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

Percentage Accurate: 27.6% → 30.7%

Time: 10.5s

Alternatives: 3

Speedup: 269.0×

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: 27.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 30.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x\_m \cdot \left(\sin \left(\left(-\frac{t\_m}{16} \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\_m\right)\right) + \frac{\pi}{2}\right) \cdot \cos \left(\left(\left(a + a\right) \cdot b\right) \cdot \frac{t\_m}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t\_m, 0.5 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+54)
    (*
     x_m
     (*
      (sin (+ (- (* (/ t_m 16.0) (* (fma 2.0 y 1.0) z_m))) (/ PI 2.0)))
      (cos (* (* (+ a a) b) (/ t_m 16.0)))))
    (* x_m (sin (fma -0.0625 (* b t_m) (* 0.5 PI)))))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+54) {
		tmp = x_m * (sin((-((t_m / 16.0) * (fma(2.0, y, 1.0) * z_m)) + (((double) M_PI) / 2.0))) * cos((((a + a) * b) * (t_m / 16.0))));
	} else {
		tmp = x_m * sin(fma(-0.0625, (b * t_m), (0.5 * ((double) M_PI))));
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+54)
		tmp = Float64(x_m * Float64(sin(Float64(Float64(-Float64(Float64(t_m / 16.0) * Float64(fma(2.0, y, 1.0) * z_m))) + Float64(pi / 2.0))) * cos(Float64(Float64(Float64(a + a) * b) * Float64(t_m / 16.0)))));
	else
		tmp = Float64(x_m * sin(fma(-0.0625, Float64(b * t_m), Float64(0.5 * pi))));
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+54], N[(x$95$m * N[(N[Sin[N[((-N[(N[(t$95$m / 16.0), $MachinePrecision] * N[(N[(2.0 * y + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(a + a), $MachinePrecision] * b), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[Sin[N[(-0.0625 * N[(b * t$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.0000000000000002e54

   1. Initial program 40.9%

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

   3. Applied rewrites 40.9%

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

   4. Taylor expanded in a around inf

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

      1. count-2-rev N/A

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

      2. lower-+.f64 40.3

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

   6. Applied rewrites 40.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. cos-neg-rev N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 40.3

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

   8. Applied rewrites 40.3%

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

   if 2.0000000000000002e54 < (*.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 17.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 19.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 23.0

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

   7. Applied rewrites 23.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cos-neg-rev N/A

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

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

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

      6. lower-sin.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-PI.f64 23.0

          \[\leadsto x \cdot \sin \left(\left(-0.0625 \cdot \left(b \cdot t\right)\right) + \frac{\pi}{2}\right) \]

   9. Applied rewrites 23.0%

      \[\leadsto x \cdot \sin \left(\left(-0.0625 \cdot \left(b \cdot t\right)\right) + \frac{\pi}{2}\right) \]

   10. Taylor expanded in t around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-PI.f64 23.0

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

   12. Applied rewrites 23.0%

       \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 30.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(\frac{\left(t\_m \cdot z\_m\right) \cdot \mathsf{fma}\left(2, y, 1\right)}{16}\right) \cdot \cos \left(\left(\left(a + a\right) \cdot b\right) \cdot \frac{t\_m}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t\_m, 0.5 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       2e+54)
    (*
     x_m
     (*
      (cos (/ (* (* t_m z_m) (fma 2.0 y 1.0)) 16.0))
      (cos (* (* (+ a a) b) (/ t_m 16.0)))))
    (* x_m (sin (fma -0.0625 (* b t_m) (* 0.5 PI)))))))

C

z_m = fabs(z);
t_m = fabs(t);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+54) {
		tmp = x_m * (cos((((t_m * z_m) * fma(2.0, y, 1.0)) / 16.0)) * cos((((a + a) * b) * (t_m / 16.0))));
	} else {
		tmp = x_m * sin(fma(-0.0625, (b * t_m), (0.5 * ((double) M_PI))));
	}
	return x_s * tmp;
}

Julia

z_m = abs(z)
t_m = abs(t)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0))) <= 2e+54)
		tmp = Float64(x_m * Float64(cos(Float64(Float64(Float64(t_m * z_m) * fma(2.0, y, 1.0)) / 16.0)) * cos(Float64(Float64(Float64(a + a) * b) * Float64(t_m / 16.0)))));
	else
		tmp = Float64(x_m * sin(fma(-0.0625, Float64(b * t_m), Float64(0.5 * pi))));
	end
	return Float64(x_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+54], N[(x$95$m * N[(N[Cos[N[(N[(N[(t$95$m * z$95$m), $MachinePrecision] * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(a + a), $MachinePrecision] * b), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[Sin[N[(-0.0625 * N[(b * t$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.0000000000000002e54

   1. Initial program 40.9%

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

   3. Applied rewrites 40.9%

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

   4. Taylor expanded in a around inf

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

      1. count-2-rev N/A

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

      2. lower-+.f64 40.3

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

   6. Applied rewrites 40.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. +-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-fma.f64 40.3

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

   8. Applied rewrites 40.3%

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

   if 2.0000000000000002e54 < (*.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 17.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 19.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 23.0

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

   7. Applied rewrites 23.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cos-neg-rev N/A

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

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

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

      6. lower-sin.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-PI.f64 23.0

          \[\leadsto x \cdot \sin \left(\left(-0.0625 \cdot \left(b \cdot t\right)\right) + \frac{\pi}{2}\right) \]

   9. Applied rewrites 23.0%

      \[\leadsto x \cdot \sin \left(\left(-0.0625 \cdot \left(b \cdot t\right)\right) + \frac{\pi}{2}\right) \]

   10. Taylor expanded in t around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-PI.f64 23.0

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

   12. Applied rewrites 23.0%

       \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, 0.5 \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 30.4% accurate, 269.0× speedup

Math

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

FPCore

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

C

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

Fortran

z_m =     private
t_m =     private
x\_m =     private
x\_s =     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_s, x_m, y, z_m, t_m, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * x_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.6%

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

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 30.7%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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))))