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

Percentage Accurate: 27.0% → 30.0%

Time: 14.2s

Alternatives: 6

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: 27.0% 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: 30.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.7 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\_m\right) \cdot t\right) \cdot -0.0625\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(b \cdot t\right) \cdot -0.0625\right) \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b)
 :precision binary64
 (if (<= z_m 4.7e-6)
   (* (cos (* (* (* (fma 2.0 y 1.0) z_m) t) -0.0625)) x)
   (* (cos (* (* b t) -0.0625)) x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	double tmp;
	if (z_m <= 4.7e-6) {
		tmp = cos((((fma(2.0, y, 1.0) * z_m) * t) * -0.0625)) * x;
	} else {
		tmp = cos(((b * t) * -0.0625)) * x;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	tmp = 0.0
	if (z_m <= 4.7e-6)
		tmp = Float64(cos(Float64(Float64(Float64(fma(2.0, y, 1.0) * z_m) * t) * -0.0625)) * x);
	else
		tmp = Float64(cos(Float64(Float64(b * t) * -0.0625)) * x);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := If[LessEqual[z$95$m, 4.7e-6], N[(N[Cos[N[(N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(N[Cos[N[(N[(b * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.69999999999999989e-6

   1. Initial program 32.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 32.9

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

   5. Applied rewrites 32.9%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

   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. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 33.0

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

   8. Applied rewrites 33.0%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. cos-neg-rev N/A

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

       4. lower-cos.f64 N/A

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

       5. distribute-lft-neg-in N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-fma.f64 34.4

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

   11. Applied rewrites 34.4%

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

   if 4.69999999999999989e-6 < z

   1. Initial program 18.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 18.8

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

   5. Applied rewrites 18.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

   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. cos-neg-rev N/A

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

      4. lower-cos.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 24.7

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

   8. Applied rewrites 24.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 25.5%

       \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.7 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot -0.0625\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(b \cdot t\right) \cdot -0.0625\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 28.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \cos \left(\left(\left(t \cdot 0.0625\right) \cdot \left(a \cdot 2\right)\right) \cdot b\right) \cdot \left(1 \cdot x\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b)
 :precision binary64
 (* (cos (* (* (* t 0.0625) (* a 2.0)) b)) (* 1.0 x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	return cos((((t * 0.0625) * (a * 2.0)) * b)) * (1.0 * x);
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = cos((((t * 0.0625d0) * (a * 2.0d0)) * b)) * (1.0d0 * x)
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t, double a, double b) {
	return Math.cos((((t * 0.0625) * (a * 2.0)) * b)) * (1.0 * x);
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t, a, b):
	return math.cos((((t * 0.0625) * (a * 2.0)) * b)) * (1.0 * x)

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	return Float64(cos(Float64(Float64(Float64(t * 0.0625) * Float64(a * 2.0)) * b)) * Float64(1.0 * x))
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m, t, a, b)
	tmp = cos((((t * 0.0625) * (a * 2.0)) * b)) * (1.0 * x);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := N[(N[Cos[N[(N[(N[(t * 0.0625), $MachinePrecision] * N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]], $MachinePrecision] * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.4%

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

4. Applied rewrites 28.2%

   \[\leadsto \left(x \cdot \color{blue}{\left(\sin \left(0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right) \cdot 0 - \cos \left(\mathsf{fma}\left(0.0625 \cdot \left(t \cdot \mathsf{fma}\left(2, y, 1\right)\right), z, \mathsf{PI}\left(\right)\right)\right) \cdot 1\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

5. Taylor expanded in z around 0

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

   1. cos-PI N/A

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

   2. metadata-eval 30.7

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

7. Applied rewrites 30.7%

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

8. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 31.6

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

10. Applied rewrites 31.6%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. div-inv N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. lower-*.f64 N/A

        \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(b \cdot \left(\mathsf{Rewrite=>}\left(lower-*.f64, \left(2 \cdot a\right)\right) \cdot \color{blue}{\left(\frac{1}{16} \cdot t\right)}\right)\right) \]

12. Applied rewrites 31.6%

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

13. Final simplification 31.6%

    \[\leadsto \cos \left(\left(\left(t \cdot 0.0625\right) \cdot \left(a \cdot 2\right)\right) \cdot b\right) \cdot \left(1 \cdot x\right) \]
14. Add Preprocessing

Alternative 3: 29.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \cos \left(\left(b \cdot t\right) \cdot -0.0625\right) \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := N[(N[Cos[N[(N[(b * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 28.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 29.3

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

5. Applied rewrites 29.3%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

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. cos-neg-rev N/A

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

   4. lower-cos.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 30.8

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

8. Applied rewrites 30.8%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 31.3%

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

12. Final simplification 31.3%

    \[\leadsto \cos \left(\left(b \cdot t\right) \cdot -0.0625\right) \cdot x \]
13. Add Preprocessing

Alternative 4: 3.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \left(\left(-0.0078125 \cdot a\right) \cdot \left(\left(t \cdot x\right) \cdot \left(\left(b \cdot b\right) \cdot t\right)\right)\right) \cdot a \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b)
 :precision binary64
 (* (* (* -0.0078125 a) (* (* t x) (* (* b b) t))) a))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	return ((-0.0078125 * a) * ((t * x) * ((b * b) * t))) * a;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (((-0.0078125d0) * a) * ((t * x) * ((b * b) * t))) * a
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t, double a, double b) {
	return ((-0.0078125 * a) * ((t * x) * ((b * b) * t))) * a;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t, a, b):
	return ((-0.0078125 * a) * ((t * x) * ((b * b) * t))) * a

