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

Percentage Accurate: 27.7% → 32.0%

Time: 6.2s

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: 27.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+258}:\\ \;\;\;\;t\_1 \cdot \sin \left(\left(b\_m \cdot t\_m\right) \cdot 0.0625 + \frac{\pi}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 1e+258)
      (* t_1 (sin (+ (* (* b_m t_m) 0.0625) (/ PI 2.0))))
      x_m))))

C

t_m = fabs(t);
b_m = fabs(b);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	double t_1 = x_m * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+258) {
		tmp = t_1 * sin((((b_m * t_m) * 0.0625) + (((double) M_PI) / 2.0)));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Java

t_m = Math.abs(t);
b_m = Math.abs(b);
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, double t_m, double a, double b_m) {
	double t_1 = x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0));
	double tmp;
	if ((t_1 * Math.cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+258) {
		tmp = t_1 * Math.sin((((b_m * t_m) * 0.0625) + (Math.PI / 2.0)));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

t_m = math.fabs(t)
b_m = math.fabs(b)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t_m, a, b_m):
	t_1 = x_m * math.cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))
	tmp = 0
	if (t_1 * math.cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+258:
		tmp = t_1 * math.sin((((b_m * t_m) * 0.0625) + (math.pi / 2.0)))
	else:
		tmp = x_m
	return x_s * tmp

Julia

t_m = abs(t)
b_m = abs(b)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t_m, a, b_m)
	t_1 = Float64(x_m * 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_m) * t_m) / 16.0))) <= 1e+258)
		tmp = Float64(t_1 * sin(Float64(Float64(Float64(b_m * t_m) * 0.0625) + Float64(pi / 2.0))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

t_m = abs(t);
b_m = abs(b);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t_m, a, b_m)
	t_1 = x_m * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0));
	tmp = 0.0;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+258)
		tmp = t_1 * sin((((b_m * t_m) * 0.0625) + (pi / 2.0)));
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $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_, t$95$m_, a_, b$95$m_] := Block[{t$95$1 = N[(x$95$m * 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[(x$95$s * If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+258], N[(t$95$1 * N[Sin[N[(N[(N[(b$95$m * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.00000000000000006e258

   1. Initial program 46.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-+.f64 46.1

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

   6. Applied rewrites 46.1%

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

   if 1.00000000000000006e258 < (*.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 5.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 14.9%

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

Alternative 2: 30.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin \left(\mathsf{fma}\left(-0.0625, t\_m \cdot z, 0.5 \cdot \pi\right)\right) \cdot x\_m\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t_m a b_m)
 :precision binary64
 (* x_s (* (sin (fma -0.0625 (* t_m z) (* 0.5 PI))) x_m)))

C

t_m = fabs(t);
b_m = fabs(b);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t_m, double a, double b_m) {
	return x_s * (sin(fma(-0.0625, (t_m * z), (0.5 * ((double) M_PI)))) * x_m);
}

Julia

t_m = abs(t)
b_m = abs(b)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t_m, a, b_m)
	return Float64(x_s * Float64(sin(fma(-0.0625, Float64(t_m * z), Float64(0.5 * pi))) * x_m))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $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_, t$95$m_, a_, b$95$m_] := N[(x$95$s * N[(N[Sin[N[(-0.0625 * N[(t$95$m * z), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

2. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 29.0

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

4. Applied rewrites 29.0%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 30.1%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-neg.f64 30.2

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 30.2

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

9. Applied rewrites 30.2%

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

10. Taylor expanded in y around 0

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

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

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

    2. metadata-eval N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lift-PI.f64 30.2

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

12. Applied rewrites 30.2%

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

Alternative 3: 31.1% accurate, 269.0× speedup

Math

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

FPCore

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

C

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

Fortran

t_m =     private
b_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, t_m, a, b_m)
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
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = x_s * x_m
end function

Java

t_m = Math.abs(t);
b_m = Math.abs(b);
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, double t_m, double a, double b_m) {
	return x_s * x_m;
}

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $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_, t$95$m_, a_, b$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 31.1%

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

Developer Target 1: 30.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025105 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))