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

Percentage Accurate: 27.8% → 32.2%

Time: 8.5s

Alternatives: 13

Speedup: 2.6×

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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: 27.8% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 32.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot \left|b\right|\right) \cdot \left|t\right|}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(0.0625, \left|t\right| \cdot \left(\left|z\right| \cdot \mathsf{fma}\left(2, y, 1\right)\right), \pi \cdot 0.5\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot \left|b\right|\right) \cdot \left|t\right|\right) \cdot 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) (fabs b)) (fabs t)) 16.0)))
      2e+290)
   (*
    (*
     x
     (sin (fma 0.0625 (* (fabs t) (* (fabs z) (fma 2.0 y 1.0))) (* PI 0.5))))
    (sin (fma 0.5 PI (* (* (* (fma a 2.0 1.0) (fabs b)) (fabs t)) 0.0625))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * fabs(b)) * fabs(t)) / 16.0))) <= 2e+290) {
		tmp = (x * sin(fma(0.0625, (fabs(t) * (fabs(z) * fma(2.0, y, 1.0))), (((double) M_PI) * 0.5)))) * sin(fma(0.5, ((double) M_PI), (((fma(a, 2.0, 1.0) * fabs(b)) * fabs(t)) * 0.0625)));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * abs(b)) * abs(t)) / 16.0))) <= 2e+290)
		tmp = Float64(Float64(x * sin(fma(0.0625, Float64(abs(t) * Float64(abs(z) * fma(2.0, y, 1.0))), Float64(pi * 0.5)))) * sin(fma(0.5, pi, Float64(Float64(Float64(fma(a, 2.0, 1.0) * abs(b)) * abs(t)) * 0.0625))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(N[(x * N[Sin[N[(0.0625 * N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. mult-flip N/A

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

      20. lower-*.f64 N/A

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

      21. lower-PI.f64 N/A

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

      22. metadata-eval 27.8

          \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(0.0625, t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right), \pi \cdot \color{blue}{0.5}\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. Applied rewrites 27.8%

      \[\leadsto \left(x \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.0625, t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right), \pi \cdot 0.5\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 27.9

         \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(0.0625, t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right), \pi \cdot 0.5\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 27.9

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 27.9

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

   5. Applied rewrites 27.9%

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

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 32.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t\_1 \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot \left(b \cdot t\right), -0.0625, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))) 2e+290)
     (* t_1 (sin (fma (* (fma a 2.0 1.0) (* b t)) -0.0625 (* PI 0.5))))
     (* x (sin (* 0.5 PI))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = t_1 * sin(fma((fma(a, 2.0, 1.0) * (b * t)), -0.0625, (((double) M_PI) * 0.5)));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290)
		tmp = Float64(t_1 * sin(fma(Float64(fma(a, 2.0, 1.0) * Float64(b * t)), -0.0625, Float64(pi * 0.5))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(t$95$1 * N[Sin[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[(b * t), $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 27.7

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

   4. Applied rewrites 27.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 28.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot \left(b \cdot t\right), -0.0625, \pi \cdot 0.5\right)\right)} \]

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 32.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left|t\right|}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot \left|b\right|\right) \cdot \left|t\right|}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t\_1 \cdot \sin \left(\mathsf{fma}\left(0.0625, \left(\left|b\right| \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \left|t\right|, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) (fabs t)) 16.0)))))
   (if (<=
        (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) (fabs b)) (fabs t)) 16.0)))
        2e+290)
     (*
      t_1
      (sin (fma 0.0625 (* (* (fabs b) (fma a 2.0 1.0)) (fabs t)) (* PI 0.5))))
     (* x (sin (* 0.5 PI))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * fabs(t)) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * fabs(b)) * fabs(t)) / 16.0))) <= 2e+290) {
		tmp = t_1 * sin(fma(0.0625, ((fabs(b) * fma(a, 2.0, 1.0)) * fabs(t)), (((double) M_PI) * 0.5)));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * abs(t)) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * abs(b)) * abs(t)) / 16.0))) <= 2e+290)
		tmp = Float64(t_1 * sin(fma(0.0625, Float64(Float64(abs(b) * fma(a, 2.0, 1.0)) * abs(t)), Float64(pi * 0.5))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(t$95$1 * N[Sin[N[(0.0625 * N[(N[(N[Abs[b], $MachinePrecision] * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 27.8

          \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.0625, \left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t, \pi \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied rewrites 27.8%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.0625, \left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t, \pi \cdot 0.5\right)\right)} \]

