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

Percentage Accurate: 99.8% → 99.8%

Time: 11.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -0.0044:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;x \cdot 1 + \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -0.0044)
     t_0
     (if (<= x 1.3e-26) (+ (* x 1.0) (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -0.0044) {
		tmp = t_0;
	} else if (x <= 1.3e-26) {
		tmp = (x * 1.0) + (sin(y) * z);
	} 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 <= (-0.0044d0)) then
        tmp = t_0
    else if (x <= 1.3d-26) then
        tmp = (x * 1.0d0) + (sin(y) * z)
    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 <= -0.0044) {
		tmp = t_0;
	} else if (x <= 1.3e-26) {
		tmp = (x * 1.0) + (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -0.0044:
		tmp = t_0
	elif x <= 1.3e-26:
		tmp = (x * 1.0) + (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -0.0044)
		tmp = t_0;
	elseif (x <= 1.3e-26)
		tmp = Float64(Float64(x * 1.0) + Float64(sin(y) * z));
	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 <= -0.0044)
		tmp = t_0;
	elseif (x <= 1.3e-26)
		tmp = (x * 1.0) + (sin(y) * z);
	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, -0.0044], t$95$0, If[LessEqual[x, 1.3e-26], N[(N[(x * 1.0), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00440000000000000027 or 1.30000000000000005e-26 < x

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   if -0.00440000000000000027 < x < 1.30000000000000005e-26

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} + z \cdot \sin y \]
   4. Step-by-step derivation

   5. Applied rewrites 86.8%

      \[\leadsto x \cdot \color{blue}{1} + z \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0044:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;x \cdot 1 + \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -0.0044:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -0.0044) t_0 (if (<= x 1.3e-26) (fma (sin y) z (* x 1.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -0.0044) {
		tmp = t_0;
	} else if (x <= 1.3e-26) {
		tmp = fma(sin(y), z, (x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -0.0044)
		tmp = t_0;
	elseif (x <= 1.3e-26)
		tmp = fma(sin(y), z, Float64(x * 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0044], t$95$0, If[LessEqual[x, 1.3e-26], N[(N[Sin[y], $MachinePrecision] * z + N[(x * 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00440000000000000027 or 1.30000000000000005e-26 < x

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   if -0.00440000000000000027 < x < 1.30000000000000005e-26

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\sin y, z, x \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.8%

      \[\leadsto \mathsf{fma}\left(\sin y, z, x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -3.25 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -3.25e-52) t_0 (if (<= x 1.15e-40) (* (sin y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -3.25e-52) {
		tmp = t_0;
	} else if (x <= 1.15e-40) {
		tmp = sin(y) * z;
	} 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 <= (-3.25d-52)) then
        tmp = t_0
    else if (x <= 1.15d-40) then
        tmp = sin(y) * z
    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 <= -3.25e-52) {
		tmp = t_0;
	} else if (x <= 1.15e-40) {
		tmp = Math.sin(y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -3.25e-52:
		tmp = t_0
	elif x <= 1.15e-40:
		tmp = math.sin(y) * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -3.25e-52)
		tmp = t_0;
	elseif (x <= 1.15e-40)
		tmp = Float64(sin(y) * z);
	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 <= -3.25e-52)
		tmp = t_0;
	elseif (x <= 1.15e-40)
		tmp = sin(y) * z;
	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, -3.25e-52], t$95$0, If[LessEqual[x, 1.15e-40], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.25e-52 or 1.15e-40 < x

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   if -3.25e-52 < x < 1.15e-40

   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 67.7

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

   5. Applied rewrites 67.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.00041:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.00041) t_0 (if (<= y 5.7e-8) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.00041) {
		tmp = t_0;
	} else if (y <= 5.7e-8) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.00041)
		tmp = t_0;
	elseif (y <= 5.7e-8)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00041], t$95$0, If[LessEqual[y, 5.7e-8], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0999999999999999e-4 or 5.70000000000000009e-8 < 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 56.6

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

   5. Applied rewrites 56.6%

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

   if -4.0999999999999999e-4 < y < 5.70000000000000009e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.2% accurate, 30.6× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma z y x))

C

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

Julia

function code(x, y, z)
	return fma(z, y, x)
end

Wolfram

code[x_, y_, z_] := N[(z * y + x), $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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 42.3

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

5. Applied rewrites 42.3%

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

Alternative 7: 16.7% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * z), $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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 42.3

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

5. Applied rewrites 42.3%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 9.9%

   \[\leadsto y \cdot \color{blue}{z} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))