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

Percentage Accurate: 99.8% → 99.8%

Time: 11.8s

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

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

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: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.68:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.68)
     t_0
     (if (<= y 0.92)
       (fma
        (fma
         (* y y)
         (fma
          y
          (* y (fma (* y y) -0.001388888888888889 0.041666666666666664))
          -0.5)
         1.0)
        x
        (*
         z
         (fma
          (fma (* y y) 0.008333333333333333 -0.16666666666666666)
          (* y (* y y))
          y)))
       (if (<= y 3.5e+62) t_0 (* z (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.68) {
		tmp = t_0;
	} else if (y <= 0.92) {
		tmp = fma(fma((y * y), fma(y, (y * fma((y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), x, (z * fma(fma((y * y), 0.008333333333333333, -0.16666666666666666), (y * (y * y)), y)));
	} else if (y <= 3.5e+62) {
		tmp = t_0;
	} else {
		tmp = z * sin(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.68)
		tmp = t_0;
	elseif (y <= 0.92)
		tmp = fma(fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), x, Float64(z * fma(fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), Float64(y * Float64(y * y)), y)));
	elseif (y <= 3.5e+62)
		tmp = t_0;
	else
		tmp = Float64(z * sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.68], t$95$0, If[LessEqual[y, 0.92], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(z * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+62], t$95$0, N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.680000000000000049 or 0.92000000000000004 < y < 3.49999999999999984e62

   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 63.4

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

   5. Applied rewrites 63.4%

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

   if -0.680000000000000049 < y < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \cos y + \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto x \cdot \cos y + \color{blue}{\left(z \cdot y + \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) \cdot \left(y \cdot {y}^{2}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{6} \cdot z\right)} \cdot \left(y \cdot {y}^{2}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot z} + \frac{-1}{6} \cdot z\right) \cdot \left(y \cdot {y}^{2}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)} \cdot \left(y \cdot {y}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{z \cdot \left(\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)\right)}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto x \cdot \cos y + \color{blue}{z \cdot \left(y + \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \cos y + \color{blue}{z \cdot \left(y + \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \cos y + z \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right) + y\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot \cos y + z \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}, y \cdot {y}^{2}, y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto x \cdot \cos y + \color{blue}{z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      3. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      5. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      14. lower-*.f64 99.6

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot x} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      4. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)} \]

   if 3.49999999999999984e62 < 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 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.68:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+62}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-178}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.2e-59)
   (fma 1.0 x (* z (sin y)))
   (if (<= z 2.7e-178) (* (cos y) x) (fma (sin y) z (* x 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.2e-59) {
		tmp = fma(1.0, x, (z * sin(y)));
	} else if (z <= 2.7e-178) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(sin(y), z, (x * 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.2e-59)
		tmp = fma(1.0, x, Float64(z * sin(y)));
	elseif (z <= 2.7e-178)
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(sin(y), z, Float64(x * 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.2e-59], N[(1.0 * x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e-178], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z + N[(x * 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.19999999999999918e-59

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 83.1%

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

   if -9.19999999999999918e-59 < z < 2.70000000000000009e-178

   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 92.0

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

   5. Applied rewrites 92.0%

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

   if 2.70000000000000009e-178 < z

   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. 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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 86.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-178}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x \cdot 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-178}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 1.0 x (* z (sin y)))))
   (if (<= z -9.2e-59) t_0 (if (<= z 2.7e-178) (* (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 <= -9.2e-59) {
		tmp = t_0;
	} else if (z <= 2.7e-178) {
		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 <= -9.2e-59)
		tmp = t_0;
	elseif (z <= 2.7e-178)
		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, -9.2e-59], t$95$0, If[LessEqual[z, 2.7e-178], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.19999999999999918e-59 or 2.70000000000000009e-178 < 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 85.0%

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

   if -9.19999999999999918e-59 < z < 2.70000000000000009e-178

   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 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-178}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, z \cdot \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.68:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.68)
     t_0
     (if (<= y 0.92)
       (fma
        (fma
         (* y y)
         (fma
          y
          (* y (fma (* y y) -0.001388888888888889 0.041666666666666664))
          -0.5)
         1.0)
        x
        (*
         z
         (fma
          (fma (* y y) 0.008333333333333333 -0.16666666666666666)
          (* y (* y y))
          y)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.68) {
		tmp = t_0;
	} else if (y <= 0.92) {
		tmp = fma(fma((y * y), fma(y, (y * fma((y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), x, (z * fma(fma((y * y), 0.008333333333333333, -0.16666666666666666), (y * (y * y)), y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.68)
		tmp = t_0;
	elseif (y <= 0.92)
		tmp = fma(fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), x, Float64(z * fma(fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), Float64(y * Float64(y * y)), y)));
	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, -0.68], t$95$0, If[LessEqual[y, 0.92], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(z * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.680000000000000049 or 0.92000000000000004 < 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 50.2

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

   5. Applied rewrites 50.2%

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

   if -0.680000000000000049 < y < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \cos y + \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto x \cdot \cos y + \color{blue}{\left(z \cdot y + \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) \cdot \left(y \cdot {y}^{2}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{6} \cdot z\right)} \cdot \left(y \cdot {y}^{2}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot z} + \frac{-1}{6} \cdot z\right) \cdot \left(y \cdot {y}^{2}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{\left(z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)} \cdot \left(y \cdot {y}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \cos y + \left(z \cdot y + \color{blue}{z \cdot \left(\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)\right)}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto x \cdot \cos y + \color{blue}{z \cdot \left(y + \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \cos y + \color{blue}{z \cdot \left(y + \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \cos y + z \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right) \cdot \left(y \cdot {y}^{2}\right) + y\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot \cos y + z \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}, y \cdot {y}^{2}, y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto x \cdot \cos y + \color{blue}{z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      3. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      5. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)}, 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      14. lower-*.f64 99.6

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot x} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \]

      4. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.68:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.7% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+114}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 6.5e+114) (* x 1.0) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.5e+114) {
		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 <= 6.5d+114) 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 <= 6.5e+114) {
		tmp = x * 1.0;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 6.5e+114:
		tmp = x * 1.0
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 6.5e+114)
		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 <= 6.5e+114)
		tmp = x * 1.0;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 6.5e+114], N[(x * 1.0), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.5000000000000001e114

   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. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\cos y + \frac{z \cdot \sin y}{x}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 95.7

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

   9. Applied rewrites 95.7%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 40.5%

       \[\leadsto x \cdot 1 \]

   if 6.5000000000000001e114 < 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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.5%

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

Alternative 7: 51.9% 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 52.0

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

5. Applied rewrites 52.0%

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

Alternative 8: 16.8% 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 52.0

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 18.6%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))