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

Percentage Accurate: 27.6% → 31.9%

Time: 14.4s

Alternatives: 6

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_1 \cdot \left(0 - \cos \left(\mathsf{fma}\left(\frac{b\_m \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t\_m, \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

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 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 2e+294)
     (* t_1 (- 0.0 (cos (fma (/ (* b_m (fma a 2.0 1.0)) 16.0) t_m (PI)))))
     (* x (sin (* 0.5 (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000013e294

   1. Initial program 48.6%

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

      1. lift-cos.f64 N/A

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

      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. sin-sum N/A

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

      4. lift-cos.f64 N/A

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

      5. sin-PI/2 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \left(\color{blue}{0} - \cos \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \mathsf{PI}\left(\right)\right)\right) \cdot 1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 50.5%

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

   if 2.00000000000000013e294 < (*.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 0.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 1.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.9

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

   7. Applied rewrites 11.9%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \left(0 - \cos \left(\mathsf{fma}\left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}, t, \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;\left(x \cdot t\_1\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\_1\right) \cdot \cos \left(\mathsf{fma}\left(\frac{b\_m}{-16} \cdot t\_m, \mathsf{fma}\left(2, a, 1\right), \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

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 (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0))))
   (if (<=
        (* (* x t_1) (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
        2e+294)
     (* (* (- x) t_1) (cos (fma (* (/ b_m -16.0) t_m) (fma 2.0 a 1.0) (PI))))
     (* x (sin (* 0.5 (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000013e294

   1. Initial program 48.6%

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

      1. lift-cos.f64 N/A

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

      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. sin-sum N/A

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

      4. lift-cos.f64 N/A

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

      5. sin-PI/2 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.2%

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

   if 2.00000000000000013e294 < (*.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 0.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 1.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.9

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

   7. Applied rewrites 11.9%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(-x\right) \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{b}{-16} \cdot t, \mathsf{fma}\left(2, a, 1\right), \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_1 \cdot \left(-\cos \left(\mathsf{fma}\left(-0.0625 \cdot b\_m, t\_m \cdot \mathsf{fma}\left(2, a, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

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 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 2e+294)
     (* t_1 (- (cos (fma (* -0.0625 b_m) (* t_m (fma 2.0 a 1.0)) (PI)))))
     (* x (sin (* 0.5 (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000013e294

   1. Initial program 48.6%

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

      1. lift-cos.f64 N/A

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

      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. sin-sum N/A

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

      4. lift-cos.f64 N/A

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

      5. sin-PI/2 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. cos-+PI N/A

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

      5. cos-neg-rev N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-+PI-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-PI.f64 50.0

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

   7. Applied rewrites 50.0%

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

   if 2.00000000000000013e294 < (*.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 0.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 1.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 11.9

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

   7. Applied rewrites 11.9%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \left(-\cos \left(\mathsf{fma}\left(-0.0625 \cdot b, t \cdot \mathsf{fma}\left(2, a, 1\right), \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+266}:\\ \;\;\;\;t\_1 \cdot \left(-\cos \left(\mathsf{fma}\left(-0.0625 \cdot b\_m, t\_m, \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

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 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 2e+266)
     (* t_1 (- (cos (fma (* -0.0625 b_m) t_m (PI)))))
     (* x (sin (* 0.5 (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.0000000000000001e266

   1. Initial program 48.6%

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

      1. lift-cos.f64 N/A

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

      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. sin-sum N/A

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

      4. lift-cos.f64 N/A

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

      5. sin-PI/2 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. cos-+PI N/A

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

      5. cos-neg-rev N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-+PI-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 49.3

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

   7. Applied rewrites 49.3%

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

   if 2.0000000000000001e266 < (*.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 1.8%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 2.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 12.5

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

   7. Applied rewrites 12.5%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \left(-\cos \left(\mathsf{fma}\left(-0.0625 \cdot b, t, \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_m \leq 122:\\ \;\;\;\;t\_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\_m\right) \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* x (sin (* 0.5 (PI))))))
   (if (<= t_m 122.0)
     (* t_1 (sin (fma 0.5 (PI) (* 0.0625 (* (* (fma 2.0 y 1.0) t_m) z)))))
     t_1)))

Derivation

1. Split input into 2 regimes

2. if t < 122

   1. Initial program 34.5%

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

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 34.7%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 34.5

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 34.5

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

   6. Applied rewrites 34.5%

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

   7. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 35.9

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

   9. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\left(x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.0625 \cdot \left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right)\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.7%

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

   if 122 < t

   1. Initial program 7.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 8.4%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 15.5

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

   7. Applied rewrites 15.5%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 122:\\ \;\;\;\;\left(x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\right) \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 30.7% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 27.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 28.2%

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

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. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Final simplification 31.2%

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

Developer Target 1: 30.4% 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 2024329 
(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))))