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

Percentage Accurate: 27.5% → 32.1%

Time: 9.3s

Alternatives: 16

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 32.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot t\_1 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\left(x \cdot \sin \left(\left(1 + \frac{-0.0625 \cdot \left(\mathsf{fma}\left(y + y, \left|z\right|, \left|z\right|\right) \cdot \left|t\right|\right)}{0.5 \cdot \pi}\right) \cdot \left(0.5 \cdot \pi\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
         t_1)
        5e+220)
     (*
      (*
       x
       (sin
        (*
         (+
          1.0
          (/
           (* -0.0625 (* (fma (+ y y) (fabs z) (fabs z)) (fabs t)))
           (* 0.5 PI)))
         (* 0.5 PI))))
      t_1)
     (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * t_1) <= 5e+220) {
		tmp = (x * sin(((1.0 + ((-0.0625 * (fma((y + y), fabs(z), fabs(z)) * fabs(t))) / (0.5 * ((double) M_PI)))) * (0.5 * ((double) M_PI))))) * t_1;
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * t_1) <= 5e+220)
		tmp = Float64(Float64(x * sin(Float64(Float64(1.0 + Float64(Float64(-0.0625 * Float64(fma(Float64(y + y), abs(z), abs(z)) * abs(t))) / Float64(0.5 * pi))) * Float64(0.5 * pi)))) * t_1);
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 5e+220], N[(N[(x * N[Sin[N[(N[(1.0 + N[(N[(-0.0625 * N[(N[(N[(y + y), $MachinePrecision] * N[Abs[z], $MachinePrecision] + N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 5.0000000000000002e220

   1. Initial program 27.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 27.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-unsound-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   if 5.0000000000000002e220 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 2: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right) \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(-0.0625 \cdot \mathsf{fma}\left(y + y, \left|z\right|, \left|z\right|\right), \left|t\right|, 0.5 \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \left|t\right|}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0)))
      5e+220)
   (*
    (*
     x
     (sin
      (fma (* -0.0625 (fma (+ y y) (fabs z) (fabs z))) (fabs t) (* 0.5 PI))))
    (cos (/ 1.0 (/ 16.0 (* (* b (fma a 2.0 1.0)) (fabs t))))))
   (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0))) <= 5e+220) {
		tmp = (x * sin(fma((-0.0625 * fma((y + y), fabs(z), fabs(z))), fabs(t), (0.5 * ((double) M_PI))))) * cos((1.0 / (16.0 / ((b * fma(a, 2.0, 1.0)) * fabs(t)))));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))) <= 5e+220)
		tmp = Float64(Float64(x * sin(fma(Float64(-0.0625 * fma(Float64(y + y), abs(z), abs(z))), abs(t), Float64(0.5 * pi)))) * cos(Float64(1.0 / Float64(16.0 / Float64(Float64(b * fma(a, 2.0, 1.0)) * abs(t))))));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+220], N[(N[(x * N[Sin[N[(N[(-0.0625 * N[(N[(y + y), $MachinePrecision] * N[Abs[z], $MachinePrecision] + N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(1.0 / N[(16.0 / N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 5.0000000000000002e220

   1. Initial program 27.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 27.5%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.5%

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 27.5%

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

   3. Applied rewrites 27.5%

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

   5. Applied rewrites 27.4%

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

   if 5.0000000000000002e220 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 3: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot t\_1 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(\left|t\right| \cdot \left(\left|z\right| \cdot \mathsf{fma}\left(2, y, 1\right)\right), -0.0625, \pi \cdot 0.5\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
         t_1)
        5e+220)
     (*
      (*
       x
       (sin
        (fma (* (fabs t) (* (fabs z) (fma 2.0 y 1.0))) -0.0625 (* PI 0.5))))
      t_1)
     (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * t_1) <= 5e+220) {
		tmp = (x * sin(fma((fabs(t) * (fabs(z) * fma(2.0, y, 1.0))), -0.0625, (((double) M_PI) * 0.5)))) * t_1;
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * t_1) <= 5e+220)
		tmp = Float64(Float64(x * sin(fma(Float64(abs(t) * Float64(abs(z) * fma(2.0, y, 1.0))), -0.0625, Float64(pi * 0.5)))) * t_1);
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 5e+220], N[(N[(x * N[Sin[N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 5.0000000000000002e220

