Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1

Percentage Accurate: 27.9% → 32.4%

Time: 12.7s

Alternatives: 6

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

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 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 27.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

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 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 32.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\\ t_2 := \frac{t}{16} \cdot b\\ t_3 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_3 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+259}:\\ \;\;\;\;t\_3 \cdot \left(\cos t\_1 \cdot \cos t\_2 - \sin t\_1 \cdot \sin t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ t 16.0) (* b (* a 2.0))))
        (t_2 (* (/ t 16.0) b))
        (t_3 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_3 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))) 2e+259)
     (* t_3 (- (* (cos t_1) (cos t_2)) (* (sin t_1) (sin t_2))))
     (* (sin (* (PI) 0.5)) x))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2e259

   1. Initial program 44.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16}\right) \]

      8. associate-*l/ N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{t}{16} \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(b \cdot \left(a \cdot 2 + 1\right)\right)}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \left(b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}\right)\right) \]

      12. distribute-lft-in N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(b \cdot \left(a \cdot 2\right) + b \cdot 1\right)}\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right) + \frac{t}{16} \cdot \left(b \cdot 1\right)\right)} \]

      14. *-rgt-identity N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right) + \frac{t}{16} \cdot \color{blue}{b}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right) + \color{blue}{b \cdot \frac{t}{16}}\right) \]

      16. cos-sum N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \cos \left(b \cdot \frac{t}{16}\right) - \sin \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \sin \left(b \cdot \frac{t}{16}\right)\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \cos \left(b \cdot \frac{t}{16}\right) - \sin \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \sin \left(b \cdot \frac{t}{16}\right)\right)} \]

   4. Applied rewrites 45.9%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \cos \left(\frac{t}{16} \cdot b\right) - \sin \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \sin \left(\frac{t}{16} \cdot b\right)\right)} \]

   if 2e259 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 6.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      18. lower-PI.f64 6.0

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   4. Applied rewrites 6.0%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 16.3

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \left(\cos \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \cos \left(\frac{t}{16} \cdot b\right) - \sin \left(\frac{t}{16} \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right) \cdot \sin \left(\frac{t}{16} \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{16} \cdot \left(\left(y \cdot 2\right) \cdot z\right)\\ t_2 := \frac{t}{16} \cdot z\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+272}:\\ \;\;\;\;\left(x \cdot \left(\cos t\_1 \cdot \cos t\_2 - \sin t\_1 \cdot \sin t\_2\right)\right) \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ t 16.0) (* (* y 2.0) z))) (t_2 (* (/ t 16.0) z)))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
        1e+272)
     (*
      (* x (- (* (cos t_1) (cos t_2)) (* (sin t_1) (sin t_2))))
      (sin (* 0.5 (PI))))
     (* (sin (* (PI) 0.5)) x))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.0000000000000001e272

   1. Initial program 45.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      18. lower-PI.f64 45.4

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   4. Applied rewrites 45.4%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{t \cdot \left(\left(y \cdot 2 + 1\right) \cdot z\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot \color{blue}{\left(\left(y \cdot 2 + 1\right) \cdot z\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot \color{blue}{\left(z \cdot \left(y \cdot 2 + 1\right)\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(t \cdot z\right) \cdot \left(y \cdot 2 + 1\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(t \cdot z\right)} \cdot \left(y \cdot 2 + 1\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(t \cdot z\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      8. lower-*.f64 45.1

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(t \cdot z\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right)} \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{y \cdot 2} + 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\color{blue}{2 \cdot y} + 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-fma.f64 45.1

          \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   6. Applied rewrites 45.1%

