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

Percentage Accurate: 70.7% → 78.4%

Time: 13.2s

Alternatives: 9

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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: 70.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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: 78.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t\_1 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x} \cdot \left(\sin \left(\mathsf{fma}\left(\frac{z}{3}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos y\right), 2, \left(\sqrt{x} \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right) \cdot 2\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{x}\right)}^{-1}} \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \end{array} \end{array} \]

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 (<= (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) t_1) 5e+149)
     (-
      (fma
       (* (sqrt x) (* (sin (fma (/ z 3.0) t (/ (PI) 2.0))) (cos y)))
       2.0
       (* (* (sqrt x) (* (sin (/ (* t z) 3.0)) (sin y))) 2.0))
      t_1)
     (- (* (/ 2.0 (pow (sqrt x) -1.0)) (cos y)) (/ (/ a 3.0) b)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 4.9999999999999999e149

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. cos-diff N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 81.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 81.3

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

   6. Applied rewrites 81.3%

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

   if 4.9999999999999999e149 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 45.9%

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

   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 82.7

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

   5. Applied rewrites 82.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.9

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

   7. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. exp-to-pow N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. flip-+ N/A

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

      10. sinh-cosh N/A

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

      11. sin-PI/2 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. sinh---cosh-rev N/A

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

      15. lift-sqrt.f64 N/A

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

      16. pow1/2 N/A

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

      17. exp-to-pow N/A

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

      18. lift-log.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. sinh-+-cosh-rev N/A

          \[\leadsto \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(\log x \cdot \frac{1}{2}\right)}} + \color{blue}{\left(\cosh \left(\log x \cdot \frac{1}{2}\right) + \sinh \left(\log x \cdot \frac{1}{2}\right)\right)}\right) \cdot \cos y - \frac{\frac{a}{3}}{b} \]

      21. flip-+ N/A

          \[\leadsto \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(\log x \cdot \frac{1}{2}\right)}} + \color{blue}{\frac{\cosh \left(\log x \cdot \frac{1}{2}\right) \cdot \cosh \left(\log x \cdot \frac{1}{2}\right) - \sinh \left(\log x \cdot \frac{1}{2}\right) \cdot \sinh \left(\log x \cdot \frac{1}{2}\right)}{\cosh \left(\log x \cdot \frac{1}{2}\right) - \sinh \left(\log x \cdot \frac{1}{2}\right)}}\right) \cdot \cos y - \frac{\frac{a}{3}}{b} \]

   9. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{1}{\sqrt{x}}}} \cdot \cos y - \frac{\frac{a}{3}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x} \cdot \left(\sin \left(\mathsf{fma}\left(\frac{z}{3}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos y\right), 2, \left(\sqrt{x} \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right) \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{x}\right)}^{-1}} \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t\_2 \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t\_1 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;t\_2 \cdot \left(\cos \left(\frac{t \cdot z}{-3}\right) \cdot \cos y - \sin y \cdot \sin \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{x}\right)}^{-1}} \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \end{array} \end{array} \]

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 (* 2.0 (sqrt x))))
   (if (<= (- (* t_2 (cos (- y (/ (* z t) 3.0)))) t_1) 5e+149)
     (-
      (*
       t_2
       (-
        (* (cos (/ (* t z) -3.0)) (cos y))
        (* (sin y) (sin (* -0.3333333333333333 (* t z))))))
      t_1)
     (- (* (/ 2.0 (pow (sqrt x) -1.0)) (cos y)) (/ (/ a 3.0) 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) {
	double t_1 = a / (b * 3.0);
	double t_2 = 2.0 * sqrt(x);
	double tmp;
	if (((t_2 * cos((y - ((z * t) / 3.0)))) - t_1) <= 5e+149) {
		tmp = (t_2 * ((cos(((t * z) / -3.0)) * cos(y)) - (sin(y) * sin((-0.3333333333333333 * (t * z)))))) - t_1;
	} else {
		tmp = ((2.0 / pow(sqrt(x), -1.0)) * cos(y)) - ((a / 3.0) / b);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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) :: tmp
    t_1 = a / (b * 3.0d0)
    t_2 = 2.0d0 * sqrt(x)
    if (((t_2 * cos((y - ((z * t) / 3.0d0)))) - t_1) <= 5d+149) then
        tmp = (t_2 * ((cos(((t * z) / (-3.0d0))) * cos(y)) - (sin(y) * sin(((-0.3333333333333333d0) * (t * z)))))) - t_1
    else
        tmp = ((2.0d0 / (sqrt(x) ** (-1.0d0))) * cos(y)) - ((a / 3.0d0) / b)
    end if
    code = tmp
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) {
	double t_1 = a / (b * 3.0);
	double t_2 = 2.0 * Math.sqrt(x);
	double tmp;
	if (((t_2 * Math.cos((y - ((z * t) / 3.0)))) - t_1) <= 5e+149) {
		tmp = (t_2 * ((Math.cos(((t * z) / -3.0)) * Math.cos(y)) - (Math.sin(y) * Math.sin((-0.3333333333333333 * (t * z)))))) - t_1;
	} else {
		tmp = ((2.0 / Math.pow(Math.sqrt(x), -1.0)) * Math.cos(y)) - ((a / 3.0) / b);
	}
	return tmp;
}

