Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Percentage Accurate: 69.3% → 76.5%

Time: 16.4s

Alternatives: 11

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 76.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \frac{t}{3} \cdot z - \frac{\mathsf{PI}\left(\right)}{2}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \cos t\_2 - \cos y \cdot \sin t\_2\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sqrt{x}\right) \cdot \frac{{\cos y}^{2}}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - y\right)} - t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (- (* (/ t 3.0) z) (/ (PI) 2.0))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (-
      (* (* 2.0 (sqrt x)) (- (* (sin y) (cos t_2)) (* (cos y) (sin t_2))))
      t_1)
     (-
      (* (* -2.0 (sqrt x)) (/ (pow (cos y) 2.0) (sin (- (* 0.5 (PI)) y))))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 2

   1. Initial program 79.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} - \frac{a}{b \cdot 3} \]

      2. flip3-- N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\frac{{y}^{3} - {\left(\frac{z \cdot t}{3}\right)}^{3}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right)} - \frac{a}{b \cdot 3} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\frac{{y}^{3} - {\left(\frac{z \cdot t}{3}\right)}^{3}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right)} - \frac{a}{b \cdot 3} \]

      4. lower--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\color{blue}{{y}^{3} - {\left(\frac{z \cdot t}{3}\right)}^{3}}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\color{blue}{{y}^{3}} - {\left(\frac{z \cdot t}{3}\right)}^{3}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right) - \frac{a}{b \cdot 3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - \color{blue}{{\left(\frac{z \cdot t}{3}\right)}^{3}}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right) - \frac{a}{b \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{\color{blue}{z \cdot t}}{3}\right)}^{3}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right) - \frac{a}{b \cdot 3} \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{\color{blue}{t \cdot z}}{3}\right)}^{3}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right) - \frac{a}{b \cdot 3} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{\color{blue}{t \cdot z}}{3}\right)}^{3}}{y \cdot y + \left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right)}\right) - \frac{a}{b \cdot 3} \]

      10. +-commutative N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\color{blue}{\left(\frac{z \cdot t}{3} \cdot \frac{z \cdot t}{3} + y \cdot \frac{z \cdot t}{3}\right) + y \cdot y}}\right) - \frac{a}{b \cdot 3} \]

      11. distribute-rgt-out N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\color{blue}{\frac{z \cdot t}{3} \cdot \left(\frac{z \cdot t}{3} + y\right)} + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\color{blue}{\frac{z \cdot t}{3}} \cdot \left(\frac{z \cdot t}{3} + y\right) + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\frac{\color{blue}{z \cdot t}}{3} \cdot \left(\frac{z \cdot t}{3} + y\right) + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

      14. associate-/l* N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\color{blue}{\left(z \cdot \frac{t}{3}\right)} \cdot \left(\frac{z \cdot t}{3} + y\right) + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

      15. *-commutative N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\color{blue}{\left(\frac{t}{3} \cdot z\right)} \cdot \left(\frac{z \cdot t}{3} + y\right) + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

      16. +-commutative N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\left(\frac{t}{3} \cdot z\right) \cdot \color{blue}{\left(y + \frac{z \cdot t}{3}\right)} + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

      17. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{{y}^{3} - {\left(\frac{t \cdot z}{3}\right)}^{3}}{\color{blue}{\frac{t}{3} \cdot \left(z \cdot \left(y + \frac{z \cdot t}{3}\right)\right)} + y \cdot y}\right) - \frac{a}{b \cdot 3} \]

   4. Applied rewrites 32.0%

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

   6. Applied rewrites 79.3%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sin y \cdot \cos \left(\frac{t}{3} \cdot z - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos y \cdot \sin \left(\frac{t}{3} \cdot z - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} - \frac{a}{b \cdot 3} \]

   if 2 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 0.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\sin \left(\color{blue}{\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\sin \left(\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      8. lower-PI.f64 0.0

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\sin \left(\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

   6. Applied rewrites 0.0%

      \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\color{blue}{\sin \left(\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

   7. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}\right)} - \frac{a}{b \cdot 3} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}} - \frac{a}{b \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}} - \frac{a}{b \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right)} \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right)} \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{x}} \cdot 2\right) \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \color{blue}{\frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}} - \frac{a}{b \cdot 3} \]

   9. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right) \cdot \frac{\left(-1 \cdot \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right) - y\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5 - \mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right)}} - \frac{a}{b \cdot 3} \]

