Float64(y * z) * -0.16666666666666666)), z), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3e+116], t$95$0, If[LessEqual[y, -0.0102], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.36], N[(y * N[(y * N[(x * -0.5 + N[(N[(y * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9999999999999999e116 or 0.35999999999999999 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if -2.9999999999999999e116 < y < -0.010200000000000001

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 51.1

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

   5. Applied rewrites 51.1%

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

   if -0.010200000000000001 < y < 0.35999999999999999

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \color{blue}{\left(z \cdot y\right)} \cdot -0.16666666666666666\right), z\right), x\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(z \cdot y\right) \cdot -0.16666666666666666\right), z\right), x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+116}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq -0.0102:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.36:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))