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

Percentage Accurate: 27.6% → 32.0%

Time: 15.4s

Alternatives: 6

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: 27.6% 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: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\\ \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot t\_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\cos \left(\frac{\mathsf{fma}\left(b, a \cdot 2, b\right)}{\frac{16}{t}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x)))
   (if (<= (* (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0)) t_1) 5e+305)
     (* (cos (/ (fma b (* a 2.0) b) (/ 16.0 t))) t_1)
     (* 1.0 (* 1.0 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x;
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 5e+305) {
		tmp = cos((fma(b, (a * 2.0), b) / (16.0 / t))) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 5e+305)
		tmp = Float64(cos(Float64(fma(b, Float64(a * 2.0), b) / Float64(16.0 / t))) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, 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), $MachinePrecision]}, 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], 5e+305], N[(N[Cos[N[(N[(b * N[(a * 2.0), $MachinePrecision] + b), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x), $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)))) < 5.00000000000000009e305

   1. Initial program 56.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

   if 5.00000000000000009e305 < (*.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 0.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 10.2%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\cos \left(\frac{\mathsf{fma}\left(b, a \cdot 2, b\right)}{\frac{16}{t}}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(\cos \left(\left(z \cdot y\right) \cdot \left(0.125 \cdot t\right)\right) \cdot x\right) \cdot \cos \left(\frac{\mathsf{fma}\left(b, a \cdot 2, b\right)}{\frac{16}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))
       (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x))
      5e+305)
   (*
    (* (cos (* (* z y) (* 0.125 t))) x)
    (cos (/ (fma b (* a 2.0) b) (/ 16.0 t))))
   (* 1.0 (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x)) <= 5e+305) {
		tmp = (cos(((z * y) * (0.125 * t))) * x) * cos((fma(b, (a * 2.0), b) / (16.0 / t)));
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x)) <= 5e+305)
		tmp = Float64(Float64(cos(Float64(Float64(z * y) * Float64(0.125 * t))) * x) * cos(Float64(fma(b, Float64(a * 2.0), b) / Float64(16.0 / t))));
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := 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] * N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], 5e+305], N[(N[(N[Cos[N[(N[(z * y), $MachinePrecision] * N[(0.125 * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0), $MachinePrecision] + b), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(1.0 * x), $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)))) < 5.00000000000000009e305

   1. Initial program 47.1%

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

      1. lift-/.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. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 46.5

         \[\leadsto \left(x \cdot \cos \left(\left(0.125 \cdot t\right) \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(b, a \cdot 2, b\right)}{\frac{16}{t}}\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(0.125 \cdot t\right) \cdot \left(y \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(b, a \cdot 2, b\right)}{\frac{16}{t}}\right) \]

   if 5.00000000000000009e305 < (*.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 0.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 10.8%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(\cos \left(\left(z \cdot y\right) \cdot \left(0.125 \cdot t\right)\right) \cdot x\right) \cdot \cos \left(\frac{\mathsf{fma}\left(b, a \cdot 2, b\right)}{\frac{16}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 30.3% 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 2024230 
(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))))