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t\_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{1}{\frac{16}{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))
   (if (<= (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1) 2e+290)
     (* (* x (cos (/ 1.0 (/ 16.0 (* t (* z (fma 2.0 y 1.0))))))) t_1)
     (* x (sin (* 0.5 PI))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 2e+290) {
		tmp = (x * cos((1.0 / (16.0 / (t * (z * fma(2.0, y, 1.0))))))) * t_1;
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 2e+290)
		tmp = Float64(Float64(x * cos(Float64(1.0 / Float64(16.0 / Float64(t * Float64(z * fma(2.0, y, 1.0))))))) * t_1);
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision], 2e+290], N[(N[(x * N[Cos[N[(1.0 / N[(16.0 / N[(t * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 27.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 27.8

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 27.8

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

   3. Applied rewrites 27.8%

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

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot \left|b\right|\right) \cdot \left|t\right|}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x \cdot \cos \left(0.0625 \cdot \left(\left|t\right| \cdot z\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot \left(0.0625 \cdot \left|b\right|\right), \left|t\right|, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) (fabs b)) (fabs t)) 16.0)))
      2e+290)
   (*
    (* x (cos (* 0.0625 (* (fabs t) z))))
    (sin (fma (* (fma a 2.0 1.0) (* 0.0625 (fabs b))) (fabs t) (* PI 0.5))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * fabs(b)) * fabs(t)) / 16.0))) <= 2e+290) {
		tmp = (x * cos((0.0625 * (fabs(t) * z)))) * sin(fma((fma(a, 2.0, 1.0) * (0.0625 * fabs(b))), fabs(t), (((double) M_PI) * 0.5)));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * abs(b)) * abs(t)) / 16.0))) <= 2e+290)
		tmp = Float64(Float64(x * cos(Float64(0.0625 * Float64(abs(t) * z)))) * sin(fma(Float64(fma(a, 2.0, 1.0) * Float64(0.0625 * abs(b))), abs(t), Float64(pi * 0.5))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(N[(x * N[Cos[N[(0.0625 * N[(N[Abs[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[(0.0625 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 27.7

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

   4. Applied rewrites 27.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 28.3

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

   7. Applied rewrites 28.3%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   9. Applied rewrites 28.9%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. metadata-eval N/A

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

       11. mult-flip N/A

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

       12. lift-PI.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lift-PI.f64 N/A

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

       17. mult-flip N/A

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

       18. metadata-eval N/A

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

   11. Applied rewrites 28.5%

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

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot \left|b\right|\right) \cdot \left|t\right|}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, \left|b\right| \cdot \left|t\right|, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(\left|t\right| \cdot z\right) \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) (fabs b)) (fabs t)) 16.0)))
      2e+290)
   (*
    x
    (*
     (sin (fma -0.0625 (* (fabs b) (fabs t)) (* 0.5 PI)))
     (cos (* (fma 2.0 y 1.0) (* (* (fabs t) z) 0.0625)))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * fabs(b)) * fabs(t)) / 16.0))) <= 2e+290) {
		tmp = x * (sin(fma(-0.0625, (fabs(b) * fabs(t)), (0.5 * ((double) M_PI)))) * cos((fma(2.0, y, 1.0) * ((fabs(t) * z) * 0.0625))));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * abs(b)) * abs(t)) / 16.0))) <= 2e+290)
		tmp = Float64(x * Float64(sin(fma(-0.0625, Float64(abs(b) * abs(t)), Float64(0.5 * pi))) * cos(Float64(fma(2.0, y, 1.0) * Float64(Float64(abs(t) * z) * 0.0625)))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(x * N[(N[Sin[N[(-0.0625 * N[(N[Abs[b], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(y \cdot \left(2 + \frac{1}{y}\right)\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      2e+290)
   (*
    x
    (*
     (cos (* 0.0625 (* b t)))
     (cos (* (* y (+ 2.0 (/ 1.0 y))) (* (* t z) 0.0625)))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = x * (cos((0.0625 * (b * t))) * cos(((y * (2.0 + (1.0 / y))) * ((t * z) * 0.0625))));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = x * (Math.cos((0.0625 * (b * t))) * Math.cos(((y * (2.0 + (1.0 / y))) * ((t * z) * 0.0625))));
	} else {
		tmp = x * Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290:
		tmp = x * (math.cos((0.0625 * (b * t))) * math.cos(((y * (2.0 + (1.0 / y))) * ((t * z) * 0.0625))))
	else:
		tmp = x * math.sin((0.5 * math.pi))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 2e+290)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t))) * cos(Float64(Float64(y * Float64(2.0 + Float64(1.0 / y))) * Float64(Float64(t * z) * 0.0625)))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290)
		tmp = x * (cos((0.0625 * (b * t))) * cos(((y * (2.0 + (1.0 / y))) * ((t * z) * 0.0625))));
	else
		tmp = x * sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e+290], N[(x * N[(N[Cos[N[(0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(y * N[(2.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 28.9

