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

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -230000000000:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -1.8e+114)
     t_0
     (if (<= y -230000000000.0)
       (* (cos y) x)
       (if (<= y 1.8)
         (fma
          (fma
           y
           (*
            y
            (fma
             y
             (* y (fma (* y y) -0.001388888888888889 0.041666666666666664))
             -0.5))
           1.0)
          x
          (* y z))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -1.8e+114) {
		tmp = t_0;
	} else if (y <= -230000000000.0) {
		tmp = cos(y) * x;
	} else if (y <= 1.8) {
		tmp = fma(fma(y, (y * fma(y, (y * fma((y * y), -0.001388888888888889, 0.041666666666666664)), -0.5)), 1.0), x, (y * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -1.8e+114)
		tmp = t_0;
	elseif (y <= -230000000000.0)
		tmp = Float64(cos(y) * x);
	elseif (y <= 1.8)
		tmp = fma(fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664)), -0.5)), 1.0), x, Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+114], t$95$0, If[LessEqual[y, -230000000000.0], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.8], N[(N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e114 or 1.80000000000000004 < y

   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 58.9

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

   5. Applied rewrites 58.9%

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

   if -1.8e114 < y < -2.3e11

   1. Initial program 99.4%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if -2.3e11 < y < 1.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)} + z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)} + z \cdot y \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right)} + z \cdot y \]

      3. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) + z \cdot y \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) + z \cdot y \]

      5. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) + z \cdot y \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) + z \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)}, 1\right) + z \cdot y \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      12. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{-1}{720} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      13. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \frac{-1}{720}\right)} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      14. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) + z \cdot y \]

      15. lower-*.f64 98.0

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) + z \cdot y \]

   8. Applied rewrites 98.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} + z \cdot y \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) + z \cdot y} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} + z \cdot y \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot x} + z \cdot y \]

      4. lower-fma.f64 98.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot y\right)} \]

   10. Applied rewrites 98.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+114}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -230000000000:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-158}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 1.0 x (* z (sin y)))))
   (if (<= z -7.8e-13) t_0 (if (<= z 7.1e-158) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(1.0, x, (z * sin(y)));
	double tmp;
	if (z <= -7.8e-13) {
		tmp = t_0;
	} else if (z <= 7.1e-158) {
		tmp = cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(1.0, x, Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -7.8e-13)
		tmp = t_0;
	elseif (z <= 7.1e-158)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 * x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e-13], t$95$0, If[LessEqual[z, 7.1e-158], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.80000000000000009e-13 or 7.10000000000000003e-158 < z

   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. Taylor expanded in y around 0

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

   7. Applied rewrites 86.3%

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

   if -7.80000000000000009e-13 < z < 7.10000000000000003e-158

   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 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-158}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -45000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -45000.0)
     t_0
     (if (<= y 3.3e-11)
       (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 <= -45000.0) {
		tmp = t_0;
	} else if (y <= 3.3e-11) {
		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 <= -45000.0)
		tmp = t_0;
	elseif (y <= 3.3e-11)
		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, -45000.0], t$95$0, If[LessEqual[y, 3.3e-11], 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 < -45000 or 3.3000000000000002e-11 < 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 48.9

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

   5. Applied rewrites 48.9%

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

   if -45000 < y < 3.3000000000000002e-11

   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 99.3

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(z \cdot y\right) \cdot -0.16666666666666666\right), z\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+37}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+248}:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+37) (* y z) (if (<= z 2.7e+248) (- (- x)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+37) {
		tmp = y * z;
	} else if (z <= 2.7e+248) {
		tmp = -(-x);
	} else {
		tmp = y * z;
	}
	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 <= (-2.1d+37)) then
        tmp = y * z
    else if (z <= 2.7d+248) then
        tmp = -(-x)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+37) {
		tmp = y * z;
	} else if (z <= 2.7e+248) {
		tmp = -(-x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+37:
		tmp = y * z
	elif z <= 2.7e+248:
		tmp = -(-x)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+37)
		tmp = Float64(y * z);
	elseif (z <= 2.7e+248)
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e+37)
		tmp = y * z;
	elseif (z <= 2.7e+248)
		tmp = -(-x);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+37], N[(y * z), $MachinePrecision], If[LessEqual[z, 2.7e+248], (-(-x)), N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e37 or 2.69999999999999989e248 < z

   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 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 40.7%

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

   if -2.1000000000000001e37 < z < 2.69999999999999989e248

   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 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{neg}\left(-1 \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.7%

       \[\leadsto -\left(-x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.4% 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 51.7

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

5. Applied rewrites 51.7%

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

Alternative 7: 39.4% accurate, 42.8× speedup

Math

\[\begin{array}{l} \\ -\left(-x\right) \end{array} \]

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(-Float64(-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. 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 51.7

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 47.5%

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

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg}\left(-1 \cdot x\right) \]
10. Step-by-step derivation

11. Applied rewrites 39.5%

    \[\leadsto -\left(-x\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))