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

Percentage Accurate: 28.2% → 32.4%

Time: 14.4s

Alternatives: 3

Speedup: 24.5×

• Could not converge on a ground truth

  1. x = 7.898791041367994e-218
  2. y = 2.423793175599403e-32
  3. z = 7.234509885009176e+247
  4. t = 2.937294930346515e+246
  5. a = 0.1941754916610055
  6. b = 8.979190551126094e+54

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: 28.2% 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: 32.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\right) \leq 10^{+150}:\\ \;\;\;\;\left(\cos \left(\frac{\left(\left(\left(\frac{-1}{\mathsf{fma}\left(-2, y, 1\right)} \cdot \mathsf{fma}\left(2, y, -1\right)\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) \cdot t}{16}\right) \cdot x\_m\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))))
   (*
    x_s
    (if (<=
         (* t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m))
         1e+150)
      (*
       (*
        (cos
         (/
          (*
           (*
            (* (* (/ -1.0 (fma -2.0 y 1.0)) (fma 2.0 y -1.0)) (fma 2.0 y 1.0))
            z)
           t)
          16.0))
        x_m)
       t_1)
      (* 1.0 (* 1.0 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((((b * ((a * 2.0) + 1.0)) * t) / 16.0));
	double tmp;
	if ((t_1 * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 1e+150) {
		tmp = (cos(((((((-1.0 / fma(-2.0, y, 1.0)) * fma(2.0, y, -1.0)) * fma(2.0, y, 1.0)) * z) * t) / 16.0)) * x_m) * t_1;
	} else {
		tmp = 1.0 * (1.0 * 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 = cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0))
	tmp = 0.0
	if (Float64(t_1 * Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)) <= 1e+150)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(Float64(Float64(Float64(-1.0 / fma(-2.0, y, 1.0)) * fma(2.0, y, -1.0)) * fma(2.0, y, 1.0)) * z) * t) / 16.0)) * x_m) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * 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[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(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]), $MachinePrecision], 1e+150], N[(N[(N[Cos[N[(N[(N[(N[(N[(N[(-1.0 / N[(-2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $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)))) < 9.99999999999999981e149

   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

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. difference-of-sqr-1 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. frac-2neg N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. metadata-eval N/A

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

      23. distribute-neg-in N/A

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

      24. metadata-eval N/A

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

      25. +-commutative N/A

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

   4. Applied rewrites 49.2%

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

   if 9.99999999999999981e149 < (*.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 5.4%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 8.8%

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

   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 13.3%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 10^{+150}:\\ \;\;\;\;\left(\cos \left(\frac{\left(\left(\left(\frac{-1}{\mathsf{fma}\left(-2, y, 1\right)} \cdot \mathsf{fma}\left(2, y, -1\right)\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) \cdot t}{16}\right) \cdot x\right) \cdot \cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 30.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(8, a, 4\right), a, 2\right), a, 1\right)} \cdot b\right) \cdot t}{16}\right) \cdot \left(1 \cdot x\_m\right)\right) \end{array} \]

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 (/ (* (* (/ 1.0 (fma (fma (fma 8.0 a 4.0) a 2.0) a 1.0)) b) t) 16.0))
   (* 1.0 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(((((1.0 / fma(fma(fma(8.0, a, 4.0), a, 2.0), a, 1.0)) * b) * t) / 16.0)) * (1.0 * 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(Float64(Float64(1.0 / fma(fma(fma(8.0, a, 4.0), a, 2.0), a, 1.0)) * b) * t) / 16.0)) * Float64(1.0 * 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[(N[(N[(1.0 / N[(N[(N[(8.0 * a + 4.0), $MachinePrecision] * a + 2.0), $MachinePrecision] * a + 1.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in t 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 27.2%

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

7. Applied rewrites 20.3%

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

8. Taylor expanded in a around 0

   \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\color{blue}{1 + a \cdot \left(2 + a \cdot \left(4 + 8 \cdot a\right)\right)}} \cdot b\right) \cdot t}{16}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\color{blue}{a \cdot \left(2 + a \cdot \left(4 + 8 \cdot a\right)\right) + 1}} \cdot b\right) \cdot t}{16}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\color{blue}{\left(2 + a \cdot \left(4 + 8 \cdot a\right)\right) \cdot a} + 1} \cdot b\right) \cdot t}{16}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(2 + a \cdot \left(4 + 8 \cdot a\right), a, 1\right)}} \cdot b\right) \cdot t}{16}\right) \]

   4. +-commutative N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\color{blue}{a \cdot \left(4 + 8 \cdot a\right) + 2}, a, 1\right)} \cdot b\right) \cdot t}{16}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\color{blue}{\left(4 + 8 \cdot a\right) \cdot a} + 2, a, 1\right)} \cdot b\right) \cdot t}{16}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(4 + 8 \cdot a, a, 2\right)}, a, 1\right)} \cdot b\right) \cdot t}{16}\right) \]

   7. +-commutative N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{8 \cdot a + 4}, a, 2\right), a, 1\right)} \cdot b\right) \cdot t}{16}\right) \]

   8. lower-fma.f64 29.1

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(8, a, 4\right)}, a, 2\right), a, 1\right)} \cdot b\right) \cdot t}{16}\right) \]

10. Applied rewrites 29.1%

    \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(8, a, 4\right), a, 2\right), a, 1\right)}} \cdot b\right) \cdot t}{16}\right) \]

11. Final simplification 29.1%

    \[\leadsto \cos \left(\frac{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(8, a, 4\right), a, 2\right), a, 1\right)} \cdot b\right) \cdot t}{16}\right) \cdot \left(1 \cdot x\right) \]
12. Add Preprocessing

Alternative 3: 31.3% accurate, 24.5× speedup

Math

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

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 (* 1.0 (* 1.0 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 * (1.0 * (1.0 * 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 * (1.0d0 * (1.0d0 * 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 * (1.0 * (1.0 * 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 * (1.0 * (1.0 * 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(1.0 * Float64(1.0 * 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 * (1.0 * (1.0 * 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[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in t 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 27.2%

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

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 29.0%

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

9. Final simplification 29.0%

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

Developer Target 1: 30.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024235 
(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))))