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

Percentage Accurate: 99.8% → 99.8%

Time: 13.6s

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: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), x\right) + t\_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)))
   (if (<= z -1.6e-60)
     (+ (fma (* y y) (* x (fma y (* y 0.041666666666666664) -0.5)) x) t_0)
     (if (<= z 7e+37) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double tmp;
	if (z <= -1.6e-60) {
		tmp = fma((y * y), (x * fma(y, (y * 0.041666666666666664), -0.5)), x) + t_0;
	} else if (z <= 7e+37) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	tmp = 0.0
	if (z <= -1.6e-60)
		tmp = Float64(fma(Float64(y * y), Float64(x * fma(y, Float64(y * 0.041666666666666664), -0.5)), x) + t_0);
	elseif (z <= 7e+37)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.6e-60], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(y * N[(y * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[z, 7e+37], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6000000000000001e-60

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -1.6000000000000001e-60 < z < 7e37

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 88.8

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

   5. Applied rewrites 88.8%

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

   if 7e37 < z

   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 70.3

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

   5. Applied rewrites 70.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), x\right) + \sin y \cdot z\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)))
   (if (<= z -3.8e+66) t_0 (if (<= z 7e+37) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double tmp;
	if (z <= -3.8e+66) {
		tmp = t_0;
	} else if (z <= 7e+37) {
		tmp = x * cos(y);
	} 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 = sin(y) * z
    if (z <= (-3.8d+66)) then
        tmp = t_0
    else if (z <= 7d+37) then
        tmp = x * cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) * z;
	double tmp;
	if (z <= -3.8e+66) {
		tmp = t_0;
	} else if (z <= 7e+37) {
		tmp = x * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * z
	tmp = 0
	if z <= -3.8e+66:
		tmp = t_0
	elif z <= 7e+37:
		tmp = x * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	tmp = 0.0
	if (z <= -3.8e+66)
		tmp = t_0;
	elseif (z <= 7e+37)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * z;
	tmp = 0.0;
	if (z <= -3.8e+66)
		tmp = t_0;
	elseif (z <= 7e+37)
		tmp = x * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e+66], t$95$0, If[LessEqual[z, 7e+37], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000002e66 or 7e37 < z

   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 69.7

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

   5. Applied rewrites 69.7%

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

   if -3.8000000000000002e66 < z < 7e37

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+66}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.061:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot z\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.061)
     t_0
     (if (<= y 3.6e-22)
       (fma y (fma y (* -0.16666666666666666 (* y z)) z) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.061) {
		tmp = t_0;
	} else if (y <= 3.6e-22) {
		tmp = fma(y, fma(y, (-0.16666666666666666 * (y * z)), 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 <= -0.061)
		tmp = t_0;
	elseif (y <= 3.6e-22)
		tmp = fma(y, fma(y, Float64(-0.16666666666666666 * Float64(y * z)), 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, -0.061], t$95$0, If[LessEqual[y, 3.6e-22], N[(y * N[(y * N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.060999999999999999 or 3.5999999999999998e-22 < 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 53.1

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

   5. Applied rewrites 53.1%

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

   if -0.060999999999999999 < y < 3.5999999999999998e-22

   1. Initial program 100.0%

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

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \cos y - z \cdot \sin y}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \cos y - z \cdot \sin y}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{x \cdot \cos y - z \cdot \sin y}}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y + z \cdot \sin y}}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y + z \cdot \sin y}}} \]

      8. lower-/.f64 99.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \cos y + z \cdot \sin y}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y + z \cdot \sin y}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y} + z \cdot \sin y}} \]

      11. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)} \]
   6. 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. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.061:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot z\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+170}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+170) (* y z) (if (<= z 3.5e+202) (* x 1.0) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+170) {
		tmp = y * z;
	} else if (z <= 3.5e+202) {
		tmp = x * 1.0;
	} 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 <= (-2.6d+170)) then
        tmp = y * z
    else if (z <= 3.5d+202) then
        tmp = x * 1.0d0
    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 <= -2.6e+170) {
		tmp = y * z;
	} else if (z <= 3.5e+202) {
		tmp = x * 1.0;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.6e+170:
		tmp = y * z
	elif z <= 3.5e+202:
		tmp = x * 1.0
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+170)
		tmp = Float64(y * z);
	elseif (z <= 3.5e+202)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.6e+170)
		tmp = y * z;
	elseif (z <= 3.5e+202)
		tmp = x * 1.0;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+170], N[(y * z), $MachinePrecision], If[LessEqual[z, 3.5e+202], N[(x * 1.0), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5999999999999998e170 or 3.49999999999999987e202 < z

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 40.3%

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

   if -2.5999999999999998e170 < z < 3.49999999999999987e202

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.8%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.3% 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 52.9

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

5. Applied rewrites 52.9%

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

Alternative 7: 17.1% 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 52.9

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

5. Applied rewrites 52.9%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 16.8%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))