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

Percentage Accurate: 27.5% → 30.7%

Time: 10.5s

Alternatives: 8

Speedup: 44.8×

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: 27.5% 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: 30.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_2 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+279}:\\ \;\;\;\;t\_2 \cdot \sin \left(\mathsf{fma}\left({t\_1}^{2}, \frac{t\_1}{2}, \frac{\mathsf{fma}\left(a \cdot 2, b\_m, b\_m\right) \cdot t\_m}{-16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-0.0625 \cdot \left(b\_m \cdot t\_m\right)\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (cbrt (PI)))
        (t_2 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (*
    x_s
    (if (<= (* t_2 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 1e+279)
      (*
       t_2
       (sin
        (fma
         (pow t_1 2.0)
         (/ t_1 2.0)
         (/ (* (fma (* a 2.0) b_m b_m) t_m) -16.0))))
      (* (cos (* -0.0625 (* b_m t_m))) x_m)))))

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)))) < 1.00000000000000006e279

   1. Initial program 47.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 47.0

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

   4. Applied rewrites 47.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{2} + \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right) \]

      6. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}} + \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}, \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

      11. lower-cbrt.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \frac{\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

      14. lower-cbrt.f64 47.0

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \frac{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}}{2}, \frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{-16}\right)\right) \]

   6. Applied rewrites 47.0%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{2}, \frac{\mathsf{fma}\left(a \cdot 2, b, b\right) \cdot t}{-16}\right)\right)} \]

   if 1.00000000000000006e279 < (*.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 3.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 12.4%

      \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 30.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+279}:\\ \;\;\;\;t\_1 \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \left(b\_m \cdot t\_m\right) \cdot \mathsf{fma}\left(-0.125, a, -0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-0.0625 \cdot \left(b\_m \cdot t\_m\right)\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 1e+279)
      (* t_1 (sin (fma (PI) 0.5 (* (* b_m t_m) (fma -0.125 a -0.0625)))))
      (* (cos (* -0.0625 (* b_m t_m))) x_m)))))

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)))) < 1.00000000000000006e279

   1. Initial program 47.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \frac{-1}{16} \cdot \left(b \cdot t\right)\right)} + \frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right) + \frac{-1}{16} \cdot \left(b \cdot t\right)\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right) + \frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right)} \]

      7. lower-PI.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{2}, \frac{-1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right) + \frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\left(\frac{-1}{8} \cdot a\right) \cdot \left(b \cdot t\right)} + \frac{-1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\left(b \cdot t\right) \cdot \left(\frac{-1}{8} \cdot a + \frac{-1}{16}\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\left(b \cdot t\right) \cdot \left(\frac{-1}{8} \cdot a + \frac{-1}{16}\right)}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\left(b \cdot t\right)} \cdot \left(\frac{-1}{8} \cdot a + \frac{-1}{16}\right)\right)\right) \]

      12. lower-fma.f64 46.5

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \left(b \cdot t\right) \cdot \color{blue}{\mathsf{fma}\left(-0.125, a, -0.0625\right)}\right)\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \left(b \cdot t\right) \cdot \mathsf{fma}\left(-0.125, a, -0.0625\right)\right)\right)} \]

   if 1.00000000000000006e279 < (*.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 3.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 12.4%

      \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 30.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 10^{+279}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\_m\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-0.0625 \cdot \left(b\_m \cdot t\_m\right)\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0))) t_1)
         1e+279)
      (* (* x_m (cos (* -0.0625 (* (* (fma 2.0 y 1.0) z) t_m)))) t_1)
      (* (cos (* -0.0625 (* b_m t_m))) x_m)))))

C

b_m = fabs(b);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * t_1) <= 1e+279) {
		tmp = (x_m * cos((-0.0625 * ((fma(2.0, y, 1.0) * z) * t_m)))) * t_1;
	} else {
		tmp = cos((-0.0625 * (b_m * t_m))) * x_m;
	}
	return x_s * tmp;
}