Python

[x, y, z, t, a, b] = sort([x, y, z, t, a, b])
def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	t_2 = 2.0 * math.sqrt(x)
	tmp = 0
	if ((t_2 * math.cos((y - ((z * t) / 3.0)))) - t_1) <= 5e+149:
		tmp = (t_2 * ((math.cos(((t * z) / -3.0)) * math.cos(y)) - (math.sin(y) * math.sin((-0.3333333333333333 * (t * z)))))) - t_1
	else:
		tmp = ((2.0 / math.pow(math.sqrt(x), -1.0)) * math.cos(y)) - ((a / 3.0) / b)
	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(a / Float64(b * 3.0))
	t_2 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (Float64(Float64(t_2 * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - t_1) <= 5e+149)
		tmp = Float64(Float64(t_2 * Float64(Float64(cos(Float64(Float64(t * z) / -3.0)) * cos(y)) - Float64(sin(y) * sin(Float64(-0.3333333333333333 * Float64(t * z)))))) - t_1);
	else
		tmp = Float64(Float64(Float64(2.0 / (sqrt(x) ^ -1.0)) * cos(y)) - Float64(Float64(a / 3.0) / b));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b = num2cell(sort([x, y, z, t, a, b])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	t_2 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (((t_2 * cos((y - ((z * t) / 3.0)))) - t_1) <= 5e+149)
		tmp = (t_2 * ((cos(((t * z) / -3.0)) * cos(y)) - (sin(y) * sin((-0.3333333333333333 * (t * z)))))) - t_1;
	else
		tmp = ((2.0 / (sqrt(x) ^ -1.0)) * cos(y)) - ((a / 3.0) / b);
	end
	tmp_2 = 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[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], 5e+149], N[(N[(t$95$2 * N[(N[(N[Cos[N[(N[(t * z), $MachinePrecision] / -3.0), $MachinePrecision]], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[N[(-0.3333333333333333 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(2.0 / N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 4.9999999999999999e149

   1. Initial program 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.3

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

   4. Applied rewrites 80.3%

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

   6. Applied rewrites 81.2%

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

   7. Taylor expanded in z around inf

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 81.6

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

   9. Applied rewrites 81.6%

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

   if 4.9999999999999999e149 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 45.9%

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

   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 82.7

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

   5. Applied rewrites 82.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.9

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

   7. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. exp-to-pow N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. flip-+ N/A

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

      10. sinh-cosh N/A

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

      11. sin-PI/2 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. sinh---cosh-rev N/A

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

      15. lift-sqrt.f64 N/A

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

      16. pow1/2 N/A

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

      17. exp-to-pow N/A

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

      18. lift-log.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. sinh-+-cosh-rev N/A

          \[\leadsto \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(\log x \cdot \frac{1}{2}\right)}} + \color{blue}{\left(\cosh \left(\log x \cdot \frac{1}{2}\right) + \sinh \left(\log x \cdot \frac{1}{2}\right)\right)}\right) \cdot \cos y - \frac{\frac{a}{3}}{b} \]

      21. flip-+ N/A

          \[\leadsto \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(\log x \cdot \frac{1}{2}\right)}} + \color{blue}{\frac{\cosh \left(\log x \cdot \frac{1}{2}\right) \cdot \cosh \left(\log x \cdot \frac{1}{2}\right) - \sinh \left(\log x \cdot \frac{1}{2}\right) \cdot \sinh \left(\log x \cdot \frac{1}{2}\right)}{\cosh \left(\log x \cdot \frac{1}{2}\right) - \sinh \left(\log x \cdot \frac{1}{2}\right)}}\right) \cdot \cos y - \frac{\frac{a}{3}}{b} \]

   9. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{1}{\sqrt{x}}}} \cdot \cos y - \frac{\frac{a}{3}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\frac{t \cdot z}{-3}\right) \cdot \cos y - \sin y \cdot \sin \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{x}\right)}^{-1}} \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x} \cdot 2, \mathsf{fma}\left(\cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right), \cos y, \sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333\right) \cdot \sin y\right), -0.3333333333333333 \cdot \frac{a}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \end{array} \end{array} \]

