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

Percentage Accurate: 28.3% → 32.6%

Time: 18.4s

Alternatives: 2

Speedup: 225.0×

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

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: 28.3% 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: 32.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := 1 + \mathsf{fma}\left(y, 2, 1\right) \cdot \left(z\_m \cdot \left(t\_m \cdot 0.0625\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{t\_m \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;x\_m \cdot \left(\left(\cos t\_1 \cdot \cos 1 + \sin t\_1 \cdot \sin 1\right) \cdot \cos \left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \frac{t\_m}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

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 (+ 1.0 (* (fma y 2.0 1.0) (* z_m (* t_m 0.0625))))))
   (*
    x_s
    (if (<=
         (*
          (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
          (cos (/ (* t_m (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
         2e+253)
      (*
       x_m
       (*
        (+ (* (cos t_1) (cos 1.0)) (* (sin t_1) (sin 1.0)))
        (cos (* (* b (fma a 2.0 1.0)) (/ t_m 16.0)))))
      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 t_1 = 1.0 + (fma(y, 2.0, 1.0) * (z_m * (t_m * 0.0625)));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+253) {
		tmp = x_m * (((cos(t_1) * cos(1.0)) + (sin(t_1) * sin(1.0))) * cos(((b * fma(a, 2.0, 1.0)) * (t_m / 16.0))));
	} 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)
	t_1 = Float64(1.0 + Float64(fma(y, 2.0, 1.0) * Float64(z_m * Float64(t_m * 0.0625))))
	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(t_m * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+253)
		tmp = Float64(x_m * Float64(Float64(Float64(cos(t_1) * cos(1.0)) + Float64(sin(t_1) * sin(1.0))) * cos(Float64(Float64(b * fma(a, 2.0, 1.0)) * Float64(t_m / 16.0)))));
	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_] := Block[{t$95$1 = N[(1.0 + N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z$95$m * N[(t$95$m * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t$95$m * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+253], N[(x$95$m * N[(N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[1.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision]), $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)))) < 1.9999999999999999e253

   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. Step-by-step derivation

      1. associate-*l* 49.0%

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

      2. *-commutative 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/l* 49.0%

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

      5. fma-define 49.0%

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

      6. associate-/l* 49.0%

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

      7. fma-define 49.0%

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

   3. Simplified 49.0%

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

      1. associate-*r* 49.0%

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

      2. div-inv 49.0%

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

      3. metadata-eval 49.0%

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

      4. expm1-log1p-u 45.8%

         \[\leadsto x \cdot \left(\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)\right)\right)} \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

   6. Applied egg-rr 45.8%

      \[\leadsto x \cdot \left(\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)\right)\right)} \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 45.8%

         \[\leadsto x \cdot \left(\cos \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)} - 1\right)} \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

      2. cos-diff 45.3%

         \[\leadsto x \cdot \left(\color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)}\right) \cdot \sin 1\right)} \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

      3. log1p-undefine 45.3%

         \[\leadsto x \cdot \left(\left(\cos \left(e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)}}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)}\right) \cdot \sin 1\right) \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

      4. rem-exp-log 44.5%

         \[\leadsto x \cdot \left(\left(\cos \color{blue}{\left(1 + \mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)} \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)}\right) \cdot \sin 1\right) \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

      5. *-commutative 44.5%

         \[\leadsto x \cdot \left(\left(\cos \left(1 + \mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \color{blue}{\left(0.0625 \cdot t\right)}\right)\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)}\right) \cdot \sin 1\right) \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \frac{t}{16}\right)\right) \]

   8. Applied egg-rr 49.0%

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

   if 1.9999999999999999e253 < (*.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 0.9%

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

      1. associate-*l* 0.9%

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

      2. *-commutative 0.9%

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

      3. *-commutative 0.9%

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

      4. associate-/l* 0.9%

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

      5. fma-define 0.9%

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

      6. associate-/l* 0.9%

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

      7. fma-define 0.9%

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

   3. Simplified 0.9%

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

   5. Taylor expanded in t around 0 9.9%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;x \cdot \left(\left(\cos \left(1 + \mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \cos 1 + \sin \left(1 + \mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \sin 1\right) \cdot \cos \left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \frac{t}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.4% accurate, 225.0× 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 = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z_m, t_m, a, b)
    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 26.3%

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

   1. associate-*l* 26.3%

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

   2. *-commutative 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. fma-define 26.3%

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

   6. associate-/l* 26.3%

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

   7. fma-define 26.3%

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

3. Simplified 26.3%

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

5. Taylor expanded in t around 0 29.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 31.0% accurate, 1.0× 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 2024144 
(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))))