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

Percentage Accurate: 27.5% → 29.6%

Time: 12.9s

Alternatives: 2

Speedup: 2.3×

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: 27.5% 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

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: 29.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot t\_1 \leq 10^{+173}:\\ \;\;\;\;\cos \left(\left(b \cdot t\right) \cdot {\left(\frac{-16}{\mathsf{fma}\left(-2, a, -1\right)}\right)}^{-1}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.125 \cdot \left(\left(a \cdot t\right) \cdot b\right)\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m)))
   (*
    x_s
    (if (<= (* (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0)) t_1) 1e+173)
      (* (cos (* (* b t) (pow (/ -16.0 (fma -2.0 a -1.0)) -1.0))) t_1)
      (* (cos (* 0.125 (* (* a t) b))) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m;
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+173) {
		tmp = cos(((b * t) * pow((-16.0 / fma(-2.0, a, -1.0)), -1.0))) * t_1;
	} else {
		tmp = cos((0.125 * ((a * t) * b))) * x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+173)
		tmp = Float64(cos(Float64(Float64(b * t) * (Float64(-16.0 / fma(-2.0, a, -1.0)) ^ -1.0))) * t_1);
	else
		tmp = Float64(cos(Float64(0.125 * Float64(Float64(a * t) * b))) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

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_, a_, b_] := Block[{t$95$1 = N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], 1e+173], N[(N[Cos[N[(N[(b * t), $MachinePrecision] * N[Power[N[(-16.0 / N[(-2.0 * a + -1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Cos[N[(0.125 * N[(N[(a * t), $MachinePrecision] * b), $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)))) < 1e173

   1. Initial program 49.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 49.5%

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

   if 1e173 < (*.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 4.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 6.7

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
   7. 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. lower-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 8.6

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

   8. Applied rewrites 8.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \cos \left(\frac{1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 9.6%

       \[\leadsto \cos \left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right) \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 10.5%

       \[\leadsto \cos \left(\left(\left(a \cdot t\right) \cdot b\right) \cdot 0.125\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 10^{+173}:\\ \;\;\;\;\cos \left(\left(b \cdot t\right) \cdot {\left(\frac{-16}{\mathsf{fma}\left(-2, a, -1\right)}\right)}^{-1}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.125 \cdot \left(\left(a \cdot t\right) \cdot b\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 29.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    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
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * (cos(((b * t) * 0.0625d0)) * x_m)
end function

Java

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, double a, double b) {
	return x_s * (Math.cos(((b * t) * 0.0625)) * x_m);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, a_, b_] := N[(x$95$s * N[(N[Cos[N[(N[(b * t), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.4%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 27.2

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

5. Applied rewrites 27.2%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
7. 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. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 26.9

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

8. Applied rewrites 26.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 28.2%

    \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]
12. 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

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 2024296 
(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))))