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

Average Error: 46.0 → 45.1

Time: 1.3m

Precision: 64

Math

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

↓

\[x \cdot \left(\sqrt[3]{\cos \left(\frac{\sqrt[3]{b}}{\frac{\frac{1}{\sqrt[3]{t}}}{\sqrt[3]{t} \cdot \sqrt[3]{b}}} \cdot \left(\frac{1}{\sqrt[3]{\frac{16.0}{\sqrt[3]{t}}} \cdot \sqrt[3]{\frac{16.0}{\sqrt[3]{t}}}} \cdot \frac{\sqrt[3]{b}}{\frac{\sqrt[3]{\frac{16.0}{\sqrt[3]{t}}}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\sqrt[3]{b}}{\frac{\frac{1}{\sqrt[3]{t}}}{\sqrt[3]{t} \cdot \sqrt[3]{b}}} \cdot \left(\frac{1}{\sqrt[3]{\frac{16.0}{\sqrt[3]{t}}} \cdot \sqrt[3]{\frac{16.0}{\sqrt[3]{t}}}} \cdot \frac{\sqrt[3]{b}}{\frac{\sqrt[3]{\frac{16.0}{\sqrt[3]{t}}}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)\right)} \cdot \sqrt[3]{\cos \left(\frac{\sqrt[3]{b}}{\frac{\frac{1}{\sqrt[3]{t}}}{\sqrt[3]{t} \cdot \sqrt[3]{b}}} \cdot \left(\frac{1}{\sqrt[3]{\frac{16.0}{\sqrt[3]{t}}} \cdot \sqrt[3]{\frac{16.0}{\sqrt[3]{t}}}} \cdot \frac{\sqrt[3]{b}}{\frac{\sqrt[3]{\frac{16.0}{\sqrt[3]{t}}}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)\right)}\right)\right)\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r40731687 = x;
        double r40731688 = y;
        double r40731689 = 2.0;
        double r40731690 = r40731688 * r40731689;
        double r40731691 = 1.0;
        double r40731692 = r40731690 + r40731691;
        double r40731693 = z;
        double r40731694 = r40731692 * r40731693;
        double r40731695 = t;
        double r40731696 = r40731694 * r40731695;
        double r40731697 = 16.0;
        double r40731698 = r40731696 / r40731697;
        double r40731699 = cos(r40731698);
        double r40731700 = r40731687 * r40731699;
        double r40731701 = a;
        double r40731702 = r40731701 * r40731689;
        double r40731703 = r40731702 + r40731691;
        double r40731704 = b;
        double r40731705 = r40731703 * r40731704;
        double r40731706 = r40731705 * r40731695;
        double r40731707 = r40731706 / r40731697;
        double r40731708 = cos(r40731707);
        double r40731709 = r40731700 * r40731708;
        return r40731709;
}

↓

double f(double x, double __attribute__((unused)) y, double __attribute__((unused)) z, double t, double a, double b) {
        double r40731710 = x;
        double r40731711 = b;
        double r40731712 = cbrt(r40731711);
        double r40731713 = 1.0;
        double r40731714 = t;
        double r40731715 = cbrt(r40731714);
        double r40731716 = r40731713 / r40731715;
        double r40731717 = r40731715 * r40731712;
        double r40731718 = r40731716 / r40731717;
        double r40731719 = r40731712 / r40731718;
        double r40731720 = 16.0;
        double r40731721 = r40731720 / r40731715;
        double r40731722 = cbrt(r40731721);
        double r40731723 = r40731722 * r40731722;
        double r40731724 = r40731713 / r40731723;
        double r40731725 = a;
        double r40731726 = 2.0;
        double r40731727 = 1.0;
        double r40731728 = fma(r40731725, r40731726, r40731727);
        double r40731729 = r40731722 / r40731728;
        double r40731730 = r40731712 / r40731729;
        double r40731731 = r40731724 * r40731730;
        double r40731732 = r40731719 * r40731731;
        double r40731733 = cos(r40731732);
        double r40731734 = cbrt(r40731733);
        double r40731735 = r40731734 * r40731734;
        double r40731736 = r40731734 * r40731735;
        double r40731737 = r40731710 * r40731736;
        return r40731737;
}

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	44.7
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 46.0

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

2. Simplified 45.9

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

3. Taylor expanded around 0 45.3

   \[\leadsto \left(\color{blue}{1} \cdot x\right) \cdot \cos \left(\frac{b}{\frac{\frac{16.0}{t}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)\]
4. Using strategy rm

5. Applied *-un-lft-identity 45.3

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

6. Applied add-cube-cbrt 45.3

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

7. Applied *-un-lft-identity 45.3

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

8. Applied times-frac 45.3

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

9. Applied times-frac 45.3

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

10. Applied add-cube-cbrt 45.3

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

11. Applied times-frac 45.1

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

12. Simplified 45.1

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

14. Applied *-un-lft-identity 45.1

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

15. Applied add-cube-cbrt 45.1

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

16. Applied times-frac 45.1

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

17. Applied *-un-lft-identity 45.1

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

18. Applied cbrt-prod 45.1

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

19. Applied times-frac 45.1

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

20. Simplified 45.1

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

22. Applied add-cube-cbrt 45.1

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

23. Final simplification 45.1

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

Reproduce

herbie shell --seed 2019168 +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))))