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

Percentage Accurate: 27.4% → 31.4%

Time: 13.2s

Alternatives: 4

Speedup: 2.4×

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: 27.4% 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: 31.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\mathsf{PI}\left(\right)}{2}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\left(x\_m \cdot \sin \left(\mathsf{fma}\left(\frac{t\_m}{16}, \mathsf{fma}\left(y, 2, 1\right) \cdot z\_m, t\_1\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, a, -1\right) \cdot b\_m}{-16}, t\_m, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b_m)
 :precision binary64
 (let* ((t_1 (/ (PI) 2.0)))
   (*
    x_s
    (if (<=
         (*
          (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
          (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
         5e-141)
      (*
       (* x_m (sin (fma (/ t_m 16.0) (* (fma y 2.0 1.0) z_m) t_1)))
       (sin (fma (/ (* (fma -2.0 a -1.0) b_m) -16.0) t_m t_1)))
      (* (sin (* (PI) 0.5)) x_m)))))

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)))) < 4.9999999999999999e-141

   1. Initial program 53.9%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 52.4%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 52.6

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 52.6

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

   6. Applied rewrites 52.6%

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

   if 4.9999999999999999e-141 < (*.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 13.3%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 20.9

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 20.9%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 31.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot t\_m\right) \cdot b\_m\right)\right) \cdot x\_m\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\_m\right) \cdot t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
       2e+118)
    (*
     (* (cos (* -0.0625 (* (* (fma 2.0 a 1.0) t_m) b_m))) x_m)
     (cos (* -0.0625 (* (* (fma 2.0 y 1.0) z_m) t_m))))
    (* (sin (* (PI) 0.5)) x_m))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.99999999999999993e118

   1. Initial program 49.7%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 50.3%

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

   if 1.99999999999999993e118 < (*.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 8.0%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 8.4%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 17.5

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 31.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(\left(t\_m \cdot z\_m\right) \cdot 0.0625\right)\right) \cdot \cos \left(\left(\left(b\_m \cdot t\_m\right) \cdot a\right) \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
       4e+42)
    (* (* x_m (cos (* (* t_m z_m) 0.0625))) (cos (* (* (* b_m t_m) a) 0.125)))
    (* (sin (* (PI) 0.5)) x_m))))

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)))) < 4.00000000000000018e42

   1. Initial program 50.1%

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

   3. Taylor expanded in 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 49.6

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

   5. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 50.0

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 50.0

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

   7. Applied rewrites 50.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.7

         \[\leadsto \left(x \cdot \cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right) \cdot \cos \left(\left(\color{blue}{\left(b \cdot t\right)} \cdot a\right) \cdot 0.125\right) \]

   10. Applied rewrites 49.7%

       \[\leadsto \left(x \cdot \cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right) \cdot \cos \color{blue}{\left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right)} \]

   if 4.00000000000000018e42 < (*.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 9.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

      6. lower-PI.f64 18.5

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 18.5%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 30.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b_m)
 :precision binary64
 (* x_s (* (sin (* (PI) 0.5)) x_m)))

Derivation

1. Initial program 28.8%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 28.4%

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

5. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

   3. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot x \]

   6. lower-PI.f64 31.5

      \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot x \]

7. Applied rewrites 31.5%

   \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x} \]
8. Add Preprocessing

Developer Target 1: 30.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(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))))