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

   9. Applied rewrites 28.9%

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

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{\frac{16}{\mathsf{fma}\left(y + y, z, z\right) \cdot t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      2e+290)
   (*
    x
    (*
     (cos (* 0.0625 (* b t)))
     (cos (/ 1.0 (/ 16.0 (* (fma (+ y y) z z) t))))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = x * (cos((0.0625 * (b * t))) * cos((1.0 / (16.0 / (fma((y + y), z, z) * t)))));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 2e+290)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t))) * cos(Float64(1.0 / Float64(16.0 / Float64(fma(Float64(y + y), z, z) * t))))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e+290], N[(x * N[(N[Cos[N[(0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(1.0 / N[(16.0 / N[(N[(N[(y + y), $MachinePrecision] * z + z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

      17. associate-/l* N/A

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

      18. lift-*.f64 N/A

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

      19. div-flip N/A

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

      20. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 28.4%

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

   if 2.00000000000000012e290 < (*.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 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-PI.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      2e+290)
   (*
    x
    (* (cos (* 0.0625 (* b t))) (cos (* (fma 2.0 y 1.0) (* (* t z) 0.0625)))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = x * (cos((0.0625 * (b * t))) * cos((fma(2.0, y, 1.0) * ((t * z) * 0.0625))));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 2e+290)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t))) * cos(Float64(fma(2.0, y, 1.0) * Float64(Float64(t * z) * 0.0625)))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e+290], N[(x * N[(N[Cos[N[(0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      24. lower-*.f64 28.9

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   6. Applied rewrites 28.9%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   if 2.00000000000000012e290 < (*.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 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(1 + 2 \cdot y\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      18. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      21. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      24. lower-*.f64 28.9

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   6. Applied rewrites 28.9%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \left(\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{-16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\pi}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      14. mult-flip N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 29.0

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   8. Applied rewrites 29.0%

      \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)}\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]
   10. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x\right) \cdot \cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(y + y, z, z\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      2e+290)
   (*
    (* (cos (* -0.0625 (* b t))) x)
    (cos (* -0.0625 (* (fma (+ y y) z z) t))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = (cos((-0.0625 * (b * t))) * x) * cos((-0.0625 * (fma((y + y), z, z) * t)));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 2e+290)
		tmp = Float64(Float64(cos(Float64(-0.0625 * Float64(b * t))) * x) * cos(Float64(-0.0625 * Float64(fma(Float64(y + y), z, z) * t))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e+290], N[(N[(N[Cos[N[(-0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(-0.0625 * N[(N[(N[(y + y), $MachinePrecision] * z + z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 28.4%

      \[\leadsto \left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x\right) \cdot \color{blue}{\cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(y + y, z, z\right) \cdot t\right)\right)} \]

   if 2.00000000000000012e290 < (*.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 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(1 + 2 \cdot y\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      18. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      21. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      24. lower-*.f64 28.9

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   6. Applied rewrites 28.9%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \left(\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{-16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\pi}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      14. mult-flip N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 29.0

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   8. Applied rewrites 29.0%

      \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)}\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]
   10. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot \left|t\right|, 0.5 \cdot \pi\right)\right) \cdot \cos \left(0.0625 \cdot \left(\left|t\right| \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0)))
      2e+290)
   (*
    x
    (*
     (sin (fma -0.0625 (* b (fabs t)) (* 0.5 PI)))
     (cos (* 0.0625 (* (fabs t) z)))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0))) <= 2e+290) {
		tmp = x * (sin(fma(-0.0625, (b * fabs(t)), (0.5 * ((double) M_PI)))) * cos((0.0625 * (fabs(t) * z))));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))) <= 2e+290)
		tmp = Float64(x * Float64(sin(fma(-0.0625, Float64(b * abs(t)), Float64(0.5 * pi))) * cos(Float64(0.0625 * Float64(abs(t) * z)))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(x * N[(N[Sin[N[(-0.0625 * N[(b * N[Abs[t], $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(N[Abs[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(1 + 2 \cdot y\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      18. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      21. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      24. lower-*.f64 28.9