   10. Taylor expanded in z around 0

       \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{x} \cdot \frac{\cos y \cdot \cos \left(\mathsf{neg}\left(y\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - y\right)}\right)} - \frac{a}{b \cdot 3} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.3%

       \[\leadsto \left(-2 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{{\cos y}^{2}}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - y\right)}} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{t \cdot z}{3}\right), \sin y, \cos \left(\frac{t \cdot z}{-3}\right) \cdot \cos y\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sqrt{x}\right) \cdot \frac{{\cos y}^{2}}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - y\right)} - t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (-
      (*
       (* 2.0 (sqrt x))
       (fma (sin (/ (* t z) 3.0)) (sin y) (* (cos (/ (* t z) -3.0)) (cos y))))
      t_1)
     (-
      (* (* -2.0 (sqrt x)) (/ (pow (cos y) 2.0) (sin (- (* 0.5 (PI)) y))))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 2

   1. Initial program 79.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} - \frac{a}{b \cdot 3} \]

      2. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} - \frac{a}{b \cdot 3} \]

      3. cos-diff N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]

      4. +-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right) + \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\sin \left(\frac{z \cdot t}{3}\right) \cdot \sin y} + \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{z \cdot t}{3}\right), \sin y, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{z \cdot t}{3}\right)}, \sin y, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{\color{blue}{z \cdot t}}{3}\right), \sin y, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{\color{blue}{t \cdot z}}{3}\right), \sin y, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{\color{blue}{t \cdot z}}{3}\right), \sin y, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \]

      11. lower-sin.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{t \cdot z}{3}\right), \color{blue}{\sin y}, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{t \cdot z}{3}\right), \sin y, \color{blue}{\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y}\right) - \frac{a}{b \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sin \left(\frac{t \cdot z}{3}\right), \sin y, \color{blue}{\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y}\right) - \frac{a}{b \cdot 3} \]

   4. Applied rewrites 79.6%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{t \cdot z}{3}\right), \sin y, \cos \left(\frac{t \cdot z}{-3}\right) \cdot \cos y\right)} - \frac{a}{b \cdot 3} \]

   if 2 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 0.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\sin \left(\color{blue}{\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{\frac{3}{2}}\right)}{\sin \left(\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

      8. lower-PI.f64 0.0

         \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\sin \left(\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

   6. Applied rewrites 0.0%

      \[\leadsto \frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\color{blue}{\sin \left(\left(-\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(x + 0\right)} - \frac{a}{b \cdot 3} \]

   7. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}\right)} - \frac{a}{b \cdot 3} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}} - \frac{a}{b \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}} - \frac{a}{b \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right)} \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right)} \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{x}} \cdot 2\right) \cdot \frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{x} \cdot 2\right) \cdot \color{blue}{\frac{\cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}} - \frac{a}{b \cdot 3} \]

   9. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right) \cdot \frac{\left(-1 \cdot \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right) - y\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5 - \mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right)}} - \frac{a}{b \cdot 3} \]

   10. Taylor expanded in z around 0

       \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{x} \cdot \frac{\cos y \cdot \cos \left(\mathsf{neg}\left(y\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - y\right)}\right)} - \frac{a}{b \cdot 3} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.3%

       \[\leadsto \left(-2 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{{\cos y}^{2}}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right) - y\right)}} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \sqrt{x} \cdot 2\\ t_2 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t\_2 \leq -5000000000 \lor \neg \left(t\_2 \leq 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (sqrt x) 2.0)) (t_2 (/ a (* b 3.0))))
   (if (or (<= t_2 -5000000000.0) (not (<= t_2 1e-83)))
     (fma -0.3333333333333333 (/ a b) t_1)
     (* t_1 (cos (fma -0.3333333333333333 (* t z) y))))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = sqrt(x) * 2.0;
	double t_2 = a / (b * 3.0);
	double tmp;
	if ((t_2 <= -5000000000.0) || !(t_2 <= 1e-83)) {
		tmp = fma(-0.3333333333333333, (a / b), t_1);
	} else {
		tmp = t_1 * cos(fma(-0.3333333333333333, (t * z), y));
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(sqrt(x) * 2.0)
	t_2 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if ((t_2 <= -5000000000.0) || !(t_2 <= 1e-83))
		tmp = fma(-0.3333333333333333, Float64(a / b), t_1);
	else
		tmp = Float64(t_1 * cos(fma(-0.3333333333333333, Float64(t * z), y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5000000000.0], N[Not[LessEqual[t$95$2, 1e-83]], $MachinePrecision]], N[(-0.3333333333333333 * N[(a / b), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 * N[Cos[N[(-0.3333333333333333 * N[(t * z), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -5e9 or 1e-83 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 77.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   4. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) \cdot 2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\cos y}\right) \cdot 2 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot \cos y, 2, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y} \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{\sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}}\right) \]

