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

Average Error: 47.2 → 44.8

Time: 41.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.202788797825691552687024499585777929908 \cdot 10^{50}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \le 2.560308249679174743120860805530103314552 \cdot 10^{-28}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{t}}{\frac{\sqrt{16}}{1 + 2 \cdot y}}\right) \cdot \left(\left(\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt{16}} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)\right)\right) \cdot \cos \left(\left(1 + a \cdot 2\right) \cdot \left(\frac{t}{16} \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1001080 = x;
        double r1001081 = y;
        double r1001082 = 2.0;
        double r1001083 = r1001081 * r1001082;
        double r1001084 = 1.0;
        double r1001085 = r1001083 + r1001084;
        double r1001086 = z;
        double r1001087 = r1001085 * r1001086;
        double r1001088 = t;
        double r1001089 = r1001087 * r1001088;
        double r1001090 = 16.0;
        double r1001091 = r1001089 / r1001090;
        double r1001092 = cos(r1001091);
        double r1001093 = r1001080 * r1001092;
        double r1001094 = a;
        double r1001095 = r1001094 * r1001082;
        double r1001096 = r1001095 + r1001084;
        double r1001097 = b;
        double r1001098 = r1001096 * r1001097;
        double r1001099 = r1001098 * r1001088;
        double r1001100 = r1001099 / r1001090;
        double r1001101 = cos(r1001100);
        double r1001102 = r1001093 * r1001101;
        return r1001102;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1001103 = t;
        double r1001104 = -1.2027887978256916e+50;
        bool r1001105 = r1001103 <= r1001104;
        double r1001106 = x;
        double r1001107 = 2.5603082496791747e-28;
        bool r1001108 = r1001103 <= r1001107;
        double r1001109 = z;
        double r1001110 = cbrt(r1001109);
        double r1001111 = cbrt(r1001103);
        double r1001112 = 16.0;
        double r1001113 = sqrt(r1001112);
        double r1001114 = 1.0;
        double r1001115 = 2.0;
        double r1001116 = y;
        double r1001117 = r1001115 * r1001116;
        double r1001118 = r1001114 + r1001117;
        double r1001119 = r1001113 / r1001118;
        double r1001120 = r1001111 / r1001119;
        double r1001121 = r1001110 * r1001120;
        double r1001122 = r1001111 * r1001111;
        double r1001123 = r1001122 / r1001113;
        double r1001124 = r1001123 * r1001110;
        double r1001125 = r1001124 * r1001110;
        double r1001126 = r1001121 * r1001125;
        double r1001127 = cos(r1001126);
        double r1001128 = r1001106 * r1001127;
        double r1001129 = a;
        double r1001130 = r1001129 * r1001115;
        double r1001131 = r1001114 + r1001130;
        double r1001132 = r1001103 / r1001112;
        double r1001133 = b;
        double r1001134 = r1001132 * r1001133;
        double r1001135 = r1001131 * r1001134;
        double r1001136 = cos(r1001135);
        double r1001137 = r1001128 * r1001136;
        double r1001138 = r1001108 ? r1001137 : r1001106;
        double r1001139 = r1001105 ? r1001106 : r1001138;
        return r1001139;
}

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	47.2
Target	45.2
Herbie	44.8

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

Derivation

1. Split input into 2 regimes

2. if t < -1.2027887978256916e+50 or 2.5603082496791747e-28 < t

   1. Initial program 59.2

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

   2. Simplified 59.2

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

   3. Taylor expanded around 0 57.9

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

   4. Taylor expanded around 0 55.3

      \[\leadsto \left(\cos 0 \cdot x\right) \cdot \color{blue}{1}\]

   if -1.2027887978256916e+50 < t < 2.5603082496791747e-28

   1. Initial program 35.8

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

   2. Simplified 35.2

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

   4. Applied add-cube-cbrt 35.3

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

   5. Applied *-un-lft-identity 35.3

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

   6. Applied add-sqr-sqrt 35.3

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

   7. Applied times-frac 35.3

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

   8. Applied times-frac 35.3

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

   9. Applied add-cube-cbrt 35.3

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

   10. Applied times-frac 34.8

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

   11. Simplified 34.8

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

   12. Simplified 34.8

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

4. Final simplification 44.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.202788797825691552687024499585777929908 \cdot 10^{50}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \le 2.560308249679174743120860805530103314552 \cdot 10^{-28}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{t}}{\frac{\sqrt{16}}{1 + 2 \cdot y}}\right) \cdot \left(\left(\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt{16}} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)\right)\right) \cdot \cos \left(\left(1 + a \cdot 2\right) \cdot \left(\frac{t}{16} \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))