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

Percentage Accurate: 27.9% → 32.4%

Time: 15.7s

Alternatives: 12

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: 32.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)}{16}\right)\right) \cdot \sin \left(\frac{\left(b\_m \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot t\_m}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+262)
   (*
    (* x (cos (/ (* (fma 2.0 y 1.0) (* t_m z)) 16.0)))
    (sin (+ (/ (* (* b_m (fma 2.0 a 1.0)) t_m) -16.0) (/ (PI) 2.0))))
   (* (sin (* (PI) 0.5)) x)))

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)))) < 2e262

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.0

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

   4. Applied rewrites 49.0%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 49.4%

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

   if 2e262 < (*.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 1.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 12.3

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

   7. Applied rewrites 12.3%

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

Alternative 2: 32.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)}{16}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0))) t_1) 2e+262)
     (* (* x (cos (/ (* (fma 2.0 y 1.0) (* t_m z)) 16.0))) t_1)
     (* (sin (* (PI) 0.5)) x))))

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)))) < 2e262

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.0

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

   4. Applied rewrites 49.0%

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

   if 2e262 < (*.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 1.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 12.3

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

   7. Applied rewrites 12.3%

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

Alternative 3: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\left(\cos \left(\frac{\left(b\_m \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot t\_m}{-16}\right) \cdot x\right) \cdot \cos \left(\frac{\left(z \cdot \mathsf{fma}\left(y, 2, 1\right)\right) \cdot t\_m}{-16}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+262)
   (*
    (* (cos (/ (* (* b_m (fma 2.0 a 1.0)) t_m) -16.0)) x)
    (cos (/ (* (* z (fma y 2.0 1.0)) t_m) -16.0)))
   (* (sin (* (PI) 0.5)) x)))

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)))) < 2e262

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.0

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

   4. Applied rewrites 49.0%

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

   6. Applied rewrites 49.0%

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

   if 2e262 < (*.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 1.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 12.3

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

   7. Applied rewrites 12.3%

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

Alternative 4: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(b\_m \cdot t\_m, -0.0625, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+262)
   (*
    (* x (cos (/ (* (fma 2.0 y 1.0) (* t_m z)) 16.0)))
    (sin (fma (* b_m t_m) -0.0625 (/ (PI) 2.0))))
   (* (sin (* (PI) 0.5)) x)))

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)))) < 2e262

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in a around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 48.1

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

   7. Applied rewrites 48.1%

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

   9. Applied rewrites 48.5%

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

   if 2e262 < (*.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 1.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 12.3

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

   7. Applied rewrites 12.3%

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

Alternative 5: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\cos \left(\left(b\_m \cdot t\_m\right) \cdot 0.0625\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right) \cdot t\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+306)
   (*
    (* (cos (* (* b_m t_m) 0.0625)) x)
    (sin (fma (PI) 0.5 (* -0.0625 (* (* (fma y 2.0 1.0) z) t_m)))))
   (* (sin (* (PI) 0.5)) x)))

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)))) < 2.00000000000000003e306

   1. Initial program 47.6%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 46.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. cos-neg-rev N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

   7. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\left(\cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]

   if 2.00000000000000003e306 < (*.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

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 11.6

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

   7. Applied rewrites 11.6%

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

Alternative 6: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(b\_m \cdot t\_m\right) \cdot -0.0625\right)\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+262)
   (*
    (* x (cos (* (* b_m t_m) -0.0625)))
    (cos (* -0.0625 (* (* (fma 2.0 y 1.0) z) t_m))))
   (* (sin (* (PI) 0.5)) x)))

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)))) < 2e262

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in a around 0

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

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

   7. Applied rewrites 48.0%

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

   if 2e262 < (*.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 1.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 12.3

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

   7. Applied rewrites 12.3%

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

Alternative 7: 31.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-50}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.0625, b\_m \cdot t\_m, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<= t_m 2.9e-50)
   (*
    (* x (cos (/ (* (fma 2.0 y 1.0) (* t_m z)) 16.0)))
    (sin (fma 0.0625 (* b_m t_m) (/ (PI) 2.0))))
   (* (sin (* (PI) 0.5)) x)))

Derivation

1. Split input into 2 regimes

2. if t < 2.90000000000000008e-50

   1. Initial program 32.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 33.5

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

   4. Applied rewrites 33.5%

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

   5. Taylor expanded in a around 0

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 33.6

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

   7. Applied rewrites 33.6%

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

   9. Applied rewrites 34.0%

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

   if 2.90000000000000008e-50 < t

   1. Initial program 15.1%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 21.3

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

   7. Applied rewrites 21.3%

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

Alternative 8: 31.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 27.7%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 27.4%

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

5. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 30.5

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

7. Applied rewrites 30.5%

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

Alternative 9: 19.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot x, -0.001953125 \cdot \left(\left(4 \cdot \left(y \cdot y\right)\right) \cdot \left(z \cdot z\right)\right), x\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
t_m = (fabs.f64 t)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (fma (* (* t_m t_m) x) (* -0.001953125 (* (* 4.0 (* y y)) (* z z))) x))

C

b_m = fabs(b);
t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	return fma(((t_m * t_m) * x), (-0.001953125 * ((4.0 * (y * y)) * (z * z))), x);
}

Julia

b_m = abs(b)
t_m = abs(t)
function code(x, y, z, t_m, a, b_m)
	return fma(Float64(Float64(t_m * t_m) * x), Float64(-0.001953125 * Float64(Float64(4.0 * Float64(y * y)) * Float64(z * z))), x)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(-0.001953125 * N[(N[(4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 28.2

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

4. Applied rewrites 28.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 13.9%

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

8. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(\left(t \cdot t\right) \cdot x, \frac{-1}{512} \cdot \left(4 \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right), x\right) \]
9. Step-by-step derivation

10. Applied rewrites 20.5%

    \[\leadsto \mathsf{fma}\left(\left(t \cdot t\right) \cdot x, -0.001953125 \cdot \left(\left(4 \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(z \cdot z\right)}\right), x\right) \]
11. Add Preprocessing

Alternative 10: 3.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -0.0078125), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 28.2

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

4. Applied rewrites 28.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 13.9%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 2.1%

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

12. Applied rewrites 3.0%

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

Alternative 11: 2.7% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := N[(N[(N[(N[(a * a), $MachinePrecision] * -0.0078125), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 28.2

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

4. Applied rewrites 28.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 13.9%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 2.1%

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

12. Applied rewrites 2.7%

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

Alternative 12: 2.6% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := N[(N[(-0.0078125 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(t$95$m * N[(N[(b$95$m * b$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 28.2

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

4. Applied rewrites 28.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 13.9%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 2.1%

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

12. Applied rewrites 2.5%

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

Developer Target 1: 30.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024319 
(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))))