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

Percentage Accurate: 27.9% → 32.3%

Time: 8.3s

Alternatives: 11

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: 27.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(x\_m \cdot \sin \left(\mathsf{fma}\left(t\_m \cdot \left(z\_m \cdot \mathsf{fma}\left(2, y, 1\right)\right), -0.0625, \pi \cdot 0.5\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0))) t_1)
         2e+275)
      (*
       (* x_m (sin (fma (* t_m (* z_m (fma 2.0 y 1.0))) -0.0625 (* PI 0.5))))
       t_1)
      (* (sin (* PI 0.5)) x_m)))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 2e+275) {
		tmp = (x_m * sin(fma((t_m * (z_m * fma(2.0, y, 1.0))), -0.0625, (((double) M_PI) * 0.5)))) * t_1;
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 2e+275)
		tmp = Float64(Float64(x_m * sin(fma(Float64(t_m * Float64(z_m * fma(2.0, y, 1.0))), -0.0625, Float64(pi * 0.5)))) * t_1);
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 2e+275], N[(N[(x$95$m * N[Sin[N[(N[(t$95$m * N[(z$95$m * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.8%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 2: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(\frac{1}{\frac{16}{t\_m \cdot \left(z\_m \cdot \mathsf{fma}\left(2, y, 1\right)\right)}}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0))) t_1)
         2e+275)
      (* (* x_m (cos (/ 1.0 (/ 16.0 (* t_m (* z_m (fma 2.0 y 1.0))))))) t_1)
      (* (sin (* PI 0.5)) x_m)))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 2e+275) {
		tmp = (x_m * cos((1.0 / (16.0 / (t_m * (z_m * fma(2.0, y, 1.0))))))) * t_1;
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 2e+275)
		tmp = Float64(Float64(x_m * cos(Float64(1.0 / Float64(16.0 / Float64(t_m * Float64(z_m * fma(2.0, y, 1.0))))))) * t_1);
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 2e+275], N[(N[(x$95$m * N[Cos[N[(1.0 / N[(16.0 / N[(t$95$m * N[(z$95$m * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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-/.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-/.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 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 3: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(\cos \left(\left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\_m\right) \cdot -0.0625\right) \cdot \cos \left(\left(t\_m \cdot \left(z\_m \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right) \cdot -0.0625\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (*
     (*
      (cos (* (* (* b (fma a 2.0 1.0)) t_m) -0.0625))
      (cos (* (* t_m (* z_m (fma 2.0 y 1.0))) -0.0625)))
     x_m)
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = (cos((((b * fma(a, 2.0, 1.0)) * t_m) * -0.0625)) * cos(((t_m * (z_m * fma(2.0, y, 1.0))) * -0.0625))) * x_m;
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(b * fma(a, 2.0, 1.0)) * t_m) * -0.0625)) * cos(Float64(Float64(t_m * Float64(z_m * fma(2.0, y, 1.0))) * -0.0625))) * x_m);
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(N[Cos[N[(N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$m * N[(z$95$m * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 27.9%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 4: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(\frac{z\_m \cdot t\_m}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\_m, -0.0625, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (*
     (* x_m (cos (/ (* z_m t_m) 16.0)))
     (sin (fma (* (* b (fma a 2.0 1.0)) t_m) -0.0625 (* PI 0.5))))
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = (x_m * cos(((z_m * t_m) / 16.0))) * sin(fma(((b * fma(a, 2.0, 1.0)) * t_m), -0.0625, (((double) M_PI) * 0.5)));
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(Float64(x_m * cos(Float64(Float64(z_m * t_m) / 16.0))) * sin(fma(Float64(Float64(b * fma(a, 2.0, 1.0)) * t_m), -0.0625, Float64(pi * 0.5))));
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(x$95$m * N[Cos[N[(N[(z$95$m * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * -0.0625 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 28.6%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 5: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(0.0625 \cdot \left(t\_m \cdot z\_m\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.0625 \cdot \mathsf{fma}\left(a + a, t\_m, t\_m\right), b, 0.5 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (*
     (* x_m (cos (* 0.0625 (* t_m z_m))))
     (sin (fma (* -0.0625 (fma (+ a a) t_m t_m)) b (* 0.5 PI))))
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = (x_m * cos((0.0625 * (t_m * z_m)))) * sin(fma((-0.0625 * fma((a + a), t_m, t_m)), b, (0.5 * ((double) M_PI))));
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(Float64(x_m * cos(Float64(0.0625 * Float64(t_m * z_m)))) * sin(fma(Float64(-0.0625 * fma(Float64(a + a), t_m, t_m)), b, Float64(0.5 * pi))));
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(x$95$m * N[Cos[N[(0.0625 * N[(t$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(-0.0625 * N[(N[(a + a), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision]), $MachinePrecision] * b + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.6

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

   4. Applied rewrites 28.6%

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

   6. Applied rewrites 28.8%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 6: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;t\_1 \cdot \cos \left(0.125 \cdot \left(a \cdot \left(b \cdot t\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (let* ((t_1 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))) 2e+275)
      (* t_1 (cos (* 0.125 (* a (* b t_m)))))
      (* (sin (* PI 0.5)) x_m)))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double t_1 = x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = t_1 * cos((0.125 * (a * (b * t_m))));
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double t_1 = x_m * Math.cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0));
	double tmp;
	if ((t_1 * Math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = t_1 * Math.cos((0.125 * (a * (b * t_m))));
	} else {
		tmp = Math.sin((Math.PI * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z_m, t_m, a, b):
	t_1 = x_m * math.cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))
	tmp = 0
	if (t_1 * math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275:
		tmp = t_1 * math.cos((0.125 * (a * (b * t_m))))
	else:
		tmp = math.sin((math.pi * 0.5)) * x_m
	return x_s * tmp