   1. Initial program 27.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 27.4%

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

   if 5.0000000000000002e220 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 4: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+277}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot \left(2 + \frac{1}{a}\right)\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))) 1e+277)
     (* t_1 (cos (/ (* (* (* a (+ 2.0 (/ 1.0 a))) b) t) 16.0)))
     (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277) {
		tmp = t_1 * cos(((((a * (2.0 + (1.0 / a))) * b) * t) / 16.0));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277) {
		tmp = t_1 * Math.cos(((((a * (2.0 + (1.0 / a))) * b) * t) / 16.0));
	} else {
		tmp = x * ((2.0 * Math.sin((0.5 * Math.PI))) / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))
	tmp = 0
	if (t_1 * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277:
		tmp = t_1 * math.cos(((((a * (2.0 + (1.0 / a))) * b) * t) / 16.0))
	else:
		tmp = x * ((2.0 * math.sin((0.5 * math.pi))) / 2.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277)
		tmp = Float64(t_1 * cos(Float64(Float64(Float64(Float64(a * Float64(2.0 + Float64(1.0 / a))) * b) * t) / 16.0)));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	tmp = 0.0;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277)
		tmp = t_1 * cos(((((a * (2.0 + (1.0 / a))) * b) * t) / 16.0));
	else
		tmp = x * ((2.0 * sin((0.5 * pi))) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+277], N[(t$95$1 * N[Cos[N[(N[(N[(N[(a * N[(2.0 + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 1e277

   1. Initial program 27.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 27.5%

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

   4. Applied rewrites 27.5%

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

   if 1e277 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 5: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+220}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{1}{\frac{\frac{16}{t}}{\mathsf{fma}\left(b, a + a, b\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))) 5e+220)
     (* t_1 (cos (/ 1.0 (/ (/ 16.0 t) (fma b (+ a a) b)))))
     (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+220) {
		tmp = t_1 * cos((1.0 / ((16.0 / t) / fma(b, (a + a), b))));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+220)
		tmp = Float64(t_1 * cos(Float64(1.0 / Float64(Float64(16.0 / t) / fma(b, Float64(a + a), b)))));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+220], N[(t$95$1 * N[Cos[N[(1.0 / N[(N[(16.0 / t), $MachinePrecision] / N[(b * N[(a + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 5.0000000000000002e220

   1. Initial program 27.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 27.5%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.5%

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 27.5%

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

   3. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 27.5%

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

   5. Applied rewrites 27.5%

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

   if 5.0000000000000002e220 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 6: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+277}:\\ \;\;\;\;\left(\cos \left(\left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right) \cdot -0.0625\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      1e+277)
   (*
    (*
     (cos (* (* (* b (fma a 2.0 1.0)) t) -0.0625))
     (cos (* (* t (* z (fma 2.0 y 1.0))) -0.0625)))
    x)
   (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277) {
		tmp = (cos((((b * fma(a, 2.0, 1.0)) * t) * -0.0625)) * cos(((t * (z * fma(2.0, y, 1.0))) * -0.0625))) * x;
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+277)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(b * fma(a, 2.0, 1.0)) * t) * -0.0625)) * cos(Float64(Float64(t * Float64(z * fma(2.0, y, 1.0))) * -0.0625))) * x);
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+277], N[(N[(N[Cos[N[(N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 1e277

   1. Initial program 27.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 27.5%

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

   if 1e277 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 7: 31.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot t\_1 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(-0.0625, \left|t\right| \cdot \left|z\right|, 0.5 \cdot \pi\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
         t_1)
        5e+220)
     (* (* x (sin (fma -0.0625 (* (fabs t) (fabs z)) (* 0.5 PI)))) t_1)
     (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * t_1) <= 5e+220) {
		tmp = (x * sin(fma(-0.0625, (fabs(t) * fabs(z)), (0.5 * ((double) M_PI))))) * t_1;
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * t_1) <= 5e+220)
		tmp = Float64(Float64(x * sin(fma(-0.0625, Float64(abs(t) * abs(z)), Float64(0.5 * pi)))) * t_1);
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 5e+220], N[(N[(x * N[Sin[N[(-0.0625 * N[(N[Abs[t], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 5.0000000000000002e220

   1. Initial program 27.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 27.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 28.2%

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

   6. Applied rewrites 28.2%

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

   if 5.0000000000000002e220 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 8: 31.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right) \leq 10^{+26}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left|t\right|\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.0625 \cdot \left|t\right|, \mathsf{fma}\left(y + y, \left|z\right|, \left|z\right|\right), 0.5 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0)))
      1e+26)
   (*
    x
    (*
     (cos (* 0.0625 (* b (fabs t))))
     (sin
      (fma (* -0.0625 (fabs t)) (fma (+ y y) (fabs z) (fabs z)) (* 0.5 PI)))))
   (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0))) <= 1e+26) {
		tmp = x * (cos((0.0625 * (b * fabs(t)))) * sin(fma((-0.0625 * fabs(t)), fma((y + y), fabs(z), fabs(z)), (0.5 * ((double) M_PI)))));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))) <= 1e+26)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * abs(t)))) * sin(fma(Float64(-0.0625 * abs(t)), fma(Float64(y + y), abs(z), abs(z)), Float64(0.5 * pi)))));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+26], N[(x * N[(N[Cos[N[(0.0625 * N[(b * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(-0.0625 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(y + y), $MachinePrecision] * N[Abs[z], $MachinePrecision] + N[Abs[z], $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 1.00000000000000005e26

