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

Percentage Accurate: 99.8% → 99.8%

Time: 13.5s

Alternatives: 8

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 99.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.059:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+226}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0019)
     t_0
     (if (<= y 0.059)
       (fma y (fma y (fma x -0.5 (* (* y z) -0.16666666666666666)) z) x)
       (if (<= y 8.2e+226) (* (sin y) z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.0019) {
		tmp = t_0;
	} else if (y <= 0.059) {
		tmp = fma(y, fma(y, fma(x, -0.5, ((y * z) * -0.16666666666666666)), z), x);
	} else if (y <= 8.2e+226) {
		tmp = 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 (y <= -0.0019)
		tmp = t_0;
	elseif (y <= 0.059)
		tmp = fma(y, fma(y, fma(x, -0.5, Float64(Float64(y * z) * -0.16666666666666666)), z), x);
	elseif (y <= 8.2e+226)
		tmp = Float64(sin(y) * z);
	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, -0.0019], t$95$0, If[LessEqual[y, 0.059], N[(y * N[(y * N[(x * -0.5 + N[(N[(y * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 8.2e+226], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0019 or 8.19999999999999971e226 < y

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 59.1

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

   5. Simplified 59.1%

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

   if -0.0019 < y < 0.058999999999999997

   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. Simplified 99.8%

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

   if 0.058999999999999997 < y < 8.19999999999999971e226

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 64.3

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

   5. Simplified 64.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0019:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.059:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+226}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+151}:\\ \;\;\;\;x + \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -0.0065) t_0 (if (<= x 7.8e+151) (+ x (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -0.0065) {
		tmp = t_0;
	} else if (x <= 7.8e+151) {
		tmp = x + (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 <= (-0.0065d0)) then
        tmp = t_0
    else if (x <= 7.8d+151) then
        tmp = x + (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 <= -0.0065) {
		tmp = t_0;
	} else if (x <= 7.8e+151) {
		tmp = x + (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 <= -0.0065:
		tmp = t_0
	elif x <= 7.8e+151:
		tmp = x + (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 <= -0.0065)
		tmp = t_0;
	elseif (x <= 7.8e+151)
		tmp = Float64(x + 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 <= -0.0065)
		tmp = t_0;
	elseif (x <= 7.8e+151)
		tmp = x + (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, -0.0065], t$95$0, If[LessEqual[x, 7.8e+151], N[(x + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0064999999999999997 or 7.79999999999999952e151 < 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 93.3

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

   5. Simplified 93.3%

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

   if -0.0064999999999999997 < x < 7.79999999999999952e151

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   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. Simplified 88.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+151}:\\ \;\;\;\;x + \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -0.0065) t_0 (if (<= x 7.8e+151) (fma (sin y) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -0.0065) {
		tmp = t_0;
	} else if (x <= 7.8e+151) {
		tmp = fma(sin(y), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -0.0065)
		tmp = t_0;
	elseif (x <= 7.8e+151)
		tmp = fma(sin(y), z, 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[x, -0.0065], t$95$0, If[LessEqual[x, 7.8e+151], N[(N[Sin[y], $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0064999999999999997 or 7.79999999999999952e151 < 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 93.3

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

   5. Simplified 93.3%

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

   if -0.0064999999999999997 < x < 7.79999999999999952e151

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   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. Simplified 88.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0019)
     t_0
     (if (<= y 1.25e-16)
       (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 = x * cos(y);
	double tmp;
	if (y <= -0.0019) {
		tmp = t_0;
	} else if (y <= 1.25e-16) {
		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(x * cos(y))
	tmp = 0.0
	if (y <= -0.0019)
		tmp = t_0;
	elseif (y <= 1.25e-16)
		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[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0019], t$95$0, If[LessEqual[y, 1.25e-16], 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 2 regimes

2. if y < -0.0019 or 1.2500000000000001e-16 < y

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 50.2

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

   5. Simplified 50.2%

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

   if -0.0019 < y < 1.2500000000000001e-16

   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. Simplified 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 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0019:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% 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 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]
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 59.0

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

5. Simplified 59.0%

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

Alternative 7: 16.9% 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 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]
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 59.0

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

5. Simplified 59.0%

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

6. Taylor expanded in z around inf

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

   1. lower-*.f64 16.4

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

8. Simplified 16.4%

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

Alternative 8: 4.8% accurate, 71.3× 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 Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]
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 59.0

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

5. Simplified 59.0%

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

   1. flip-+ N/A

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

   2. lift-*.f64 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

7. Applied egg-rr 21.2%

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

8. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{y \cdot \left(-1 \cdot \left(x \cdot \left(y \cdot {z}^{2}\right)\right) - {x}^{2} \cdot z\right) - -1 \cdot {x}^{3}}}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\color{blue}{y \cdot \left(-1 \cdot \left(x \cdot \left(y \cdot {z}^{2}\right)\right) - {x}^{2} \cdot z\right) + \left(\mathsf{neg}\left(-1 \cdot {x}^{3}\right)\right)}}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{y \cdot \left(-1 \cdot \left(x \cdot \left(y \cdot {z}^{2}\right)\right) - {x}^{2} \cdot z\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{3}\right)\right)}\right)\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   3. remove-double-neg N/A

      \[\leadsto \frac{y \cdot \left(-1 \cdot \left(x \cdot \left(y \cdot {z}^{2}\right)\right) - {x}^{2} \cdot z\right) + \color{blue}{{x}^{3}}}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, -1 \cdot \left(x \cdot \left(y \cdot {z}^{2}\right)\right) - {x}^{2} \cdot z, {x}^{3}\right)}}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot {z}^{2}\right)\right) + \left(\mathsf{neg}\left({x}^{2} \cdot z\right)\right)}, {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(y \cdot {z}^{2}\right)\right)\right)} + \left(\mathsf{neg}\left({x}^{2} \cdot z\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   7. distribute-neg-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(\left(x \cdot \left(y \cdot {z}^{2}\right) + {x}^{2} \cdot z\right)\right)}, {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   8. lower-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(\left(x \cdot \left(y \cdot {z}^{2}\right) + {x}^{2} \cdot z\right)\right)}, {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(x, y \cdot {z}^{2}, {x}^{2} \cdot z\right)}\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, \color{blue}{y \cdot {z}^{2}}, {x}^{2} \cdot z\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, y \cdot \color{blue}{\left(z \cdot z\right)}, {x}^{2} \cdot z\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, y \cdot \color{blue}{\left(z \cdot z\right)}, {x}^{2} \cdot z\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, y \cdot \left(z \cdot z\right), \color{blue}{z \cdot {x}^{2}}\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, y \cdot \left(z \cdot z\right), \color{blue}{z \cdot {x}^{2}}\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, y \cdot \left(z \cdot z\right), z \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{neg}\left(\mathsf{fma}\left(x, y \cdot \left(z \cdot z\right), z \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), {x}^{3}\right)}{\left(y \cdot z - x\right) \cdot \left(y \cdot z - x\right)} \]

   17. cube-mult N/A

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

   18. unpow2 N/A

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

10. Simplified 14.4%

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

11. Taylor expanded in y around inf

    \[\leadsto \color{blue}{-1 \cdot x} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

    2. lower-neg.f64 4.5

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

13. Simplified 4.5%

    \[\leadsto \color{blue}{-x} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024210 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))