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

Percentage Accurate: 26.8% → 30.9%

Time: 7.3s

Alternatives: 5

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 26.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 30.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+164}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(t\_m + t\_m\right) \cdot \left(z \cdot y\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\_m, \frac{t\_m}{16}, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right) \cdot x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      1e+164)
   (*
    (* x (cos (/ (* (+ t_m t_m) (* z y)) 16.0)))
    (sin (fma (* (fma a 2.0 1.0) b_m) (/ t_m 16.0) (/ PI 2.0))))
   (* (sin (* 0.5 PI)) x)))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+164) {
		tmp = (x * cos((((t_m + t_m) * (z * y)) / 16.0))) * sin(fma((fma(a, 2.0, 1.0) * b_m), (t_m / 16.0), (((double) M_PI) / 2.0)));
	} else {
		tmp = sin((0.5 * ((double) M_PI))) * x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+164)
		tmp = Float64(Float64(x * cos(Float64(Float64(Float64(t_m + t_m) * Float64(z * y)) / 16.0))) * sin(fma(Float64(fma(a, 2.0, 1.0) * b_m), Float64(t_m / 16.0), Float64(pi / 2.0))));
	else
		tmp = Float64(sin(Float64(0.5 * pi)) * x);
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+164], N[(N[(x * N[Cos[N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * x), $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)))) < 1e164

   1. Initial program 26.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 27.0%

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

   4. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.8

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

   6. Applied rewrites 26.8%

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

   if 1e164 < (*.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 26.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 27.0%

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

   4. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 30.0

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

   6. Applied rewrites 30.0%

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

Alternative 2: 30.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+164}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\_m\right) \cdot 0.125\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right), b\_m \cdot \frac{t\_m}{16}, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right) \cdot x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      1e+164)
   (*
    (* x (cos (* (* (* z y) t_m) 0.125)))
    (sin (fma (fma a 2.0 1.0) (* b_m (/ t_m 16.0)) (/ PI 2.0))))
   (* (sin (* 0.5 PI)) x)))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+164) {
		tmp = (x * cos((((z * y) * t_m) * 0.125))) * sin(fma(fma(a, 2.0, 1.0), (b_m * (t_m / 16.0)), (((double) M_PI) / 2.0)));
	} else {
		tmp = sin((0.5 * ((double) M_PI))) * x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+164)
		tmp = Float64(Float64(x * cos(Float64(Float64(Float64(z * y) * t_m) * 0.125))) * sin(fma(fma(a, 2.0, 1.0), Float64(b_m * Float64(t_m / 16.0)), Float64(pi / 2.0))));
	else
		tmp = Float64(sin(Float64(0.5 * pi)) * x);
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+164], N[(N[(x * N[Cos[N[(N[(N[(z * y), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[(b$95$m * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * x), $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)))) < 1e164

   1. Initial program 26.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{\frac{1}{8}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{\frac{1}{8}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(y \cdot z\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(y \cdot z\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. lower-*.f64 26.6

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \color{blue}{\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)} \cdot t}{16}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\color{blue}{\left(a \cdot 2 + 1\right)} \cdot b\right) \cdot t}{16}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \color{blue}{\left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \frac{t}{16}\right)} \]

      8. sin-+PI/2 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \color{blue}{\sin \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \frac{t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \color{blue}{\sin \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \frac{t}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \frac{t}{16} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \frac{t}{16} + \frac{\color{blue}{\pi}}{2}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \color{blue}{\frac{t}{16}} + \frac{\pi}{2}\right) \]

      13. associate-*l* N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \left(\color{blue}{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot \frac{t}{16}\right)} + \frac{\pi}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \left(\left(\color{blue}{2 \cdot a} + 1\right) \cdot \left(b \cdot \frac{t}{16}\right) + \frac{\pi}{2}\right) \]

      15. +-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \left(\color{blue}{\left(1 + 2 \cdot a\right)} \cdot \left(b \cdot \frac{t}{16}\right) + \frac{\pi}{2}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(1 + 2 \cdot a, b \cdot \frac{t}{16}, \frac{\pi}{2}\right)\right)} \]

   6. Applied rewrites 27.2%

      \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right), b \cdot \frac{t}{16}, \frac{\pi}{2}\right)\right)} \]

   if 1e164 < (*.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 26.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 27.0%

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

   4. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 30.0

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

   6. Applied rewrites 30.0%

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

Alternative 3: 30.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 4 \cdot 10^{+269}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\_m\right) \cdot 0.125\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a + a, b\_m, b\_m\right) \cdot t\_m}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right) \cdot x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      4e+269)
   (*
    (* x (cos (* (* (* z y) t_m) 0.125)))
    (cos (/ (* (fma (+ a a) b_m b_m) t_m) 16.0)))
   (* (sin (* 0.5 PI)) x)))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 4e+269) {
		tmp = (x * cos((((z * y) * t_m) * 0.125))) * cos(((fma((a + a), b_m, b_m) * t_m) / 16.0));
	} else {
		tmp = sin((0.5 * ((double) M_PI))) * x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 4e+269)
		tmp = Float64(Float64(x * cos(Float64(Float64(Float64(z * y) * t_m) * 0.125))) * cos(Float64(Float64(fma(Float64(a + a), b_m, b_m) * t_m) / 16.0)));
	else
		tmp = Float64(sin(Float64(0.5 * pi)) * x);
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+269], N[(N[(x * N[Cos[N[(N[(N[(z * y), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(a + a), $MachinePrecision] * b$95$m + b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * x), $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)))) < 4.0000000000000002e269

