Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Average Error: 20.4 → 17.6

Time: 16.1s

Precision: 64

Math

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 1.0112129461852768 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) + \sin y \cdot \log \left(e^{\sin \left(\frac{z}{\sqrt{3}} \cdot \left(\left(\sqrt[3]{\frac{t}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right) \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r747691 = 2.0;
        double r747692 = x;
        double r747693 = sqrt(r747692);
        double r747694 = r747691 * r747693;
        double r747695 = y;
        double r747696 = z;
        double r747697 = t;
        double r747698 = r747696 * r747697;
        double r747699 = 3.0;
        double r747700 = r747698 / r747699;
        double r747701 = r747695 - r747700;
        double r747702 = cos(r747701);
        double r747703 = r747694 * r747702;
        double r747704 = a;
        double r747705 = b;
        double r747706 = r747705 * r747699;
        double r747707 = r747704 / r747706;
        double r747708 = r747703 - r747707;
        return r747708;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r747709 = z;
        double r747710 = t;
        double r747711 = r747709 * r747710;
        double r747712 = -inf.0;
        bool r747713 = r747711 <= r747712;
        double r747714 = 1.0112129461852768e+290;
        bool r747715 = r747711 <= r747714;
        double r747716 = !r747715;
        bool r747717 = r747713 || r747716;
        double r747718 = 2.0;
        double r747719 = x;
        double r747720 = sqrt(r747719);
        double r747721 = r747718 * r747720;
        double r747722 = 1.0;
        double r747723 = 0.5;
        double r747724 = y;
        double r747725 = 2.0;
        double r747726 = pow(r747724, r747725);
        double r747727 = r747723 * r747726;
        double r747728 = r747722 - r747727;
        double r747729 = r747721 * r747728;
        double r747730 = a;
        double r747731 = b;
        double r747732 = 3.0;
        double r747733 = r747731 * r747732;
        double r747734 = r747730 / r747733;
        double r747735 = r747729 - r747734;
        double r747736 = cos(r747724);
        double r747737 = sqrt(r747732);
        double r747738 = r747709 / r747737;
        double r747739 = r747710 / r747737;
        double r747740 = r747738 * r747739;
        double r747741 = cos(r747740);
        double r747742 = r747736 * r747741;
        double r747743 = sin(r747724);
        double r747744 = cbrt(r747739);
        double r747745 = r747744 * r747744;
        double r747746 = r747745 * r747744;
        double r747747 = r747738 * r747746;
        double r747748 = sin(r747747);
        double r747749 = exp(r747748);
        double r747750 = log(r747749);
        double r747751 = r747743 * r747750;
        double r747752 = r747742 + r747751;
        double r747753 = r747721 * r747752;
        double r747754 = r747753 - r747734;
        double r747755 = r747717 ? r747735 : r747754;
        return r747755;
}

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	20.4
Target	18.3
Herbie	17.6

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 1.0112129461852768e+290 < (* z t)

   1. Initial program 62.4

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

   2. Taylor expanded around 0 44.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

   if -inf.0 < (* z t) < 1.0112129461852768e+290

   1. Initial program 13.9

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 14.0

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

   4. Applied times-frac 13.9

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

   6. Applied cos-diff 13.5

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

   8. Applied add-cube-cbrt 13.5

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

   10. Applied add-log-exp 13.5

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 1.0112129461852768 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) + \sin y \cdot \log \left(e^{\sin \left(\frac{z}{\sqrt{3}} \cdot \left(\left(\sqrt[3]{\frac{t}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right) \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))