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

Percentage Accurate: 27.9% → 31.9%

Time: 12.5s

Alternatives: 6

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: 27.9% 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: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \sin \left(\left(\frac{z\_m \cdot t\_m}{-16} - \left(\left(\frac{t\_m}{16} \cdot z\_m\right) \cdot 2\right) \cdot y\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+297)
   (*
    (*
     x
     (sin
      (+
       (- (/ (* z_m t_m) -16.0) (* (* (* (/ t_m 16.0) z_m) 2.0) y))
       (/ (PI) 2.0))))
    (cos (* (* b t_m) 0.0625)))
   (* x (sin (* 0.5 (PI))))))

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)))) < 2e297

   1. Initial program 48.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 49.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 49.0

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

   7. Applied rewrites 49.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-lft-out N/A

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

      16. cos-sum-rev N/A

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

      17. *-commutative N/A

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

   9. Applied rewrites 49.2%

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

   if 2e297 < (*.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. cos-sum N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. cos-sum-rev N/A

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

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

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

      10. lower-sin.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.7

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

   9. Applied rewrites 11.7%

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

Alternative 2: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(\frac{z\_m \cdot \mathsf{fma}\left(y, 2, 1\right)}{-16}, t\_m, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+297)
   (*
    (* x (sin (fma (/ (* z_m (fma y 2.0 1.0)) -16.0) t_m (/ (PI) 2.0))))
    (cos (* (* b t_m) 0.0625)))
   (* x (sin (* 0.5 (PI))))))

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)))) < 2e297

   1. Initial program 48.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 49.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 49.0

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

   7. Applied rewrites 49.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. cos-neg-rev N/A

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

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

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

      14. lower-sin.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   9. Applied rewrites 49.4%

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

   if 2e297 < (*.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. cos-sum N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. cos-sum-rev N/A

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

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

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

      10. lower-sin.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.7

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

   9. Applied rewrites 11.7%

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

Alternative 3: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(\cos \left(0.0625 \cdot \left(b \cdot t\_m\right)\right) \cdot \cos \left(\mathsf{fma}\left(-0.125, \left(t\_m \cdot z\_m\right) \cdot y, \left(z\_m \cdot t\_m\right) \cdot -0.0625\right)\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+297)
   (*
    (*
     (cos (* 0.0625 (* b t_m)))
     (cos (fma -0.125 (* (* t_m z_m) y) (* (* z_m t_m) -0.0625))))
    x)
   (* x (sin (* 0.5 (PI))))))

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)))) < 2e297

   1. Initial program 48.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 48.8

         \[\leadsto \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(-0.125, \left(z \cdot y\right) \cdot t, \color{blue}{\left(z \cdot t\right)} \cdot -0.0625\right)\right)\right) \cdot x \]

   10. Applied rewrites 48.8%

       \[\leadsto \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(-0.125, \left(z \cdot y\right) \cdot t, \left(z \cdot t\right) \cdot -0.0625\right)\right)}\right) \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 49.0%

       \[\leadsto \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(-0.125, \left(t \cdot z\right) \cdot \color{blue}{y}, \left(z \cdot t\right) \cdot -0.0625\right)\right)\right) \cdot x \]

   if 2e297 < (*.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. cos-sum N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. cos-sum-rev N/A

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

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

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

      10. lower-sin.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.7

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

   9. Applied rewrites 11.7%

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

Alternative 4: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \cos \left(\mathsf{fma}\left(0.0625 \cdot t\_m, z\_m, \left(\left(z\_m \cdot y\right) \cdot t\_m\right) \cdot 0.125\right)\right)\right) \cdot \cos \left(\left(b \cdot t\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+297)
   (*
    (* x (cos (fma (* 0.0625 t_m) z_m (* (* (* z_m y) t_m) 0.125))))
    (cos (* (* b t_m) 0.0625)))
   (* x (sin (* 0.5 (PI))))))

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)))) < 2e297

   1. Initial program 48.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 48.8

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

   8. Applied rewrites 48.8%

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

   if 2e297 < (*.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. cos-sum N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. cos-sum-rev N/A

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

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

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

      10. lower-sin.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.7

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

   9. Applied rewrites 11.7%

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

Alternative 5: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(b \cdot t\_m\right) \cdot -0.0625\right)\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\_m\right) \cdot t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+297)
   (*
    (* x (cos (* (* b t_m) -0.0625)))
    (cos (* -0.0625 (* (* (fma 2.0 y 1.0) z_m) t_m))))
   (* x (sin (* 0.5 (PI))))))

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)))) < 2e297

   1. Initial program 48.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 48.7%

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

   if 2e297 < (*.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. cos-sum N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. cos-sum-rev N/A

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

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

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

      10. lower-sin.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.7

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

   9. Applied rewrites 11.7%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(b \cdot t\right) \cdot -0.0625\right)\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
(FPCore (x y z_m t_m a b) :precision binary64 (* x (sin (* 0.5 (PI)))))

Derivation

1. Initial program 29.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

   13. cos-sum N/A

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

   14. lower--.f64 N/A

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

4. Applied rewrites 29.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. cos-sum-rev N/A

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

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

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

   10. lower-sin.f64 N/A

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

6. Applied rewrites 29.6%

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

7. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 32.5

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

9. Applied rewrites 32.5%

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

Developer Target 1: 30.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024326 
(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))))