Julia

b_m = abs(b)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t_m, a, b_m)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * t_1) <= 1e+279)
		tmp = Float64(Float64(x_m * cos(Float64(-0.0625 * Float64(Float64(fma(2.0, y, 1.0) * z) * t_m)))) * t_1);
	else
		tmp = Float64(cos(Float64(-0.0625 * Float64(b_m * t_m))) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t$95$m_, a_, b$95$m_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+279], N[(N[(x$95$m * N[Cos[N[(-0.0625 * N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Cos[N[(-0.0625 * N[(b$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]]

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)))) < 1.00000000000000006e279

   1. Initial program 47.1%

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

   3. Taylor expanded in y around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 47.1

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

   5. Applied rewrites 47.1%

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

   if 1.00000000000000006e279 < (*.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 3.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 12.4%

      \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 30.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+279}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\_m\right)\right) \cdot x\_m\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\_m\right) \cdot b\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-0.0625 \cdot \left(b\_m \cdot t\_m\right)\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
       1e+279)
    (*
     (* (cos (* -0.0625 (* (* (fma 2.0 y 1.0) z) t_m))) x_m)
     (sin (fma (PI) 0.5 (* -0.0625 (* (* (fma a 2.0 1.0) t_m) b_m)))))
    (* (cos (* -0.0625 (* b_m t_m))) x_m))))

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)))) < 1.00000000000000006e279

   1. Initial program 47.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 46.6%

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

   if 1.00000000000000006e279 < (*.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 3.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 12.4%

      \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 30.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(-0.0625 \cdot \left(b\_m \cdot t\_m\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\left(x\_m \cdot t\_1\right) \cdot \cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (cos (* -0.0625 (* b_m t_m)))))
   (*
    x_s
    (if (<=
         (*
          (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
          (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
         2e+261)
      (* (* x_m t_1) (cos (* -0.0625 (* (fma 2.0 y 1.0) (* t_m z)))))
      (* t_1 x_m)))))

C

b_m = fabs(b);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	double t_1 = cos((-0.0625 * (b_m * t_m)));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+261) {
		tmp = (x_m * t_1) * cos((-0.0625 * (fma(2.0, y, 1.0) * (t_m * z))));
	} else {
		tmp = t_1 * x_m;
	}
	return x_s * tmp;
}

Julia

b_m = abs(b)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t_m, a, b_m)
	t_1 = cos(Float64(-0.0625 * Float64(b_m * t_m)))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+261)
		tmp = Float64(Float64(x_m * t_1) * cos(Float64(-0.0625 * Float64(fma(2.0, y, 1.0) * Float64(t_m * z)))));
	else
		tmp = Float64(t_1 * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t$95$m_, a_, b$95$m_] := Block[{t$95$1 = N[Cos[N[(-0.0625 * N[(b$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+261], N[(N[(x$95$m * t$95$1), $MachinePrecision] * N[Cos[N[(-0.0625 * N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(t$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * x$95$m), $MachinePrecision]]), $MachinePrecision]]

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)))) < 1.9999999999999999e261

   1. Initial program 46.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lower-cos.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-fma.f64 45.6

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

   5. Applied rewrites 45.6%

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

   7. Applied rewrites 46.1%

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

   if 1.9999999999999999e261 < (*.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. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 14.9%

      \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 29.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.0625 \cdot \left(b\_m \cdot t\_m\right)\right)\right) \cdot x\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (* x_s (* (sin (fma 0.5 (PI) (* 0.0625 (* b_m t_m)))) x_m)))

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cos-neg-rev N/A

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

   4. lower-cos.f64 N/A

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

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

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 30.6

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

5. Applied rewrites 30.6%

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

6. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 30.9%

   \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 30.5%

    \[\leadsto \sin \left(\mathsf{fma}\left(0.0625, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

