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

Percentage Accurate: 27.0% → 31.5%

Time: 10.6s

Alternatives: 6

Speedup: 13.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 31.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+286}:\\ \;\;\;\;t\_1 \cdot \sin \left(\mathsf{fma}\left(-0.0625, \mathsf{fma}\left(b, a + a, b\right) \cdot t\_m, 0.5 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))) 2e+286)
     (* t_1 (sin (fma -0.0625 (* (fma b (+ a a) b) t_m) (* 0.5 PI))))
     (* x (/ 2.0 2.0)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286) {
		tmp = t_1 * sin(fma(-0.0625, (fma(b, (a + a), b) * t_m), (0.5 * ((double) M_PI))));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286)
		tmp = Float64(t_1 * sin(fma(-0.0625, Float64(fma(b, Float64(a + a), b) * t_m), Float64(0.5 * pi))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, 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$95$m), $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$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+286], N[(t$95$1 * N[Sin[N[(-0.0625 * N[(N[(b * N[(a + a), $MachinePrecision] + b), $MachinePrecision] * t$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 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)))) < 2.00000000000000007e286

   1. Initial program 27.0%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mult-flip-rev N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 27.0%

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

   if 2.00000000000000007e286 < (*.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 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 26.3%

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.4%

       \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 31.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t\_m \cdot z\right) \cdot 0.0625\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(-0.0625 \cdot t\_m, b, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+286)
   (*
    (* (cos (* (fma 2.0 y 1.0) (* (* t_m z) 0.0625))) x)
    (sin (fma (* -0.0625 t_m) b (* PI 0.5))))
   (* x (/ 2.0 2.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286) {
		tmp = (cos((fma(2.0, y, 1.0) * ((t_m * z) * 0.0625))) * x) * sin(fma((-0.0625 * t_m), b, (((double) M_PI) * 0.5)));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286)
		tmp = Float64(Float64(cos(Float64(fma(2.0, y, 1.0) * Float64(Float64(t_m * z) * 0.0625))) * x) * sin(fma(Float64(-0.0625 * t_m), b, Float64(pi * 0.5))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+286], N[(N[(N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(N[(t$95$m * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Sin[N[(N[(-0.0625 * t$95$m), $MachinePrecision] * b + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 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)))) < 2.00000000000000007e286

   1. Initial program 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-+.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-fma.f64 N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 28.1%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lift-PI.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-PI.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip-rev N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. mult-flip-rev N/A

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

      20. metadata-eval N/A

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

      21. lift-*.f64 28.2

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

   10. Applied rewrites 28.2%

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

   if 2.00000000000000007e286 < (*.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 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 26.3%

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.4%

       \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 31.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 10^{+273}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\_m\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(t\_m \cdot z\right) \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      1e+273)
   (*
    x
    (*
     (cos (* 0.0625 (* b t_m)))
     (cos (* (fma 2.0 y 1.0) (* (* t_m z) 0.0625)))))
   (* x (/ 2.0 2.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+273) {
		tmp = x * (cos((0.0625 * (b * t_m))) * cos((fma(2.0, y, 1.0) * ((t_m * z) * 0.0625))));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 1e+273)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t_m))) * cos(Float64(fma(2.0, y, 1.0) * Float64(Float64(t_m * z) * 0.0625)))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+273], N[(x * N[(N[Cos[N[(0.0625 * N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(N[(t$95$m * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 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)))) < 9.99999999999999945e272

   1. Initial program 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 28.1

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

   6. Applied rewrites 28.1%

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

   if 9.99999999999999945e272 < (*.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 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 26.3%

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.4%

       \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 31.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\left(\cos \left(\left(0.0625 \cdot \mathsf{fma}\left(y + y, z, z\right)\right) \cdot t\_m\right) \cdot x\right) \cdot \cos \left(-0.0625 \cdot \left(b \cdot t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+286)
   (*
    (* (cos (* (* 0.0625 (fma (+ y y) z z)) t_m)) x)
    (cos (* -0.0625 (* b t_m))))
   (* x (/ 2.0 2.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286) {
		tmp = (cos(((0.0625 * fma((y + y), z, z)) * t_m)) * x) * cos((-0.0625 * (b * t_m)));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286)
		tmp = Float64(Float64(cos(Float64(Float64(0.0625 * fma(Float64(y + y), z, z)) * t_m)) * x) * cos(Float64(-0.0625 * Float64(b * t_m))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+286], N[(N[(N[Cos[N[(N[(0.0625 * N[(N[(y + y), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(-0.0625 * N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 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)))) < 2.00000000000000007e286

   1. Initial program 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.7%

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

   if 2.00000000000000007e286 < (*.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 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 26.3%

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.4%

       \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 30.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+286}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\_m\right)\right) \cdot \cos \left(0.125 \cdot \left(t\_m \cdot \left(y \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
      2e+286)
   (* x (* (cos (* 0.0625 (* b t_m))) (cos (* 0.125 (* t_m (* y z))))))
   (* x (/ 2.0 2.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286) {
		tmp = x * (cos((0.0625 * (b * t_m))) * cos((0.125 * (t_m * (y * z)))));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Fortran

t_m =     private
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_m, 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_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t_m) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t_m) / 16.0d0))) <= 2d+286) then
        tmp = x * (cos((0.0625d0 * (b * t_m))) * cos((0.125d0 * (t_m * (y * z)))))
    else
        tmp = x * (2.0d0 / 2.0d0)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286) {
		tmp = x * (Math.cos((0.0625 * (b * t_m))) * Math.cos((0.125 * (t_m * (y * z)))));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286:
		tmp = x * (math.cos((0.0625 * (b * t_m))) * math.cos((0.125 * (t_m * (y * z)))))
	else:
		tmp = x * (2.0 / 2.0)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * t_m))) * cos(Float64(0.125 * Float64(t_m * Float64(y * z))))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+286)
		tmp = x * (cos((0.0625 * (b * t_m))) * cos((0.125 * (t_m * (y * z)))));
	else
		tmp = x * (2.0 / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+286], N[(x * N[(N[Cos[N[(0.0625 * N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.125 * N[(t$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 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)))) < 2.00000000000000007e286

   1. Initial program 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 27.6

         \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)\right) \]

   if 2.00000000000000007e286 < (*.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 27.0%

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

   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 27.7

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 26.3%

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.4%

       \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 30.4% accurate, 13.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \cdot \frac{2}{2} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b) :precision binary64 (* x (/ 2.0 2.0)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return x * (2.0 / 2.0);
}

Fortran

t_m =     private
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_m, 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_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * (2.0d0 / 2.0d0)
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	return x * (2.0 / 2.0);
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return x * (2.0 / 2.0)

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(x * Float64(2.0 / 2.0))
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = x * (2.0 / 2.0);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(x * N[(2.0 / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.0%

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

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 27.7

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

4. Applied rewrites 27.7%

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

6. Applied rewrites 26.3%

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

8. Applied rewrites 27.6%

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

9. Taylor expanded in t around 0

   \[\leadsto x \cdot \frac{2}{2} \]
10. Step-by-step derivation

11. Applied rewrites 30.4%

    \[\leadsto x \cdot \frac{2}{2} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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))))