      11. lower-/.f64 88.4

          \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

   7. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{a}{b}}, \sqrt{x} \cdot 2\right) \]

   if -5e9 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1e-83

   1. Initial program 56.1%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. distribute-lft-neg-in N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5000000000 \lor \neg \left(\frac{a}{b \cdot 3} \leq 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, \sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \sqrt{x} \cdot 2\\ t_2 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t\_2 \leq -5000000000 \lor \neg \left(t\_2 \leq 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (sqrt x) 2.0)) (t_2 (/ a (* b 3.0))))
   (if (or (<= t_2 -5000000000.0) (not (<= t_2 1e-83)))
     (fma -0.3333333333333333 (/ a b) t_1)
     (* t_1 (cos y)))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = sqrt(x) * 2.0;
	double t_2 = a / (b * 3.0);
	double tmp;
	if ((t_2 <= -5000000000.0) || !(t_2 <= 1e-83)) {
		tmp = fma(-0.3333333333333333, (a / b), t_1);
	} else {
		tmp = t_1 * cos(y);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	t_1 = Float64(sqrt(x) * 2.0)
	t_2 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if ((t_2 <= -5000000000.0) || !(t_2 <= 1e-83))
		tmp = fma(-0.3333333333333333, Float64(a / b), t_1);
	else
		tmp = Float64(t_1 * cos(y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5000000000.0], N[Not[LessEqual[t$95$2, 1e-83]], $MachinePrecision]], N[(-0.3333333333333333 * N[(a / b), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -5e9 or 1e-83 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 77.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   4. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) \cdot 2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\cos y}\right) \cdot 2 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot \cos y, 2, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y} \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{\sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}}\right) \]

      11. lower-/.f64 88.4

          \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

   7. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{a}{b}}, \sqrt{x} \cdot 2\right) \]

   if -5e9 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1e-83

   1. Initial program 56.1%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   4. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) \cdot 2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\cos y}\right) \cdot 2 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot \cos y, 2, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y} \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{\sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}}\right) \]

      11. lower-/.f64 55.5

          \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

   7. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5000000000 \lor \neg \left(\frac{a}{b \cdot 3} \leq 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, \sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{b}}{3} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos y)) (/ (/ a b) 3.0)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y)) - ((a / b) / 3.0);
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos(y)) - ((a / b) / 3.0d0)
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos(y)) - ((a / b) / 3.0);
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos(y)) - ((a / b) / 3.0)

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(Float64(a / b) / 3.0))
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos(y)) - ((a / b) / 3.0);
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 74.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Applied rewrites 74.3%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{a}{b \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{\color{blue}{b \cdot 3}} \]

   3. associate-/r* N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{\frac{a}{b}}}{3} \]

   5. lower-/.f64 74.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]

7. Applied rewrites 74.3%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{\frac{a}{b}}{3}} \]
8. Add Preprocessing

Alternative 6: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos y)) (/ a (* b 3.0))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y)) - (a / (b * 3.0));
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos(y)) - (a / (b * 3.0d0))
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos(y)) - (a / (b * 3.0));
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos(y)) - (a / (b * 3.0))

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(a / Float64(b * 3.0)))
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos(y)) - (a / (b * 3.0));
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 74.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Applied rewrites 74.3%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
6. Add Preprocessing

Alternative 7: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \left(\cos y \cdot 2\right) \cdot \sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (fma a (/ -0.3333333333333333 b) (* (* (cos y) 2.0) (sqrt x))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma(a, (-0.3333333333333333 / b), ((cos(y) * 2.0) * sqrt(x)));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(a, Float64(-0.3333333333333333 / b), Float64(Float64(cos(y) * 2.0) * sqrt(x)))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(a * N[(-0.3333333333333333 / b), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

4. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
6. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) \cdot 2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   3. cos-neg-rev N/A

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\cos y}\right) \cdot 2 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   4. metadata-eval N/A

      \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot \cos y, 2, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   8. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y} \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{\sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}}\right) \]

   11. lower-/.f64 74.3

       \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

7. Applied rewrites 74.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 74.3%

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.3333333333333333}{b}}, \left(\cos y \cdot 2\right) \cdot \sqrt{x}\right) \]
10. Add Preprocessing