   1. Initial program 26.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{\frac{1}{8}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(t \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{\frac{1}{8}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(y \cdot z\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(y \cdot z\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. lower-*.f64 26.6

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(2 \cdot \left(a \cdot b\right) + \color{blue}{b}\right) \cdot t}{16}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\left(\left(2 \cdot a\right) \cdot b + b\right) \cdot t}{16}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot \frac{1}{8}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2 \cdot a, \color{blue}{b}, b\right) \cdot t}{16}\right) \]

      4. count-2-rev N/A

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

      5. lift-+.f64 26.6

         \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a + a, b, b\right) \cdot t}{16}\right) \]

   7. Applied rewrites 26.6%

      \[\leadsto \left(x \cdot \cos \left(\left(\left(z \cdot y\right) \cdot t\right) \cdot 0.125\right)\right) \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(a + a, b, b\right)} \cdot t}{16}\right) \]

   if 4.0000000000000002e269 < (*.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 26.8%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 27.0%

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

   4. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 30.0

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

   6. Applied rewrites 30.0%

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

Alternative 4: 30.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	return sin((0.5 * ((double) M_PI))) * x;
}

Java

t_m = Math.abs(t);
b_m = Math.abs(b);
public static double code(double x, double y, double z, double t_m, double a, double b_m) {
	return Math.sin((0.5 * Math.PI)) * x;
}

Python

t_m = math.fabs(t)
b_m = math.fabs(b)
def code(x, y, z, t_m, a, b_m):
	return math.sin((0.5 * math.pi)) * x

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	return Float64(sin(Float64(0.5 * pi)) * x)
end

MATLAB

t_m = abs(t);
b_m = abs(b);
function tmp = code(x, y, z, t_m, a, b_m)
	tmp = sin((0.5 * pi)) * x;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 26.8%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
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. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

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

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

   8. lower-sin.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 N/A

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

3. Applied rewrites 27.0%

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

4. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 30.0

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

6. Applied rewrites 30.0%

   \[\leadsto \color{blue}{\sin \left(0.5 \cdot \pi\right) \cdot x} \]
7. Add Preprocessing

Alternative 5: 25.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \mathsf{fma}\left(b\_m \cdot \left(b\_m \cdot \left(t\_m \cdot t\_m\right)\right), -0.001953125, 1\right) \cdot x \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (* (fma (* b_m (* b_m (* t_m t_m))) -0.001953125 1.0) x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	return fma((b_m * (b_m * (t_m * t_m))), -0.001953125, 1.0) * x;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	return Float64(fma(Float64(b_m * Float64(b_m * Float64(t_m * t_m))), -0.001953125, 1.0) * x)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := N[(N[(N[(b$95$m * N[(b$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.001953125 + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 26.8%

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

2. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 27.9

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

4. Applied rewrites 27.9%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 19.2%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot {t}^{2}, \frac{-1}{512}, 1\right) \cdot x \]

   2. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot {t}^{2}, \frac{-1}{512}, 1\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot {t}^{2}, \frac{-1}{512}, 1\right) \cdot x \]

   4. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(t \cdot t\right), \frac{-1}{512}, 1\right) \cdot x \]

   5. lift-*.f64 23.6

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(t \cdot t\right), -0.001953125, 1\right) \cdot x \]

10. Applied rewrites 23.6%

    \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(t \cdot t\right), -0.001953125, 1\right) \cdot x \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(t \cdot t\right), \frac{-1}{512}, 1\right) \cdot x \]

    2. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(t \cdot t\right), \frac{-1}{512}, 1\right) \cdot x \]

    3. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(t \cdot t\right), \frac{-1}{512}, 1\right) \cdot x \]

    4. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot {t}^{2}, \frac{-1}{512}, 1\right) \cdot x \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \left(b \cdot {t}^{2}\right), \frac{-1}{512}, 1\right) \cdot x \]

    6. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \left(b \cdot {t}^{2}\right), \frac{-1}{512}, 1\right) \cdot x \]

    7. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \left(b \cdot {t}^{2}\right), \frac{-1}{512}, 1\right) \cdot x \]

    8. pow2 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \left(b \cdot \left(t \cdot t\right)\right), \frac{-1}{512}, 1\right) \cdot x \]

    9. lift-*.f64 25.9

       \[\leadsto \mathsf{fma}\left(b \cdot \left(b \cdot \left(t \cdot t\right)\right), -0.001953125, 1\right) \cdot x \]

12. Applied rewrites 25.9%

    \[\leadsto \mathsf{fma}\left(b \cdot \left(b \cdot \left(t \cdot t\right)\right), -0.001953125, 1\right) \cdot x \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(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))))