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

Average Error: 46.6 → 44.5

Time: 20.7s

Precision: binary64

Cost: 64

Math

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

↓

\[x \]

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))))

↓

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

↓

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 * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

↓

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 * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

↓

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

↓

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

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

↓

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

↓

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

↓

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

Target

Original	46.6
Target	44.7
Herbie	44.5

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

Derivation

1. Initial program 46.6

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

2. Simplified 46.1

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\frac{a}{8} + 0.0625\right)\right)\right)} \]
   Proof

   [Start] 46.6	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   associate-*l* [=>] 46.6	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]
   *-lft-identity [<=] 46.6	\[ x \cdot \left(\cos \color{blue}{\left(1 \cdot \frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-*l* [=>] 46.3	\[ x \cdot \left(\cos \left(1 \cdot \frac{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-/l* [=>] 46.3	\[ x \cdot \left(\cos \left(1 \cdot \color{blue}{\frac{y \cdot 2 + 1}{\frac{16}{z \cdot t}}}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-*r/ [=>] 46.3	\[ x \cdot \left(\cos \color{blue}{\left(\frac{1 \cdot \left(y \cdot 2 + 1\right)}{\frac{16}{z \cdot t}}\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-*l/ [<=] 46.3	\[ x \cdot \left(\cos \color{blue}{\left(\frac{1}{\frac{16}{z \cdot t}} \cdot \left(y \cdot 2 + 1\right)\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-/r/ [=>] 46.3	\[ x \cdot \left(\cos \left(\color{blue}{\left(\frac{1}{16} \cdot \left(z \cdot t\right)\right)} \cdot \left(y \cdot 2 + 1\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   *-commutative [=>] 46.3	\[ x \cdot \left(\cos \left(\color{blue}{\left(\left(z \cdot t\right) \cdot \frac{1}{16}\right)} \cdot \left(y \cdot 2 + 1\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-*r* [<=] 46.3	\[ x \cdot \left(\cos \color{blue}{\left(\left(z \cdot t\right) \cdot \left(\frac{1}{16} \cdot \left(y \cdot 2 + 1\right)\right)\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   *-commutative [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \color{blue}{\left(\left(y \cdot 2 + 1\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   distribute-rgt1-in [<=] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \color{blue}{\left(\frac{1}{16} + \left(y \cdot 2\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   *-commutative [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(\frac{1}{16} + \color{blue}{\frac{1}{16} \cdot \left(y \cdot 2\right)}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-/r/ [<=] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(\frac{1}{16} + \color{blue}{\frac{1}{\frac{16}{y \cdot 2}}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-/l* [<=] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(\frac{1}{16} + \color{blue}{\frac{1 \cdot \left(y \cdot 2\right)}{16}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-*r/ [<=] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(\frac{1}{16} + \color{blue}{1 \cdot \frac{y \cdot 2}{16}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   *-lft-identity [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(\frac{1}{16} + \color{blue}{\frac{y \cdot 2}{16}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   metadata-eval [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(\color{blue}{0.0625} + \frac{y \cdot 2}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   associate-/l* [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \color{blue}{\frac{y}{\frac{16}{2}}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   metadata-eval [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{\color{blue}{8}}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
   *-lft-identity [<=] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(1 \cdot \frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)}\right) \]
   associate-*r/ [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(\frac{1 \cdot \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\right)}{16}\right)}\right) \]
   associate-/l* [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{16}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}\right)}\right) \]
   associate-/r/ [=>] 46.3	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\right)\right)}\right) \]
   associate-*l* [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)\right)}\right)\right) \]
   associate-*r* [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(\left(\frac{1}{16} \cdot \left(a \cdot 2 + 1\right)\right) \cdot \left(b \cdot t\right)\right)}\right) \]
   *-commutative [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot \frac{1}{16}\right)} \cdot \left(b \cdot t\right)\right)\right) \]
   distribute-lft1-in [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(a \cdot 2\right) \cdot \frac{1}{16} + \frac{1}{16}\right)} \cdot \left(b \cdot t\right)\right)\right) \]
   *-commutative [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(\color{blue}{\frac{1}{16} \cdot \left(a \cdot 2\right)} + \frac{1}{16}\right) \cdot \left(b \cdot t\right)\right)\right) \]
   associate-/r/ [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(\color{blue}{\frac{1}{\frac{16}{a \cdot 2}}} + \frac{1}{16}\right) \cdot \left(b \cdot t\right)\right)\right) \]
   associate-/l* [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(\color{blue}{\frac{1 \cdot \left(a \cdot 2\right)}{16}} + \frac{1}{16}\right) \cdot \left(b \cdot t\right)\right)\right) \]
   associate-*r/ [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(\color{blue}{1 \cdot \frac{a \cdot 2}{16}} + \frac{1}{16}\right) \cdot \left(b \cdot t\right)\right)\right) \]
   *-lft-identity [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(\color{blue}{\frac{a \cdot 2}{16}} + \frac{1}{16}\right) \cdot \left(b \cdot t\right)\right)\right) \]
   *-commutative [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \left(\frac{a \cdot 2}{16} + \frac{1}{16}\right)\right)}\right) \]
   +-commutative [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \color{blue}{\left(\frac{1}{16} + \frac{a \cdot 2}{16}\right)}\right)\right) \]
   *-commutative [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\color{blue}{\left(t \cdot b\right)} \cdot \left(\frac{1}{16} + \frac{a \cdot 2}{16}\right)\right)\right) \]
   +-commutative [<=] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \color{blue}{\left(\frac{a \cdot 2}{16} + \frac{1}{16}\right)}\right)\right) \]
   associate-/l* [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\color{blue}{\frac{a}{\frac{16}{2}}} + \frac{1}{16}\right)\right)\right) \]
   metadata-eval [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\frac{a}{\color{blue}{8}} + \frac{1}{16}\right)\right)\right) \]
   metadata-eval [=>] 46.1	\[ x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\frac{a}{8} + \color{blue}{0.0625}\right)\right)\right) \]

3. Taylor expanded in t around 0 45.5

   \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

4. Taylor expanded in z around 0 44.5

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

5. Final simplification 44.5

   \[\leadsto x \]

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* 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))))