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

Average Error: 45.8 → 44.7

Time: 30.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 8.227621728721539868314394994500716678162 \cdot 10^{-310}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(b \cdot \frac{t}{16}\right) \cdot \left(2 \cdot a + 1\right)\right) \cdot \left(\cos \left(\left(\frac{\sqrt{t}}{\frac{\sqrt[3]{16}}{\sqrt[3]{1 + y \cdot 2}}} \cdot z\right) \cdot \frac{\sqrt{t}}{\frac{\sqrt[3]{16}}{\sqrt[3]{1 + y \cdot 2}} \cdot \frac{\sqrt[3]{16}}{\sqrt[3]{1 + y \cdot 2}}}\right) \cdot x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r668278 = x;
        double r668279 = y;
        double r668280 = 2.0;
        double r668281 = r668279 * r668280;
        double r668282 = 1.0;
        double r668283 = r668281 + r668282;
        double r668284 = z;
        double r668285 = r668283 * r668284;
        double r668286 = t;
        double r668287 = r668285 * r668286;
        double r668288 = 16.0;
        double r668289 = r668287 / r668288;
        double r668290 = cos(r668289);
        double r668291 = r668278 * r668290;
        double r668292 = a;
        double r668293 = r668292 * r668280;
        double r668294 = r668293 + r668282;
        double r668295 = b;
        double r668296 = r668294 * r668295;
        double r668297 = r668296 * r668286;
        double r668298 = r668297 / r668288;
        double r668299 = cos(r668298);
        double r668300 = r668291 * r668299;
        return r668300;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r668301 = t;
        double r668302 = 8.22762172872154e-310;
        bool r668303 = r668301 <= r668302;
        double r668304 = x;
        double r668305 = b;
        double r668306 = 16.0;
        double r668307 = r668301 / r668306;
        double r668308 = r668305 * r668307;
        double r668309 = 2.0;
        double r668310 = a;
        double r668311 = r668309 * r668310;
        double r668312 = 1.0;
        double r668313 = r668311 + r668312;
        double r668314 = r668308 * r668313;
        double r668315 = cos(r668314);
        double r668316 = sqrt(r668301);
        double r668317 = cbrt(r668306);
        double r668318 = y;
        double r668319 = r668318 * r668309;
        double r668320 = r668312 + r668319;
        double r668321 = cbrt(r668320);
        double r668322 = r668317 / r668321;
        double r668323 = r668316 / r668322;
        double r668324 = z;
        double r668325 = r668323 * r668324;
        double r668326 = r668322 * r668322;
        double r668327 = r668316 / r668326;
        double r668328 = r668325 * r668327;
        double r668329 = cos(r668328);
        double r668330 = r668329 * r668304;
        double r668331 = r668315 * r668330;
        double r668332 = r668303 ? r668304 : r668331;
        return r668332;
}

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	45.8
Target	44.1
Herbie	44.7

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

Derivation

1. Split input into 2 regimes

2. if t < 8.22762172872154e-310

   1. Initial program 45.7

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

   2. Simplified 45.4

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

   3. Taylor expanded around 0 44.9

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

   4. Taylor expanded around 0 44.0

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

   if 8.22762172872154e-310 < t

   1. Initial program 45.9

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

   2. Simplified 45.6

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

   4. Applied *-un-lft-identity 45.6

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

   5. Applied add-cube-cbrt 45.6

      \[\leadsto \left(\cos \left(\frac{t}{\frac{\frac{16}{\color{blue}{\left(\sqrt[3]{2 \cdot y + 1} \cdot \sqrt[3]{2 \cdot y + 1}\right) \cdot \sqrt[3]{2 \cdot y + 1}}}}{1 \cdot z}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   6. Applied add-cube-cbrt 45.6

      \[\leadsto \left(\cos \left(\frac{t}{\frac{\frac{\color{blue}{\left(\sqrt[3]{16} \cdot \sqrt[3]{16}\right) \cdot \sqrt[3]{16}}}{\left(\sqrt[3]{2 \cdot y + 1} \cdot \sqrt[3]{2 \cdot y + 1}\right) \cdot \sqrt[3]{2 \cdot y + 1}}}{1 \cdot z}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   7. Applied times-frac 45.6

      \[\leadsto \left(\cos \left(\frac{t}{\frac{\color{blue}{\frac{\sqrt[3]{16} \cdot \sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1} \cdot \sqrt[3]{2 \cdot y + 1}} \cdot \frac{\sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1}}}}{1 \cdot z}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   8. Applied times-frac 45.6

      \[\leadsto \left(\cos \left(\frac{t}{\color{blue}{\frac{\frac{\sqrt[3]{16} \cdot \sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1} \cdot \sqrt[3]{2 \cdot y + 1}}}{1} \cdot \frac{\frac{\sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1}}}{z}}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   9. Applied add-sqr-sqrt 45.6

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{\frac{\frac{\sqrt[3]{16} \cdot \sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1} \cdot \sqrt[3]{2 \cdot y + 1}}}{1} \cdot \frac{\frac{\sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1}}}{z}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   10. Applied times-frac 45.4

       \[\leadsto \left(\cos \color{blue}{\left(\frac{\sqrt{t}}{\frac{\frac{\sqrt[3]{16} \cdot \sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1} \cdot \sqrt[3]{2 \cdot y + 1}}}{1}} \cdot \frac{\sqrt{t}}{\frac{\frac{\sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1}}}{z}}\right)} \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   11. Simplified 45.5

       \[\leadsto \left(\cos \left(\color{blue}{\frac{\sqrt{t}}{\frac{\sqrt[3]{16}}{\sqrt[3]{y \cdot 2 + 1}} \cdot \frac{\sqrt[3]{16}}{\sqrt[3]{y \cdot 2 + 1}}}} \cdot \frac{\sqrt{t}}{\frac{\frac{\sqrt[3]{16}}{\sqrt[3]{2 \cdot y + 1}}}{z}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]

   12. Simplified 45.4

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

4. Final simplification 44.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 8.227621728721539868314394994500716678162 \cdot 10^{-310}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(b \cdot \frac{t}{16}\right) \cdot \left(2 \cdot a + 1\right)\right) \cdot \left(\cos \left(\left(\frac{\sqrt{t}}{\frac{\sqrt[3]{16}}{\sqrt[3]{1 + y \cdot 2}}} \cdot z\right) \cdot \frac{\sqrt{t}}{\frac{\sqrt[3]{16}}{\sqrt[3]{1 + y \cdot 2}} \cdot \frac{\sqrt[3]{16}}{\sqrt[3]{1 + y \cdot 2}}}\right) \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :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))))