Alternative 8: 75.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (fma (* 2.0 (cos y)) (sqrt x) (* -0.3333333333333333 (/ a b))))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma((2.0 * cos(y)), sqrt(x), (-0.3333333333333333 * (a / b)));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(Float64(2.0 * cos(y)), sqrt(x), Float64(-0.3333333333333333 * Float64(a / b)))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)} \cdot \frac{a}{b} \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \frac{-1}{3} \cdot \frac{a}{b}} \]

   3. *-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\cos y \cdot \sqrt{x}\right)} + \frac{-1}{3} \cdot \frac{a}{b} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \cos y\right) \cdot \sqrt{x}} + \frac{-1}{3} \cdot \frac{a}{b} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \cos y}, \sqrt{x}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   7. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\cos y}, \sqrt{x}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \cos y, \color{blue}{\sqrt{x}}, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}}\right) \]

   10. lower-/.f64 74.3

       \[\leadsto \mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

5. Applied rewrites 74.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]
6. Add Preprocessing

Alternative 9: 64.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \mathsf{fma}\left(-0.3333333333333333, \frac{a}{b}, \sqrt{x} \cdot 2\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (fma -0.3333333333333333 (/ a b) (* (sqrt x) 2.0)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return fma(-0.3333333333333333, (a / b), (sqrt(x) * 2.0));
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return fma(-0.3333333333333333, Float64(a / b), Float64(sqrt(x) * 2.0))
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(-0.3333333333333333 * N[(a / b), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

4. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{\left(\cos \left(\frac{t \cdot z}{3} - y\right) \cdot \cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right)\right) \cdot \left(2 \cdot {x}^{1.5}\right)}{\cos \left(\mathsf{fma}\left(\frac{t}{3}, z, y\right)\right) \cdot \left(x + 0\right)}} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
6. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \cos \left(\mathsf{neg}\left(y\right)\right)\right) \cdot 2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   3. cos-neg-rev N/A

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\cos y}\right) \cdot 2 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   4. metadata-eval N/A

      \[\leadsto \left(\sqrt{x} \cdot \cos y\right) \cdot 2 + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot \cos y, 2, \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y \cdot \sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   8. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y} \cdot \sqrt{x}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{\sqrt{x}}, 2, \frac{-1}{3} \cdot \frac{a}{b}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}}\right) \]

   11. lower-/.f64 74.3

       \[\leadsto \mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

7. Applied rewrites 74.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \sqrt{x}, 2, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{-1}{3} \cdot \frac{a}{b} + \color{blue}{2 \cdot \sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 62.1%

    \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{a}{b}}, \sqrt{x} \cdot 2\right) \]
11. Add Preprocessing

Alternative 10: 49.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \frac{-0.3333333333333333 \cdot a}{b} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (/ (* -0.3333333333333333 a) b))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return (-0.3333333333333333 * a) / b;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((-0.3333333333333333d0) * a) / b
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return (-0.3333333333333333 * a) / b;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return (-0.3333333333333333 * a) / b

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(Float64(-0.3333333333333333 * a) / b)
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (-0.3333333333333333 * a) / b;
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]

   2. lower-/.f64 47.8

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}} \]

5. Applied rewrites 47.8%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 47.8%

   \[\leadsto \frac{-0.3333333333333333 \cdot a}{\color{blue}{b}} \]
8. Add Preprocessing

Alternative 11: 49.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ -0.3333333333333333 \cdot \frac{a}{b} \end{array} \]

FPCore

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (* -0.3333333333333333 (/ a b)))

C

assert(x < y && y < z && z < t && t < a && a < b);
double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / b);
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-0.3333333333333333d0) * (a / b)
end function

Java

assert x < y && y < z && z < t && t < a && a < b;
public static double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / b);
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	return -0.3333333333333333 * (a / b)

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	return Float64(-0.3333333333333333 * Float64(a / b))
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = -0.3333333333333333 * (a / b);
end

Wolfram

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]

   2. lower-/.f64 47.8

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}} \]

5. Applied rewrites 47.8%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
6. Add Preprocessing

Developer Target 1: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t\_3 \cdot \cos \left(\frac{1}{y} - t\_1\right) - t\_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t\_1\right) \cdot t\_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024329 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -1379333748723514100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 3333333333333333/10000000000000000 z) t)))) (/ (/ a 3) b)) (if (< z 35162906135559870000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 3333333333333333/10000000000000000 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3)))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))