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

Percentage Accurate: 27.9% → 31.6%

Time: 13.0s

Alternatives: 3

Speedup: 24.5×

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

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (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: 31.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;\cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right) \cdot \left(\cos \left(0.0625 \cdot \left(z \cdot t\_m\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 2.35e+29)
   (* (cos (* (* b t_m) 0.0625)) (* (cos (* 0.0625 (* z t_m))) x))
   (* 1.0 (* 1.0 x))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 2.35e+29) {
		tmp = cos(((b * t_m) * 0.0625)) * (cos((0.0625 * (z * t_m))) * x);
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    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
    real(8) :: tmp
    if (t_m <= 2.35d+29) then
        tmp = cos(((b * t_m) * 0.0625d0)) * (cos((0.0625d0 * (z * t_m))) * x)
    else
        tmp = 1.0d0 * (1.0d0 * x)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 2.35e+29) {
		tmp = Math.cos(((b * t_m) * 0.0625)) * (Math.cos((0.0625 * (z * t_m))) * x);
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	tmp = 0
	if t_m <= 2.35e+29:
		tmp = math.cos(((b * t_m) * 0.0625)) * (math.cos((0.0625 * (z * t_m))) * x)
	else:
		tmp = 1.0 * (1.0 * x)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 2.35e+29)
		tmp = Float64(cos(Float64(Float64(b * t_m) * 0.0625)) * Float64(cos(Float64(0.0625 * Float64(z * t_m))) * x));
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m, a, b)
	tmp = 0.0;
	if (t_m <= 2.35e+29)
		tmp = cos(((b * t_m) * 0.0625)) * (cos((0.0625 * (z * t_m))) * x);
	else
		tmp = 1.0 * (1.0 * x);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 2.35e+29], N[(N[Cos[N[(N[(b * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(0.0625 * N[(z * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.3500000000000001e29

   1. Initial program 37.5%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.6

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(t \cdot z\right)} \cdot 0.0625\right)\right) \cdot 1 \]

   8. Applied rewrites 39.6%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right) \cdot 1 \]

   9. Taylor expanded in a around 0

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

       1. lower-cos.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 39.7

          \[\leadsto \left(x \cdot \cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot 0.0625\right) \]

   11. Applied rewrites 39.7%

       \[\leadsto \left(x \cdot \cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right) \cdot \color{blue}{\cos \left(\left(b \cdot t\right) \cdot 0.0625\right)} \]

   if 2.3500000000000001e29 < t

   1. Initial program 8.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 16.0%

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;\cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot \left(\cos \left(0.0625 \cdot \left(z \cdot t\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 30.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ 1 \cdot \left(\cos \left(0.0625 \cdot \left(z \cdot t\_m\right)\right) \cdot x\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (* 1.0 (* (cos (* 0.0625 (* z t_m))) x)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return 1.0 * (cos((0.0625 * (z * t_m))) * x);
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    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
    code = 1.0d0 * (cos((0.0625d0 * (z * t_m))) * x)
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	return 1.0 * (Math.cos((0.0625 * (z * t_m))) * x);
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return 1.0 * (math.cos((0.0625 * (z * t_m))) * x)

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(1.0 * Float64(cos(Float64(0.0625 * Float64(z * t_m))) * x))
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = 1.0 * (cos((0.0625 * (z * t_m))) * x);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(1.0 * N[(N[Cos[N[(0.0625 * N[(z * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.4%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 30.8%

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

6. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 32.3

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(t \cdot z\right)} \cdot 0.0625\right)\right) \cdot 1 \]

8. Applied rewrites 32.3%

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right) \cdot 1 \]

9. Final simplification 32.3%

   \[\leadsto 1 \cdot \left(\cos \left(0.0625 \cdot \left(z \cdot t\right)\right) \cdot x\right) \]
10. Add Preprocessing

Alternative 3: 31.3% accurate, 24.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ 1 \cdot \left(1 \cdot x\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b) :precision binary64 (* 1.0 (* 1.0 x)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return 1.0 * (1.0 * x);
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    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
    code = 1.0d0 * (1.0d0 * x)
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	return 1.0 * (1.0 * x);
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return 1.0 * (1.0 * x)

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(1.0 * Float64(1.0 * x))
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = 1.0 * (1.0 * x);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.4%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 30.8%

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

6. Taylor expanded in z around 0

   \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 32.2%

   \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot 1 \]

9. Final simplification 32.2%

   \[\leadsto 1 \cdot \left(1 \cdot x\right) \]
10. Add Preprocessing

Developer Target 1: 30.9% 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

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 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 2024283 
(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))))