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

Percentage Accurate: 27.7% → 30.5%

Time: 12.6s

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: 27.7% 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: 30.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+288}:\\ \;\;\;\;\left(\cos \left(\left(t \cdot \left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right) \cdot -0.0625\right) \cdot x\_m\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(t \cdot b\right) \cdot 0.0625\right) \cdot x\_m\\ \end{array} \end{array} \]

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
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
       1e+288)
    (*
     (* (cos (* (* t (* b (fma 2.0 a 1.0))) -0.0625)) x_m)
     (cos (* 0.0625 (* z t))))
    (* (cos (* (* t b) 0.0625)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+288) {
		tmp = (cos(((t * (b * fma(2.0, a, 1.0))) * -0.0625)) * x_m) * cos((0.0625 * (z * t)));
	} else {
		tmp = cos(((t * b) * 0.0625)) * 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)
	tmp = 0.0
	if (Float64(Float64(x_m * 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))) <= 1e+288)
		tmp = Float64(Float64(cos(Float64(Float64(t * Float64(b * fma(2.0, a, 1.0))) * -0.0625)) * x_m) * cos(Float64(0.0625 * Float64(z * t))));
	else
		tmp = Float64(cos(Float64(Float64(t * b) * 0.0625)) * 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_] := 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), $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], 1e+288], N[(N[(N[Cos[N[(N[(t * N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * N[Cos[N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(t * b), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e288

   1. Initial program 43.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 43.8

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

   5. Applied rewrites 43.8%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 43.8%

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

   if 1e288 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 6.1%

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

Alternative 2: 28.5% accurate, 0.8× speedup

Math

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

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
  (*
   (* x_m (cos (pow (/ 16.0 (* z t)) -1.0)))
   (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))))

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 * ((x_m * cos(pow((16.0 / (z * t)), -1.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0)));
}

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 * ((x_m * cos(((16.0d0 / (z * t)) ** (-1.0d0)))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0)))
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 * ((x_m * Math.cos(Math.pow((16.0 / (z * t)), -1.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0)));
}

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 * ((x_m * math.cos(math.pow((16.0 / (z * t)), -1.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0)))

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(Float64(x_m * cos((Float64(16.0 / Float64(z * t)) ^ -1.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))))
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 * ((x_m * cos(((16.0 / (z * t)) ^ -1.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0)));
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[(x$95$m * N[Cos[N[Power[N[(16.0 / N[(z * t), $MachinePrecision]), $MachinePrecision], -1.0], $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]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

4. Applied rewrites 13.2%

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

5. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 25.7

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

7. Applied rewrites 25.7%

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

Alternative 3: 28.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\cos \left(\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\_m\right) \cdot \cos \left(\left(z \cdot t\right) \cdot 0.0625\right)\right) \end{array} \]

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 (* (* (* (fma 2.0 a 1.0) t) b) 0.0625)) x_m)
   (cos (* (* z t) 0.0625)))))

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((((fma(2.0, a, 1.0) * t) * b) * 0.0625)) * x_m) * cos(((z * t) * 0.0625)));
}

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(Float64(cos(Float64(Float64(Float64(fma(2.0, a, 1.0) * t) * b) * 0.0625)) * x_m) * cos(Float64(Float64(z * t) * 0.0625))))
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[(N[Cos[N[(N[(N[(N[(2.0 * a + 1.0), $MachinePrecision] * t), $MachinePrecision] * b), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * N[Cos[N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 25.5

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

5. Applied rewrites 25.5%

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

6. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 25.3

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

8. Applied rewrites 25.3%

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

Alternative 4: 28.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\cos \left(\left(t \cdot \left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right) \cdot -0.0625\right) \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (* x_s (* (cos (* (* t (* b (fma 2.0 a 1.0))) -0.0625)) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (cos(((t * (b * fma(2.0, a, 1.0))) * -0.0625)) * x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(cos(Float64(Float64(t * Float64(b * fma(2.0, a, 1.0))) * -0.0625)) * x_m))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(N[Cos[N[(N[(t * N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 25.5

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

5. Applied rewrites 25.5%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 24.9

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

8. Applied rewrites 24.9%

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

10. Applied rewrites 25.1%

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

Alternative 5: 30.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (Math.cos(((t * b) * 0.0625)) * x_m);
}

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(N[Cos[N[(N[(t * b), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 25.5

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

5. Applied rewrites 25.5%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 24.9

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

8. Applied rewrites 24.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 24.7%

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

Alternative 6: 24.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (* x_s (* (fma (* (* b b) -0.001953125) (* t t) 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 * (fma(((b * b) * -0.001953125), (t * t), 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(fma(Float64(Float64(b * b) * -0.001953125), Float64(t * t), 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[(N[(N[(b * b), $MachinePrecision] * -0.001953125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 25.5

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

5. Applied rewrites 25.5%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 24.9

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

8. Applied rewrites 24.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 24.7%

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

12. Taylor expanded in t around 0

    \[\leadsto \left(1 + \frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right)\right) \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 20.7%

    \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]
15. Add Preprocessing

Developer Target 1: 30.6% 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 2024324 
(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))))