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   7. Taylor expanded in t around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-PI.f64 44.7

         \[\leadsto \left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   9. Applied rewrites 44.7%

      \[\leadsto \left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}{16}\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \mathsf{PI}\left(\right)\right)} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(x \cdot \color{blue}{\cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}{16}\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}{16}\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot z\right)}}{16}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \frac{t \cdot z}{16}\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \frac{\color{blue}{t \cdot z}}{16}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       6. associate-*l/ N/A

          \[\leadsto \left(x \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \color{blue}{\left(\frac{t}{16} \cdot z\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\color{blue}{\frac{t}{16}} \cdot z\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(\frac{t}{16} \cdot z\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{t}{16} \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \left(x \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(x \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       12. lift-fma.f64 N/A

           \[\leadsto \left(x \cdot \cos \left(\frac{t}{16} \cdot \left(z \cdot \color{blue}{\left(2 \cdot y + 1\right)}\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       13. distribute-lft-in N/A

           \[\leadsto \left(x \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(z \cdot \left(2 \cdot y\right) + z \cdot 1\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       14. distribute-lft-in N/A

           \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{t}{16} \cdot \left(z \cdot \left(2 \cdot y\right)\right) + \frac{t}{16} \cdot \left(z \cdot 1\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \left(x \cdot \cos \left(\frac{t}{16} \cdot \color{blue}{\left(\left(2 \cdot y\right) \cdot z\right)} + \frac{t}{16} \cdot \left(z \cdot 1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       16. *-rgt-identity N/A

           \[\leadsto \left(x \cdot \cos \left(\frac{t}{16} \cdot \left(\left(2 \cdot y\right) \cdot z\right) + \frac{t}{16} \cdot \color{blue}{z}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       17. cos-sum N/A

           \[\leadsto \left(x \cdot \color{blue}{\left(\cos \left(\frac{t}{16} \cdot \left(\left(2 \cdot y\right) \cdot z\right)\right) \cdot \cos \left(\frac{t}{16} \cdot z\right) - \sin \left(\frac{t}{16} \cdot \left(\left(2 \cdot y\right) \cdot z\right)\right) \cdot \sin \left(\frac{t}{16} \cdot z\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   11. Applied rewrites 45.8%

       \[\leadsto \left(x \cdot \color{blue}{\left(\cos \left(\frac{t}{16} \cdot \left(\left(y \cdot 2\right) \cdot z\right)\right) \cdot \cos \left(\frac{t}{16} \cdot z\right) - \sin \left(\frac{t}{16} \cdot \left(\left(y \cdot 2\right) \cdot z\right)\right) \cdot \sin \left(\frac{t}{16} \cdot z\right)\right)}\right) \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \]

   if 1.0000000000000001e272 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      18. lower-PI.f64 4.0

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   4. Applied rewrites 4.0%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 14.6

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+272}:\\ \;\;\;\;\left(x \cdot \left(\cos \left(\frac{t}{16} \cdot \left(\left(y \cdot 2\right) \cdot z\right)\right) \cdot \cos \left(\frac{t}{16} \cdot z\right) - \sin \left(\frac{t}{16} \cdot \left(\left(y \cdot 2\right) \cdot z\right)\right) \cdot \sin \left(\frac{t}{16} \cdot z\right)\right)\right) \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+272}:\\ \;\;\;\;\left(x \cdot \sin \left(\frac{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      1e+272)
   (*
    (* x (sin (+ (/ (* t (* z (fma 2.0 y 1.0))) -16.0) (/ (PI) 2.0))))
    (cos (* (* b t) 0.0625)))
   (* (sin (* (PI) 0.5)) x)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.0000000000000001e272

   1. Initial program 45.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \frac{1}{16}\right)} \]

      3. lower-*.f64 45.2

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   5. Applied rewrites 45.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot 0.0625\right)} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

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

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(x \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \]

   7. Applied rewrites 43.8%

      \[\leadsto \left(x \cdot \color{blue}{\sin \left(\frac{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \]

   if 1.0000000000000001e272 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      18. lower-PI.f64 4.0

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   4. Applied rewrites 4.0%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 14.6

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+272}:\\ \;\;\;\;\left(x \cdot \sin \left(\frac{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+272}:\\ \;\;\;\;\left(x \cdot \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      1e+272)
   (*
    (* x (cos (* -0.0625 (* b t))))
    (sin (fma 0.5 (PI) (* -0.0625 (* t (* z (fma 2.0 y 1.0)))))))
   (* (sin (* (PI) 0.5)) x)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.0000000000000001e272

