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

Percentage Accurate: 27.0% → 31.2%

Time: 10.6s

Alternatives: 6

Speedup: 269.0×

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: 27.0% 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: 31.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 10^{+27}:\\ \;\;\;\;\left(x\_m \cdot \sin \left(\frac{t\_m \cdot \left(z\_m \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m t_m a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0))) t_1)
         1e+27)
      (*
       (* x_m (sin (+ (/ (* t_m (* z_m (fma 2.0 y 1.0))) -16.0) (/ (PI) 2.0))))
       t_1)
      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)))) < 1e27

   1. Initial program 49.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 50.1%

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

   if 1e27 < (*.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 13.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 21.0%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+27}:\\ \;\;\;\;\left(x \cdot \sin \left(\frac{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+195}:\\ \;\;\;\;\left(x\_m \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(2, y, 1\right), \frac{t\_m \cdot z\_m}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot b\right) \cdot t\_m\right) \cdot 2}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       1e+195)
    (*
     (* x_m (sin (fma (fma 2.0 y 1.0) (/ (* t_m z_m) 16.0) (/ (PI) 2.0))))
     (cos (/ (* (* (* a b) t_m) 2.0) 16.0)))
    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)))) < 9.99999999999999977e194

   1. Initial program 49.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-fma.f64 49.6

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 49.6

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

   7. Applied rewrites 49.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 49.9

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

   9. Applied rewrites 49.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 50.8

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

   11. Applied rewrites 50.8%

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

   if 9.99999999999999977e194 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 13.7%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+195}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(2, y, 1\right), \frac{t \cdot z}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot b\right) \cdot t\right) \cdot 2}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+149}:\\ \;\;\;\;\left(x\_m \cdot \sin \left(\mathsf{fma}\left(\frac{t\_m}{16}, \mathsf{fma}\left(2, y, 1\right) \cdot z\_m, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot b\right) \cdot t\_m\right) \cdot 2}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       1e+149)
    (*
     (* x_m (sin (fma (/ t_m 16.0) (* (fma 2.0 y 1.0) z_m) (/ (PI) 2.0))))
     (cos (/ (* (* (* a b) t_m) 2.0) 16.0)))
    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)))) < 1.00000000000000005e149

   1. Initial program 49.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-fma.f64 49.6

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 49.6

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

   7. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.5

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

   9. Applied rewrites 50.5%

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

   if 1.00000000000000005e149 < (*.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 7.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 16.1%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+149}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(\frac{t}{16}, \mathsf{fma}\left(2, y, 1\right) \cdot z, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot b\right) \cdot t\right) \cdot 2}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 30.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+27}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot -0.0625\right) \cdot t\_m\right) \cdot x\_m\right) \cdot \sin \left(\mathsf{fma}\left(t\_m \cdot z\_m, -0.0625, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       1e+27)
    (*
     (* (cos (* (* (* (fma a 2.0 1.0) b) -0.0625) t_m)) x_m)
     (sin (fma (* t_m z_m) -0.0625 (/ (PI) 2.0))))
    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)))) < 1e27

   1. Initial program 49.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{x} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.3%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 49.3

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

   10. Applied rewrites 49.3%

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

   if 1e27 < (*.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 13.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 21.0%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+27}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot -0.0625\right) \cdot t\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(t \cdot z, -0.0625, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq -1 \cdot 10^{-199}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot -0.0625\right) \cdot t\_m\right) \cdot x\_m\right) \cdot \cos \left(\left(-0.0625 \cdot t\_m\right) \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       -1e-199)
    (*
     (* (cos (* (* (* (fma a 2.0 1.0) b) -0.0625) t_m)) x_m)
     (cos (* (* -0.0625 t_m) (* (fma 2.0 y 1.0) z_m))))
    x_m)))

C

z_m = fabs(z);
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_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= -1e-199) {
		tmp = (cos((((fma(a, 2.0, 1.0) * b) * -0.0625) * t_m)) * x_m) * cos(((-0.0625 * t_m) * (fma(2.0, y, 1.0) * z_m)));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= -1e-199)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(fma(a, 2.0, 1.0) * b) * -0.0625) * t_m)) * x_m) * cos(Float64(Float64(-0.0625 * t_m) * Float64(fma(2.0, y, 1.0) * z_m))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $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$95$m_, t$95$m_, a_, b_] := 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$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-199], N[(N[(N[Cos[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$m), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * N[Cos[N[(N[(-0.0625 * t$95$m), $MachinePrecision] * N[(N[(2.0 * y + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $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)))) < -9.99999999999999982e-200

   1. Initial program 47.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 44.1%

      \[\leadsto \color{blue}{x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.7%

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

   if -9.99999999999999982e-200 < (*.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 24.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 29.6%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq -1 \cdot 10^{-199}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot -0.0625\right) \cdot t\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 30.1% accurate, 269.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m t_m a b) :precision binary64 (* x_s x_m))

C

z_m = fabs(z);
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_m, double t_m, double a, double b) {
	return x_s * x_m;
}

Fortran

z_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_m, t_m, a, b)
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_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * x_m
end function

Java

z_m = Math.abs(z);
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_m, double t_m, double a, double b) {
	return x_s * x_m;
}

Python

z_m = math.fabs(z)
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_m, t_m, a, b):
	return x_s * x_m

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $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$95$m_, t$95$m_, a_, b_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 33.0%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 29.8% 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 2025026 
(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))))