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   6. Applied rewrites 28.9%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \left(\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{-16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\pi}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      14. mult-flip N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 29.0

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   8. Applied rewrites 29.0%

      \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right) \]

       2. lower-*.f64 29.3

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]

   11. Applied rewrites 29.3%

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]

   if 2.00000000000000012e290 < (*.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 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(1 + 2 \cdot y\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      18. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      21. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      24. lower-*.f64 28.9

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   6. Applied rewrites 28.9%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \left(\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{-16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\pi}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      14. mult-flip N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 29.0

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   8. Applied rewrites 29.0%

      \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)}\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]
   10. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 31.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      2e+290)
   (* x (* (cos (* 0.0625 (* b t))) (cos (* 0.0625 (* t z)))))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = x * (cos((0.0625 * (b * t))) * cos((0.0625 * (t * z))));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290) {
		tmp = x * (Math.cos((0.0625 * (b * t))) * Math.cos((0.0625 * (t * z))));
	} else {
		tmp = x * Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290:
		tmp = x * (math.cos((0.0625 * (b * t))) * math.cos((0.0625 * (t * z))))
	else:
		tmp = x * math.sin((0.5 * math.pi))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 2e+290)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t))) * cos(Float64(0.0625 * Float64(t * z)))));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 2e+290)
		tmp = x * (cos((0.0625 * (b * t))) * cos((0.0625 * (t * z))));
	else
		tmp = x * sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e+290], N[(x * N[(N[Cos[N[(0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000012e290

   1. Initial program 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 29.2%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]

   if 2.00000000000000012e290 < (*.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 27.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      11. lower-*.f64 28.4

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(1 + 2 \cdot y\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      18. lift-+.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      21. lift-fma.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      24. lower-*.f64 28.9

          \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   6. Applied rewrites 28.9%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \left(\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \left(\sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{-16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\pi}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      14. mult-flip N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

      17. lower-*.f64 29.0

          \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

   8. Applied rewrites 29.0%

      \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)}\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]
   10. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

       3. lower-PI.f64 30.9

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   11. Applied rewrites 30.9%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 30.9% accurate, 2.6× speedup

Math

\[x \cdot \sin \left(0.5 \cdot \pi\right) \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x (sin (* 0.5 PI))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * sin((0.5 * ((double) M_PI)));
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.sin((0.5 * Math.PI));
}

Python

def code(x, y, z, t, a, b):
	return x * math.sin((0.5 * math.pi))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * sin(Float64(0.5 * pi)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * sin((0.5 * pi));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.8%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{16}} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   11. lower-*.f64 28.4

       \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

4. Applied rewrites 28.4%

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right) \cdot \frac{1}{16}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(1 + 2 \cdot y\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \left(2 \cdot y + 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   8. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   11. lift-fma.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   12. lift-*.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(\left(1 + 2 \cdot y\right) \cdot z\right) \cdot t\right) \cdot \frac{1}{16}\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(\left(1 + 2 \cdot y\right) \cdot \left(z \cdot t\right)\right) \cdot \frac{1}{16}\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

   17. lower-*.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

   18. lift-+.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(1 + 2 \cdot y\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

   19. +-commutative N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

   20. lift-*.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\left(2 \cdot y + 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

   21. lift-fma.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)\right)\right) \]

   22. *-commutative N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   23. lower-*.f64 N/A

       \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   24. lower-*.f64 28.9

       \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

6. Applied rewrites 28.9%

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

   2. cos-neg-rev N/A

      \[\leadsto x \cdot \left(\cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

   3. sin-+PI/2-rev N/A

      \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto x \cdot \left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto x \cdot \left(\sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto x \cdot \left(\sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto x \cdot \left(\sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{fma}\left(2, y, 1\right)} \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{-16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\color{blue}{2}, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   13. lift-PI.f64 N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{\pi}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   14. mult-flip N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \frac{1}{2} \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)\right)\right) \]

   17. lower-*.f64 29.0

       \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, \color{blue}{1}\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)\right) \]

8. Applied rewrites 29.0%

   \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)}\right) \]

9. Taylor expanded in t around 0

   \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]
10. Step-by-step derivation

    1. lower-sin.f64 N/A

       \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    3. lower-PI.f64 30.9

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

11. Applied rewrites 30.9%

    \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64
  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))