   1. Initial program 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 28.2%

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

   if 1.00000000000000005e26 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 9: 31.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right) \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(-0.0625, \left|t\right| \cdot \left|z\right|, 0.5 \cdot \pi\right)\right)\right) \cdot \cos \left(0.125 \cdot \left(a \cdot \left(b \cdot \left|t\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0)))
      2e+252)
   (*
    (* x (sin (fma -0.0625 (* (fabs t) (fabs z)) (* 0.5 PI))))
    (cos (* 0.125 (* a (* b (fabs t))))))
   (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0))) <= 2e+252) {
		tmp = (x * sin(fma(-0.0625, (fabs(t) * fabs(z)), (0.5 * ((double) M_PI))))) * cos((0.125 * (a * (b * fabs(t)))));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))) <= 2e+252)
		tmp = Float64(Float64(x * sin(fma(-0.0625, Float64(abs(t) * abs(z)), Float64(0.5 * pi)))) * cos(Float64(0.125 * Float64(a * Float64(b * abs(t))))));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+252], N[(N[(x * N[Sin[N[(-0.0625 * N[(N[Abs[t], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.125 * N[(a * N[(b * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.0000000000000002e252

   1. Initial program 27.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 27.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 28.2%

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

   6. Applied rewrites 28.2%

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

   7. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(-0.0625, t \cdot z, 0.5 \cdot \pi\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 28.3%

         \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(-0.0625, t \cdot z, 0.5 \cdot \pi\right)\right)\right) \cdot \cos \left(0.125 \cdot \left(a \cdot \left(b \cdot \color{blue}{t}\right)\right)\right) \]

   9. Applied rewrites 28.3%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(-0.0625, t \cdot z, 0.5 \cdot \pi\right)\right)\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)} \]

   if 2.0000000000000002e252 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 10: 31.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 10^{+26}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x\right) \cdot \cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(y + y, z, z\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      1e+26)
   (*
    (* (cos (* -0.0625 (* b t))) x)
    (cos (* -0.0625 (* (fma (+ y y) z z) t))))
   (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+26) {
		tmp = (cos((-0.0625 * (b * t))) * x) * cos((-0.0625 * (fma((y + y), z, z) * t)));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 1e+26)
		tmp = Float64(Float64(cos(Float64(-0.0625 * Float64(b * t))) * x) * cos(Float64(-0.0625 * Float64(fma(Float64(y + y), z, z) * t))));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+26], N[(N[(N[Cos[N[(-0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(-0.0625 * N[(N[(N[(y + y), $MachinePrecision] * z + z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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)))) < 1.00000000000000005e26

   1. Initial program 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 28.3%

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

   if 1.00000000000000005e26 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 11: 31.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.1 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(a \cdot \mathsf{fma}\left(2, \left|t\right|, \frac{\left|t\right|}{a}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (fabs t) 4.1e-45)
   (* x (cos (* 0.0625 (* b (* a (fma 2.0 (fabs t) (/ (fabs t) a)))))))
   (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (fabs(t) <= 4.1e-45) {
		tmp = x * cos((0.0625 * (b * (a * fma(2.0, fabs(t), (fabs(t) / a))))));
	} else {
		tmp = x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (abs(t) <= 4.1e-45)
		tmp = Float64(x * cos(Float64(0.0625 * Float64(b * Float64(a * fma(2.0, abs(t), Float64(abs(t) / a)))))));
	else
		tmp = Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Abs[t], $MachinePrecision], 4.1e-45], N[(x * N[Cos[N[(0.0625 * N[(b * N[(a * N[(2.0 * N[Abs[t], $MachinePrecision] + N[(N[Abs[t], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0999999999999999e-45

   1. Initial program 27.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 28.6%

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(a \cdot \left(2 \cdot t + \frac{t}{a}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 27.3%

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

   7. Applied rewrites 27.3%

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

   if 4.0999999999999999e-45 < t

   1. Initial program 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.3%

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 30.7%

         \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

   9. Applied rewrites 30.7%

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

Alternative 12: 30.7% accurate, 1.1× speedup

Math

\[x \cdot \left(\sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left|b\right| \cdot \left|t\right|\right) \cdot 0.0625\right)\right) \cdot \cos \left(0.0625 \cdot \left(\left|t\right| \cdot z\right)\right)\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  x
  (*
   (sin (fma PI 0.5 (* (* (fabs b) (fabs t)) 0.0625)))
   (cos (* 0.0625 (* (fabs t) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * (sin(fma(((double) M_PI), 0.5, ((fabs(b) * fabs(t)) * 0.0625))) * cos((0.0625 * (fabs(t) * z))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(sin(fma(pi, 0.5, Float64(Float64(abs(b) * abs(t)) * 0.0625))) * cos(Float64(0.0625 * Float64(abs(t) * z)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(N[Sin[N[(Pi * 0.5 + N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(N[Abs[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 28.3%