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	return Float64(Float64(Float64(-0.0078125 * a) * Float64(Float64(t * x) * Float64(Float64(b * b) * t))) * a)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m, t, a, b)
	tmp = ((-0.0078125 * a) * ((t * x) * ((b * b) * t))) * a;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := N[(N[(N[(-0.0078125 * a), $MachinePrecision] * N[(N[(t * x), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 28.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 29.3

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

5. Applied rewrites 29.3%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x + {t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) + x} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left({t}^{2} \cdot x\right) \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({t}^{2} \cdot x, \frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right), x\right)} \]

8. Applied rewrites 13.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot x, -0.001953125 \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(2, a, 1\right)\right)}^{2}, b \cdot b, {\left(\mathsf{fma}\left(2, y, 1\right)\right)}^{2} \cdot \left(z \cdot z\right)\right), x\right)} \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{128} \cdot \color{blue}{\left({a}^{2} \cdot \left({b}^{2} \cdot \left({t}^{2} \cdot x\right)\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 2.0%

    \[\leadsto \left(-0.0078125 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(b \cdot b\right)\right)} \]
12. Step-by-step derivation

13. Applied rewrites 2.9%

    \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot t\right) \cdot \left(x \cdot t\right)\right) \cdot \left(-0.0078125 \cdot a\right)\right) \cdot a \]

14. Final simplification 2.9%

    \[\leadsto \left(\left(-0.0078125 \cdot a\right) \cdot \left(\left(t \cdot x\right) \cdot \left(\left(b \cdot b\right) \cdot t\right)\right)\right) \cdot a \]
15. Add Preprocessing

Alternative 5: 2.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot -0.0078125\right)\right) \cdot b \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b)
 :precision binary64
 (* (* (* (* (* t t) x) b) (* (* a a) -0.0078125)) b))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	return ((((t * t) * x) * b) * ((a * a) * -0.0078125)) * b;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((t * t) * x) * b) * ((a * a) * (-0.0078125d0))) * b
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t, double a, double b) {
	return ((((t * t) * x) * b) * ((a * a) * -0.0078125)) * b;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t, a, b):
	return ((((t * t) * x) * b) * ((a * a) * -0.0078125)) * b

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	return Float64(Float64(Float64(Float64(Float64(t * t) * x) * b) * Float64(Float64(a * a) * -0.0078125)) * b)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m, t, a, b)
	tmp = ((((t * t) * x) * b) * ((a * a) * -0.0078125)) * b;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := N[(N[(N[(N[(N[(t * t), $MachinePrecision] * x), $MachinePrecision] * b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.0078125), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 28.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 29.3

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

5. Applied rewrites 29.3%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x + {t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) + x} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left({t}^{2} \cdot x\right) \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({t}^{2} \cdot x, \frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right), x\right)} \]

8. Applied rewrites 13.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot x, -0.001953125 \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(2, a, 1\right)\right)}^{2}, b \cdot b, {\left(\mathsf{fma}\left(2, y, 1\right)\right)}^{2} \cdot \left(z \cdot z\right)\right), x\right)} \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{128} \cdot \color{blue}{\left({a}^{2} \cdot \left({b}^{2} \cdot \left({t}^{2} \cdot x\right)\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 2.0%

    \[\leadsto \left(-0.0078125 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(b \cdot b\right)\right)} \]
12. Step-by-step derivation

13. Applied rewrites 2.6%

    \[\leadsto \left(\left(\left(a \cdot a\right) \cdot -0.0078125\right) \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot b\right)\right) \cdot b \]

14. Final simplification 2.6%

    \[\leadsto \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot -0.0078125\right)\right) \cdot b \]
15. Add Preprocessing

Alternative 6: 2.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \left(\left(\left(\left(b \cdot b\right) \cdot x\right) \cdot t\right) \cdot t\right) \cdot \left(\left(a \cdot a\right) \cdot -0.0078125\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b)
 :precision binary64
 (* (* (* (* (* b b) x) t) t) (* (* a a) -0.0078125)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	return ((((b * b) * x) * t) * t) * ((a * a) * -0.0078125);
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((b * b) * x) * t) * t) * ((a * a) * (-0.0078125d0))
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t, double a, double b) {
	return ((((b * b) * x) * t) * t) * ((a * a) * -0.0078125);
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t, a, b):
	return ((((b * b) * x) * t) * t) * ((a * a) * -0.0078125)

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	return Float64(Float64(Float64(Float64(Float64(b * b) * x) * t) * t) * Float64(Float64(a * a) * -0.0078125))
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m, t, a, b)
	tmp = ((((b * b) * x) * t) * t) * ((a * a) * -0.0078125);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := N[(N[(N[(N[(N[(b * b), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.0078125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 29.3

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

5. Applied rewrites 29.3%

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x + {t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) + x} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left({t}^{2} \cdot x\right) \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({t}^{2} \cdot x, \frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right), x\right)} \]

8. Applied rewrites 13.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot x, -0.001953125 \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(2, a, 1\right)\right)}^{2}, b \cdot b, {\left(\mathsf{fma}\left(2, y, 1\right)\right)}^{2} \cdot \left(z \cdot z\right)\right), x\right)} \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{128} \cdot \color{blue}{\left({a}^{2} \cdot \left({b}^{2} \cdot \left({t}^{2} \cdot x\right)\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 2.0%

    \[\leadsto \left(-0.0078125 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(b \cdot b\right)\right)} \]
12. Step-by-step derivation

13. Applied rewrites 2.3%

    \[\leadsto \left(-0.0078125 \cdot \left(a \cdot a\right)\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot x\right)}\right)\right) \]

14. Final simplification 2.3%

    \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot x\right) \cdot t\right) \cdot t\right) \cdot \left(\left(a \cdot a\right) \cdot -0.0078125\right) \]
15. Add Preprocessing

Developer Target 1: 29.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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))))