11. Taylor expanded in a around 0

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

13. Applied rewrites 30.5%

    \[\leadsto \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.0625 \cdot \left(b \cdot t\right)\right)\right) \cdot x \]
14. Add Preprocessing

Alternative 7: 29.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\cos \left(-0.0625 \cdot \left(b\_m \cdot t\_m\right)\right) \cdot x\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (* x_s (* (cos (* -0.0625 (* b_m t_m))) x_m)))

C

b_m = fabs(b);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	return x_s * (cos((-0.0625 * (b_m * t_m))) * x_m);
}

Fortran

b_m =     private
t_m =     private
x\_m =     private
x\_s =     private
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_s, x_m, y, z, t_m, a, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = x_s * (cos(((-0.0625d0) * (b_m * t_m))) * x_m)
end function

Java

b_m = Math.abs(b);
t_m = Math.abs(t);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	return x_s * (Math.cos((-0.0625 * (b_m * t_m))) * x_m);
}

Python

b_m = math.fabs(b)
t_m = math.fabs(t)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t_m, a, b_m):
	return x_s * (math.cos((-0.0625 * (b_m * t_m))) * x_m)

Julia

b_m = abs(b)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t_m, a, b_m)
	return Float64(x_s * Float64(cos(Float64(-0.0625 * Float64(b_m * t_m))) * x_m))
end

MATLAB

b_m = abs(b);
t_m = abs(t);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t_m, a, b_m)
	tmp = x_s * (cos((-0.0625 * (b_m * t_m))) * x_m);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t$95$m_, a_, b$95$m_] := N[(x$95$s * N[(N[Cos[N[(-0.0625 * N[(b$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cos-neg-rev N/A

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

   4. lower-cos.f64 N/A

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

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

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 30.6

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

5. Applied rewrites 30.6%

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

6. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{-1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 30.9%

   \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
9. Add Preprocessing

Alternative 8: 30.6% accurate, 44.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(1 \cdot x\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m) :precision binary64 (* x_s (* 1.0 x_m)))

C

b_m = fabs(b);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	return x_s * (1.0 * x_m);
}

Fortran

b_m =     private
t_m =     private
x\_m =     private
x\_s =     private
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_s, x_m, y, z, t_m, a, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = x_s * (1.0d0 * x_m)
end function

Java

b_m = Math.abs(b);
t_m = Math.abs(t);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	return x_s * (1.0 * x_m);
}

Python

b_m = math.fabs(b)
t_m = math.fabs(t)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t_m, a, b_m):
	return x_s * (1.0 * x_m)

Julia

b_m = abs(b)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t_m, a, b_m)
	return Float64(x_s * Float64(1.0 * x_m))
end

MATLAB

b_m = abs(b);
t_m = abs(t);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t_m, a, b_m)
	tmp = x_s * (1.0 * x_m);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t$95$m_, a_, b$95$m_] := N[(x$95$s * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. cos-neg-rev N/A

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

   5. lower-cos.f64 N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f64 N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-fma.f64 29.4

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

5. Applied rewrites 29.4%

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

7. Applied rewrites 25.7%

   \[\leadsto \left(x \cdot \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \left(-0.0625 \cdot \frac{\mathsf{fma}\left(y \cdot y, 4, -1\right) \cdot \left(t \cdot z\right)}{\mathsf{fma}\left(2, y, -1\right)}\right) \]

8. Taylor expanded in b around 0

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

10. Applied rewrites 24.7%

    \[\leadsto \cos \left(\frac{0.0625 \cdot \left(\left(\left(\left(y \cdot y\right) \cdot 4 - 1\right) \cdot z\right) \cdot t\right)}{2 \cdot y - 1}\right) \cdot \color{blue}{x} \]

11. Taylor expanded in z around 0

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

13. Applied rewrites 30.5%

    \[\leadsto 1 \cdot x \]
14. Add Preprocessing

Developer Target 1: 30.2% 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 2025015 
(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))))