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

Percentage Accurate: 27.2% → 30.4%

Time: 20.8s

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: 27.2% 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: 30.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(e^{\log 2 + {\left(b \cdot \left(-t\right)\right)}^{2} \cdot -0.0009765625} + -1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (+ (exp (+ (log 2.0) (* (pow (* b (- t)) 2.0) -0.0009765625))) -1.0) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (exp((log(2.0) + (pow((b * -t), 2.0) * -0.0009765625))) + -1.0) * x;
}

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 = (exp((log(2.0d0) + (((b * -t) ** 2.0d0) * (-0.0009765625d0)))) + (-1.0d0)) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (Math.exp((Math.log(2.0) + (Math.pow((b * -t), 2.0) * -0.0009765625))) + -1.0) * x;
}

Python

def code(x, y, z, t, a, b):
	return (math.exp((math.log(2.0) + (math.pow((b * -t), 2.0) * -0.0009765625))) + -1.0) * x

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(exp(Float64(log(2.0) + Float64((Float64(b * Float64(-t)) ^ 2.0) * -0.0009765625))) + -1.0) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (exp((log(2.0) + (((b * -t) ^ 2.0) * -0.0009765625))) + -1.0) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(b * (-t)), $MachinePrecision], 2.0], $MachinePrecision] * -0.0009765625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 29.3%

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

5. Taylor expanded in z around 0 30.8%

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

6. Taylor expanded in a around 0 31.9%

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

   1. *-commutative 31.9%

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

   2. associate-*r* 31.9%

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

8. Simplified 31.9%

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

   1. expm1-log1p-u 31.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)\right)} \cdot x \]

   2. expm1-undefine 31.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)} - 1\right)} \cdot x \]

   3. *-commutative 31.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(t \cdot \left(-0.0625 \cdot b\right)\right)}\right)} - 1\right) \cdot x \]

   4. *-commutative 31.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \left(t \cdot \color{blue}{\left(b \cdot -0.0625\right)}\right)\right)} - 1\right) \cdot x \]

10. Applied egg-rr 31.9%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(t \cdot \left(b \cdot -0.0625\right)\right)\right)} - 1\right)} \cdot x \]

11. Taylor expanded in t around 0 31.6%

    \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({b}^{2} \cdot {t}^{2}\right)}} - 1\right) \cdot x \]
12. Step-by-step derivation

    1. *-commutative 31.6%

       \[\leadsto \left(e^{\log 2 + \color{blue}{\left({b}^{2} \cdot {t}^{2}\right) \cdot -0.0009765625}} - 1\right) \cdot x \]

    2. *-commutative 31.6%

       \[\leadsto \left(e^{\log 2 + \color{blue}{\left({t}^{2} \cdot {b}^{2}\right)} \cdot -0.0009765625} - 1\right) \cdot x \]

    3. unpow2 31.6%

       \[\leadsto \left(e^{\log 2 + \left(\color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}\right) \cdot -0.0009765625} - 1\right) \cdot x \]

    4. unpow2 31.6%

       \[\leadsto \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.0009765625} - 1\right) \cdot x \]

    5. sqr-neg 31.6%

       \[\leadsto \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)}\right) \cdot -0.0009765625} - 1\right) \cdot x \]

    6. swap-sqr 33.8%

       \[\leadsto \left(e^{\log 2 + \color{blue}{\left(\left(t \cdot \left(-b\right)\right) \cdot \left(t \cdot \left(-b\right)\right)\right)} \cdot -0.0009765625} - 1\right) \cdot x \]

    7. unpow1 33.8%

       \[\leadsto \left(e^{\log 2 + \left(\color{blue}{{\left(t \cdot \left(-b\right)\right)}^{1}} \cdot \left(t \cdot \left(-b\right)\right)\right) \cdot -0.0009765625} - 1\right) \cdot x \]

    8. pow-plus 33.8%

       \[\leadsto \left(e^{\log 2 + \color{blue}{{\left(t \cdot \left(-b\right)\right)}^{\left(1 + 1\right)}} \cdot -0.0009765625} - 1\right) \cdot x \]

    9. distribute-rgt-neg-in 33.8%

       \[\leadsto \left(e^{\log 2 + {\color{blue}{\left(-t \cdot b\right)}}^{\left(1 + 1\right)} \cdot -0.0009765625} - 1\right) \cdot x \]

    10. *-commutative 33.8%

        \[\leadsto \left(e^{\log 2 + {\left(-\color{blue}{b \cdot t}\right)}^{\left(1 + 1\right)} \cdot -0.0009765625} - 1\right) \cdot x \]

    11. distribute-rgt-neg-in 33.8%

        \[\leadsto \left(e^{\log 2 + {\color{blue}{\left(b \cdot \left(-t\right)\right)}}^{\left(1 + 1\right)} \cdot -0.0009765625} - 1\right) \cdot x \]

    12. metadata-eval 33.8%

        \[\leadsto \left(e^{\log 2 + {\left(b \cdot \left(-t\right)\right)}^{\color{blue}{2}} \cdot -0.0009765625} - 1\right) \cdot x \]

13. Simplified 33.8%

    \[\leadsto \left(e^{\color{blue}{\log 2 + {\left(b \cdot \left(-t\right)\right)}^{2} \cdot -0.0009765625}} - 1\right) \cdot x \]

14. Final simplification 33.8%

    \[\leadsto \left(e^{\log 2 + {\left(b \cdot \left(-t\right)\right)}^{2} \cdot -0.0009765625} + -1\right) \cdot x \]
15. Add Preprocessing

Alternative 2: 30.3% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 28.9%

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

3. Simplified 28.9%

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

5. Taylor expanded in t around 0 33.2%

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

Developer Target 1: 29.9% 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 2024131 
(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))))