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
 (if (<= (cos (- y (/ (* z t) 3.0))) 0.9999)
   (fma
    (* (sqrt x) 2.0)
    (fma
     (cos (* -0.3333333333333333 (* t z)))
     (cos y)
     (* (sin (* (* t z) 0.3333333333333333)) (sin y)))
    (* -0.3333333333333333 (/ a b)))
   (- (* (* 2.0 (sqrt x)) (cos y)) (/ (/ a 3.0) 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) {
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.9999) {
		tmp = fma((sqrt(x) * 2.0), fma(cos((-0.3333333333333333 * (t * z))), cos(y), (sin(((t * z) * 0.3333333333333333)) * sin(y))), (-0.3333333333333333 * (a / b)));
	} else {
		tmp = ((2.0 * sqrt(x)) * cos(y)) - ((a / 3.0) / b);
	}
	return tmp;
}

Julia

x, y, z, t, a, b = sort([x, y, z, t, a, b])
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.9999)
		tmp = fma(Float64(sqrt(x) * 2.0), fma(cos(Float64(-0.3333333333333333 * Float64(t * z))), cos(y), Float64(sin(Float64(Float64(t * z) * 0.3333333333333333)) * sin(y))), Float64(-0.3333333333333333 * Float64(a / b)));
	else
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(Float64(a / 3.0) / b));
	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_] := If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.9999], N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Cos[N[(-0.3333333333333333 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[N[(N[(t * z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 0.99990000000000001

   1. Initial program 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. cos-diff N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 78.1%

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

   if 0.99990000000000001 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 63.7%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   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 88.4

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

   5. Applied rewrites 88.4%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

Alternative 4: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b] = \mathsf{sort}([x, y, z, t, a, b])\\ \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-93} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{-0.3333333333333333 \cdot a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)\\ \end{array} \end{array} \]

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 (/ a (* b 3.0))))
   (if (or (<= t_1 -1e-93) (not (<= t_1 2e-37)))
     (/ (* -0.3333333333333333 a) b)
     (* (* (sqrt x) 2.0) (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 = a / (b * 3.0);
	double tmp;
	if ((t_1 <= -1e-93) || !(t_1 <= 2e-37)) {
		tmp = (-0.3333333333333333 * a) / b;
	} else {
		tmp = (sqrt(x) * 2.0) * 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(a / Float64(b * 3.0))
	tmp = 0.0
	if ((t_1 <= -1e-93) || !(t_1 <= 2e-37))
		tmp = Float64(Float64(-0.3333333333333333 * a) / b);
	else
		tmp = Float64(Float64(sqrt(x) * 2.0) * 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[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-93], N[Not[LessEqual[t$95$1, 2e-37]], $MachinePrecision]], N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * 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))) < -9.999999999999999e-94 or 2.00000000000000013e-37 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 78.2%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   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 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.4%

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

   if -9.999999999999999e-94 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.00000000000000013e-37

   1. Initial program 57.6%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   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. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-cos.f64 N/A

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

   5. Applied rewrites 54.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -1 \cdot 10^{-93} \lor \neg \left(\frac{a}{b \cdot 3} \leq 2 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{-0.3333333333333333 \cdot a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 1.1× speedup

Math

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

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
 (- (* (* 2.0 (sqrt x)) (cos y)) (/ (/ a 3.0) 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 ((2.0 * sqrt(x)) * cos(y)) - ((a / 3.0) / b);
}

Fortran

NOTE: x, y, z, t, a, and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 / 3.0d0) / 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 ((2.0 * Math.sqrt(x)) * Math.cos(y)) - ((a / 3.0) / b);
}

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 / 3.0) / 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(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(Float64(a / 3.0) / 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 = ((2.0 * sqrt(x)) * cos(y)) - ((a / 3.0) / 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[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

3. Taylor expanded in 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 80.2

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

5. Applied rewrites 80.2%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
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. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 80.3

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

7. Applied rewrites 80.3%

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

Alternative 6: 77.6% 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 72.1%

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

3. Taylor expanded in 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 80.2

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

5. Applied rewrites 80.2%

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

Alternative 7: 77.6% 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 72.1%

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

3. Taylor expanded in 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 80.2

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

5. Applied rewrites 80.2%

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

Alternative 8: 51.2% 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 72.1%

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

3. Taylor expanded in 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 58.3

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

5. Applied rewrites 58.3%

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

7. Applied rewrites 58.3%

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

Alternative 9: 51.1% 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 72.1%

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

3. Taylor expanded in 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 58.3

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

5. Applied rewrites 58.3%

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

Developer Target 1: 74.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 2024364 
(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))))