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	t_1 = Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(t_1 * cos(Float64(0.125 * Float64(a * Float64(b * t_m)))));
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

z_m = abs(z);
t_m = abs(t);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z_m, t_m, a, b)
	t_1 = x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0));
	tmp = 0.0;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = t_1 * cos((0.125 * (a * (b * t_m))));
	else
		tmp = sin((pi * 0.5)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := Block[{t$95$1 = N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(t$95$1 * N[Cos[N[(0.125 * N[(a * N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 27.9

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

   4. Applied rewrites 27.9%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 7: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(b, a + a, b\right) \cdot t\_m\right)\right) \cdot \cos \left(\left(t\_m \cdot z\_m\right) \cdot 0.0625\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (*
     (*
      (cos (* -0.0625 (* (fma b (+ a a) b) t_m)))
      (cos (* (* t_m z_m) 0.0625)))
     x_m)
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = (cos((-0.0625 * (fma(b, (a + a), b) * t_m))) * cos(((t_m * z_m) * 0.0625))) * x_m;
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(Float64(cos(Float64(-0.0625 * Float64(fma(b, Float64(a + a), b) * t_m))) * cos(Float64(Float64(t_m * z_m) * 0.0625))) * x_m);
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(N[Cos[N[(-0.0625 * N[(N[(b * N[(a + a), $MachinePrecision] + b), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$m * z$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.6

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

   4. Applied rewrites 28.6%

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

   6. Applied rewrites 28.6%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 8: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(b, a + a, b\right) \cdot t\_m\right)\right) \cdot x\_m\right) \cdot \cos \left(\left(t\_m \cdot z\_m\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (*
     (* (cos (* -0.0625 (* (fma b (+ a a) b) t_m))) x_m)
     (cos (* (* t_m z_m) 0.0625)))
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = (cos((-0.0625 * (fma(b, (a + a), b) * t_m))) * x_m) * cos(((t_m * z_m) * 0.0625));
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(Float64(cos(Float64(-0.0625 * Float64(fma(b, Float64(a + a), b) * t_m))) * x_m) * cos(Float64(Float64(t_m * z_m) * 0.0625)));
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(N[Cos[N[(-0.0625 * N[(N[(b * N[(a + a), $MachinePrecision] + b), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * N[Cos[N[(N[(t$95$m * z$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.6

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

   4. Applied rewrites 28.6%

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

   6. Applied rewrites 28.6%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 9: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(0.0625 \cdot \left(t\_m \cdot z\_m\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(a + a, t\_m, t\_m\right) \cdot \left(b \cdot 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (*
     (* x_m (cos (* 0.0625 (* t_m z_m))))
     (cos (* (fma (+ a a) t_m t_m) (* b 0.0625))))
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = (x_m * cos((0.0625 * (t_m * z_m)))) * cos((fma((a + a), t_m, t_m) * (b * 0.0625)));
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(Float64(x_m * cos(Float64(0.0625 * Float64(t_m * z_m)))) * cos(Float64(fma(Float64(a + a), t_m, t_m) * Float64(b * 0.0625))));
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(x$95$m * N[Cos[N[(0.0625 * N[(t$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(a + a), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision] * N[(b * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.6

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

   4. Applied rewrites 28.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-lft-identity N/A

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

      19. associate-*l/ N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 28.8%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 10: 31.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+275}:\\ \;\;\;\;x\_m \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot \left(t\_m \cdot \left(1 + 2 \cdot a\right)\right), 0.5 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot 0.5\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+275)
    (* x_m (sin (fma -0.0625 (* b (* t_m (+ 1.0 (* 2.0 a)))) (* 0.5 PI))))
    (* (sin (* PI 0.5)) x_m))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275) {
		tmp = x_m * sin(fma(-0.0625, (b * (t_m * (1.0 + (2.0 * a)))), (0.5 * ((double) M_PI))));
	} else {
		tmp = sin((((double) M_PI) * 0.5)) * x_m;
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+275)
		tmp = Float64(x_m * sin(fma(-0.0625, Float64(b * Float64(t_m * Float64(1.0 + Float64(2.0 * a)))), Float64(0.5 * pi))));
	else
		tmp = Float64(sin(Float64(pi * 0.5)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $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), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+275], N[(x$95$m * N[Sin[N[(-0.0625 * N[(b * N[(t$95$m * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $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)))) < 1.99999999999999992e275

   1. Initial program 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 28.9

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

   6. Applied rewrites 28.9%

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

   if 1.99999999999999992e275 < (*.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 27.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 31.0

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

   6. Applied rewrites 31.0%

      \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

Alternative 11: 31.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot x\_m\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (* (sin (* PI 0.5)) x_m)))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * (sin((((double) M_PI) * 0.5)) * x_m);
}

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * (Math.sin((Math.PI * 0.5)) * x_m);
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z_m, t_m, a, b):
	return x_s * (math.sin((math.pi * 0.5)) * x_m)

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	return Float64(x_s * Float64(sin(Float64(pi * 0.5)) * x_m))
end

MATLAB

z_m = abs(z);
t_m = abs(t);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = x_s * (sin((pi * 0.5)) * x_m);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * N[(N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

3. Applied rewrites 27.9%

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

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. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 31.0

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

6. Applied rewrites 31.0%

   \[\leadsto \color{blue}{x \cdot \sin \left(0.5 \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 31.0

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 31.0

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

8. Applied rewrites 31.0%

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

Reproduce

herbie shell --seed 2025156 
(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))))