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

Average Error: 46.0 → 43.2

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \le 6.4513146009910377 \cdot 10^{296}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\\ \end{array}\]

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) {
	double VAR;
	if ((((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 6.451314600991038e+296)) {
		VAR = ((x * cos(((((y * 2.0) + 1.0) * (z * t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0)));
	} else {
		VAR = ((x * cos((0.0 / 16.0))) * cos((0.0 / 16.0)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	46.0
Target	44.3
Herbie	43.2

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

Derivation

1. Split input into 2 regimes

2. if (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))) < 6.451314600991038e+296

   1. Initial program 33.5

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

   3. Applied associate-*l* 33.4

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

   if 6.451314600991038e+296 < (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))

   1. Initial program 63.7

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

   2. Taylor expanded around 0 60.8

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

   3. Taylor expanded around 0 56.8

      \[\leadsto \left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \le 6.4513146009910377 \cdot 10^{296}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2))))))

  (* (* x (cos (/ (* (* (+ (* y 2) 1) z) t) 16))) (cos (/ (* (* (+ (* a 2) 1) b) t) 16))))