   1. Initial program 45.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. lower-*.f64 43.3

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(t \cdot z\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

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

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(x \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      8. lower-neg.f64 44.0

         \[\leadsto \left(x \cdot \sin \left(\color{blue}{\left(-\left(t \cdot z\right) \cdot 0.0625\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   7. Applied rewrites 44.0%

      \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(-\left(t \cdot z\right) \cdot 0.0625\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

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

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\frac{-1}{16}} \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{-1}{16} \cdot \color{blue}{\left(b \cdot t\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

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

          \[\leadsto \left(x \cdot \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

   10. Applied rewrites 43.8%

       \[\leadsto \color{blue}{\left(x \cdot \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)} \]

   if 1.0000000000000001e272 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      18. lower-PI.f64 4.0

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   4. Applied rewrites 4.0%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 14.6

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+272}:\\ \;\;\;\;\left(x \cdot \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      5e+189)
   (* x (sin (fma 0.5 (PI) (* -0.0625 (* t (* z (fma 2.0 y 1.0)))))))
   (* (sin (* (PI) 0.5)) x)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000004e189

   1. Initial program 44.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. lower-*.f64 42.2

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(t \cdot z\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 42.2%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

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

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \sin \left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(x \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      8. lower-neg.f64 43.1

         \[\leadsto \left(x \cdot \sin \left(\color{blue}{\left(-\left(t \cdot z\right) \cdot 0.0625\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \left(x \cdot \color{blue}{\sin \left(\left(-\left(t \cdot z\right) \cdot 0.0625\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto x \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

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

         \[\leadsto x \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{PI}\left(\right)}, \frac{-1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \color{blue}{\frac{-1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-1}{16} \cdot \color{blue}{\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-1}{16} \cdot \left(t \cdot \color{blue}{\left(z \cdot \left(1 + 2 \cdot y\right)\right)}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-1}{16} \cdot \left(t \cdot \left(z \cdot \color{blue}{\left(2 \cdot y + 1\right)}\right)\right)\right)\right) \]

      11. lower-fma.f64 43.8

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(2, y, 1\right)}\right)\right)\right)\right) \]

   10. Applied rewrites 43.8%

       \[\leadsto \color{blue}{x \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)} \]

   if 5.0000000000000004e189 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 9.7%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      18. lower-PI.f64 9.6

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   4. Applied rewrites 9.6%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 19.2

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (sin (* (PI) 0.5)) x))

Derivation

1. Initial program 29.2%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

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

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. associate-/l* N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{t \cdot \frac{\left(a \cdot 2 + 1\right) \cdot b}{16}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{\left(a \cdot 2 + 1\right) \cdot b}{16}}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot b}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\color{blue}{b \cdot \left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\left(a \cdot 2 + 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \left(\color{blue}{a \cdot 2} + 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \color{blue}{\mathsf{fma}\left(a, 2, 1\right)}}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   18. lower-PI.f64 29.2

       \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

4. Applied rewrites 29.2%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

5. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

   3. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

   6. lower-PI.f64 31.4

      \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

7. Applied rewrites 31.4%

   \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]

8. Final simplification 31.4%

   \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x \]
9. Add Preprocessing

Developer Target 1: 30.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + pow((a * 2.0), 2.0)))));
}

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 = x * cos(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + Math.pow((a * 2.0), 2.0)))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + math.pow((a * 2.0), 2.0)))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * cos(Float64(Float64(b / 16.0) * Float64(t / Float64(Float64(1.0 - Float64(a * 2.0)) + (Float64(a * 2.0) ^ 2.0))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + ((a * 2.0) ^ 2.0)))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Cos[N[(N[(b / 16.0), $MachinePrecision] * N[(t / N[(N[(1.0 - N[(a * 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(a * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))