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

Percentage Accurate: 27.9% → 30.7%

Time: 25.7s

Alternatives: 2

Speedup: 225.0×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \cdot \cos \left(\left(e^{z\_m \cdot \left(1 + \left(2 \cdot y + -0.5 \cdot \left(z\_m \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} + -1\right) \cdot \frac{t}{16}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b)
 :precision binary64
 (*
  x
  (cos
   (*
    (+
     (exp
      (*
       z_m
       (+ 1.0 (+ (* 2.0 y) (* -0.5 (* z_m (pow (+ 1.0 (* 2.0 y)) 2.0)))))))
     -1.0)
    (/ t 16.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	return x * cos(((exp((z_m * (1.0 + ((2.0 * y) + (-0.5 * (z_m * pow((1.0 + (2.0 * y)), 2.0))))))) + -1.0) * (t / 16.0)));
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * cos(((exp((z_m * (1.0d0 + ((2.0d0 * y) + ((-0.5d0) * (z_m * ((1.0d0 + (2.0d0 * y)) ** 2.0d0))))))) + (-1.0d0)) * (t / 16.0d0)))
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t, double a, double b) {
	return x * Math.cos(((Math.exp((z_m * (1.0 + ((2.0 * y) + (-0.5 * (z_m * Math.pow((1.0 + (2.0 * y)), 2.0))))))) + -1.0) * (t / 16.0)));
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t, a, b):
	return x * math.cos(((math.exp((z_m * (1.0 + ((2.0 * y) + (-0.5 * (z_m * math.pow((1.0 + (2.0 * y)), 2.0))))))) + -1.0) * (t / 16.0)))

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	return Float64(x * cos(Float64(Float64(exp(Float64(z_m * Float64(1.0 + Float64(Float64(2.0 * y) + Float64(-0.5 * Float64(z_m * (Float64(1.0 + Float64(2.0 * y)) ^ 2.0))))))) + -1.0) * Float64(t / 16.0))))
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m, t, a, b)
	tmp = x * cos(((exp((z_m * (1.0 + ((2.0 * y) + (-0.5 * (z_m * ((1.0 + (2.0 * y)) ^ 2.0))))))) + -1.0) * (t / 16.0)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := N[(x * N[Cos[N[(N[(N[Exp[N[(z$95$m * N[(1.0 + N[(N[(2.0 * y), $MachinePrecision] + N[(-0.5 * N[(z$95$m * N[Power[N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(t / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.5%

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

3. Simplified 24.5%

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

5. Taylor expanded in b around 0 26.8%

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

   1. expm1-log1p-u 22.1%

      \[\leadsto x \cdot \left(\cos \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)\right)} \cdot \frac{t}{16}\right) \cdot 1\right) \]

   2. expm1-undefine 22.1%

      \[\leadsto x \cdot \left(\cos \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)} - 1\right)} \cdot \frac{t}{16}\right) \cdot 1\right) \]

7. Applied egg-rr 22.1%

   \[\leadsto x \cdot \left(\cos \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)} - 1\right)} \cdot \frac{t}{16}\right) \cdot 1\right) \]

8. Taylor expanded in z around 0 28.8%

   \[\leadsto x \cdot \left(\cos \left(\left(e^{\color{blue}{z \cdot \left(1 + \left(-0.5 \cdot \left(z \cdot {\left(1 + 2 \cdot y\right)}^{2}\right) + 2 \cdot y\right)\right)}} - 1\right) \cdot \frac{t}{16}\right) \cdot 1\right) \]

9. Final simplification 28.8%

   \[\leadsto x \cdot \cos \left(\left(e^{z \cdot \left(1 + \left(2 \cdot y + -0.5 \cdot \left(z \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} + -1\right) \cdot \frac{t}{16}\right) \]
10. Add Preprocessing

Alternative 2: 30.8% accurate, 225.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t a b) :precision binary64 x)

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t, double a, double b) {
	return x;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

z_m = math.fabs(z)
def code(x, y, z_m, t, a, b):
	return x

Julia

z_m = abs(z)
function code(x, y, z_m, t, a, b)
	return x
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m, t, a, b)
	tmp = x;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_, a_, b_] := x

Derivation

1. Initial program 24.5%

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

3. Simplified 24.5%

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

5. Taylor expanded in b around 0 26.8%

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

6. Taylor expanded in z around 0 28.2%

   \[\leadsto \color{blue}{x} \]

7. Final simplification 28.2%

   \[\leadsto x \]
8. Add Preprocessing

Developer target: 30.5% accurate, 1.0× 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

real(8) function code(x, y, z, t, a, b)
    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 2024130 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

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