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

Average Error: 46.1 → 44.1

Time: 11.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.2536304482267679 \cdot 10^{-95} \lor \neg \left(t \le 1.8923341242733495 \cdot 10^{-169}\right):\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (x * ((double) cos(((double) (((double) (((double) (((double) (((double) (y * 2.0)) + 1.0)) * z)) * t)) / 16.0)))))) * ((double) cos(((double) (((double) (((double) (((double) (((double) (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 (((t <= -3.253630448226768e-95) || !(t <= 1.8923341242733495e-169))) {
		VAR = ((double) (x * 1.0));
	} else {
		VAR = ((double) (((double) (x * ((double) cos(((double) (((double) (((double) (((double) (((double) (y * 2.0)) + 1.0)) * z)) * t)) / 16.0)))))) * ((double) cos(((double) (((double) (((double) (((double) (a * 2.0)) + 1.0)) * ((double) (b * t)))) / 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	46.1
Target	44.6
Herbie	44.1

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

Derivation

1. Split input into 2 regimes

2. if t < -3.253630448226768e-95 or 1.8923341242733495e-169 < t

   1. Initial program 53.9

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

   2. Taylor expanded around 0 53.0

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

   3. Taylor expanded around 0 51.3

      \[\leadsto \color{blue}{x} \cdot 1\]

   if -3.253630448226768e-95 < t < 1.8923341242733495e-169

   1. Initial program 26.9

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

   3. Applied associate-*l* 26.1

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}}{16}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.2536304482267679 \cdot 10^{-95} \lor \neg \left(t \le 1.8923341242733495 \cdot 10^{-169}\right):\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))))