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

Average Error: 45.9 → 45.1

Time: 1.0m

Precision: 64

Math

\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

↓

\[x \cdot \cos \left(\frac{\left(\left(z \cdot \left(\left(\sqrt{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right) \cdot \mathsf{fma}\left(2.0, y, 1.0\right)\right) \cdot \sqrt[3]{t}}{16.0}\right)\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r39819119 = x;
        double r39819120 = y;
        double r39819121 = 2.0;
        double r39819122 = r39819120 * r39819121;
        double r39819123 = 1.0;
        double r39819124 = r39819122 + r39819123;
        double r39819125 = z;
        double r39819126 = r39819124 * r39819125;
        double r39819127 = t;
        double r39819128 = r39819126 * r39819127;
        double r39819129 = 16.0;
        double r39819130 = r39819128 / r39819129;
        double r39819131 = cos(r39819130);
        double r39819132 = r39819119 * r39819131;
        double r39819133 = a;
        double r39819134 = r39819133 * r39819121;
        double r39819135 = r39819134 + r39819123;
        double r39819136 = b;
        double r39819137 = r39819135 * r39819136;
        double r39819138 = r39819137 * r39819127;
        double r39819139 = r39819138 / r39819129;
        double r39819140 = cos(r39819139);
        double r39819141 = r39819132 * r39819140;
        return r39819141;
}

↓

double f(double x, double y, double z, double t, double __attribute__((unused)) a, double __attribute__((unused)) b) {
        double r39819142 = x;
        double r39819143 = z;
        double r39819144 = t;
        double r39819145 = cbrt(r39819144);
        double r39819146 = r39819145 * r39819145;
        double r39819147 = cbrt(r39819146);
        double r39819148 = sqrt(r39819147);
        double r39819149 = r39819148 * r39819148;
        double r39819150 = r39819147 * r39819147;
        double r39819151 = r39819149 * r39819150;
        double r39819152 = r39819143 * r39819151;
        double r39819153 = 2.0;
        double r39819154 = y;
        double r39819155 = 1.0;
        double r39819156 = fma(r39819153, r39819154, r39819155);
        double r39819157 = r39819152 * r39819156;
        double r39819158 = r39819157 * r39819145;
        double r39819159 = 16.0;
        double r39819160 = r39819158 / r39819159;
        double r39819161 = cos(r39819160);
        double r39819162 = r39819142 * r39819161;
        return r39819162;
}

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	45.9
Target	44.7
Herbie	45.1

\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

1. Initial program 45.9

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

2. Taylor expanded around 0 45.3

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \color{blue}{1}\]
3. Using strategy rm

4. Applied add-cube-cbrt 45.3

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}{16.0}\right)\right) \cdot 1\]

5. Applied associate-*r* 45.3

   \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t}}}{16.0}\right)\right) \cdot 1\]

6. Simplified 45.1

   \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\mathsf{fma}\left(2.0, y, 1.0\right) \cdot \left(z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)} \cdot \sqrt[3]{t}}{16.0}\right)\right) \cdot 1\]
7. Using strategy rm

8. Applied add-cube-cbrt 45.1

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\mathsf{fma}\left(2.0, y, 1.0\right) \cdot \left(z \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}\right)\right) \cdot \sqrt[3]{t}}{16.0}\right)\right) \cdot 1\]
9. Using strategy rm

10. Applied add-sqr-sqrt 45.1

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\mathsf{fma}\left(2.0, y, 1.0\right) \cdot \left(z \cdot \left(\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)}\right)\right)\right) \cdot \sqrt[3]{t}}{16.0}\right)\right) \cdot 1\]

11. Final simplification 45.1

    \[\leadsto x \cdot \cos \left(\frac{\left(\left(z \cdot \left(\left(\sqrt{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right) \cdot \mathsf{fma}\left(2.0, y, 1.0\right)\right) \cdot \sqrt[3]{t}}{16.0}\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(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))))))

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