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

4. Applied rewrites 28.3%

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

5. Taylor expanded in y around 0

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 29.2%

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-PI.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 29.2%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 29.2%

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

9. Applied rewrites 29.2%

   \[\leadsto x \cdot \left(\sin \left(\mathsf{fma}\left(\pi, 0.5, \left(b \cdot t\right) \cdot 0.0625\right)\right) \cdot \cos \color{blue}{\left(0.0625 \cdot \left(t \cdot z\right)\right)}\right) \]
10. Add Preprocessing

Alternative 13: 29.7% accurate, 1.1× speedup

Math

\[x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left|t\right|\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.0625, \left|z\right| \cdot \left|t\right|, \pi \cdot 0.5\right)\right)\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  x
  (*
   (cos (* 0.0625 (* b (fabs t))))
   (sin (fma -0.0625 (* (fabs z) (fabs t)) (* PI 0.5))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * (cos((0.0625 * (b * fabs(t)))) * sin(fma(-0.0625, (fabs(z) * fabs(t)), (((double) M_PI) * 0.5))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(cos(Float64(0.0625 * Float64(b * abs(t)))) * sin(fma(-0.0625, Float64(abs(z) * abs(t)), Float64(pi * 0.5)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(N[Cos[N[(0.0625 * N[(b * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.0625 * N[(N[Abs[z], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 28.3%

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

4. Applied rewrites 28.3%

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

5. Taylor expanded in y around 0

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 29.2%

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 29.1%

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

9. Applied rewrites 29.1%

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.0625, z \cdot t, \pi \cdot 0.5\right)\right)\right) \]
10. Add Preprocessing

Alternative 14: 29.2% accurate, 1.2× speedup

Math

\[x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (* (cos (* 0.0625 (* b t))) (cos (* 0.0625 (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * (cos((0.0625 * (b * t))) * cos((0.0625 * (t * z))));
}

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((0.0625d0 * (b * t))) * cos((0.0625d0 * (t * z))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * (Math.cos((0.0625 * (b * t))) * Math.cos((0.0625 * (t * z))));
}

Python

def code(x, y, z, t, a, b):
	return x * (math.cos((0.0625 * (b * t))) * math.cos((0.0625 * (t * z))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t))) * cos(Float64(0.0625 * Float64(t * z)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * (cos((0.0625 * (b * t))) * cos((0.0625 * (t * z))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(N[Cos[N[(0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 28.3%

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

4. Applied rewrites 28.3%

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

5. Taylor expanded in y around 0

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 29.2%

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right) \]
8. Add Preprocessing

Alternative 15: 29.2% accurate, 2.2× speedup

Math

\[x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (/ (* 2.0 (sin (* 0.5 PI))) 2.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * ((2.0 * sin((0.5 * ((double) M_PI)))) / 2.0);
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * ((2.0 * Math.sin((0.5 * Math.PI))) / 2.0);
}

Python

def code(x, y, z, t, a, b):
	return x * ((2.0 * math.sin((0.5 * math.pi))) / 2.0)

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(Float64(2.0 * sin(Float64(0.5 * pi))) / 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * ((2.0 * sin((0.5 * pi))) / 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(N[(2.0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 28.3%

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

4. Applied rewrites 28.3%

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

6. Applied rewrites 26.8%

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

7. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 30.7%

      \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]

9. Applied rewrites 30.7%

   \[\leadsto x \cdot \frac{2 \cdot \sin \left(0.5 \cdot \pi\right)}{2} \]
10. Add Preprocessing

Alternative 16: 29.2% accurate, 2.4× speedup

Math

\[\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (cos (* -0.0625 (* b t))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return cos((-0.0625 * (b * t))) * x;
}

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 = cos(((-0.0625d0) * (b * t))) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return Math.cos((-0.0625 * (b * t))) * x;
}

Python

def code(x, y, z, t, a, b):
	return math.cos((-0.0625 * (b * t))) * x

Julia

function code(x, y, z, t, a, b)
	return Float64(cos(Float64(-0.0625 * Float64(b * t))) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = cos((-0.0625 * (b * t))) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Cos[N[(-0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 28.3%

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

4. Applied rewrites 28.3%

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

5. Taylor expanded in z around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 29.7%

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]

7. Applied rewrites 29.7%

   \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 29.7%

      \[\leadsto \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{x} \]

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

      \[\leadsto \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot x \]

   6. lift-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \cdot x \]

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

      \[\leadsto \cos \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right)\right) \cdot x \]

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 29.7%

       \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot x \]

9. Applied rewrites 29.7%

   \[\leadsto \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))))