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

Average Error: 21.3 → 18.5

Time: 37.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999915456516674794329446740448475:\\ \;\;\;\;\left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \log \left(e^{\sin \left(\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt{\sqrt[3]{3}}} \cdot \frac{\sqrt[3]{t}}{\sqrt{\sqrt[3]{3}}}\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r543784 = 2.0;
        double r543785 = x;
        double r543786 = sqrt(r543785);
        double r543787 = r543784 * r543786;
        double r543788 = y;
        double r543789 = z;
        double r543790 = t;
        double r543791 = r543789 * r543790;
        double r543792 = 3.0;
        double r543793 = r543791 / r543792;
        double r543794 = r543788 - r543793;
        double r543795 = cos(r543794);
        double r543796 = r543787 * r543795;
        double r543797 = a;
        double r543798 = b;
        double r543799 = r543798 * r543792;
        double r543800 = r543797 / r543799;
        double r543801 = r543796 - r543800;
        return r543801;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r543802 = y;
        double r543803 = z;
        double r543804 = t;
        double r543805 = r543803 * r543804;
        double r543806 = 3.0;
        double r543807 = r543805 / r543806;
        double r543808 = r543802 - r543807;
        double r543809 = cos(r543808);
        double r543810 = 0.9999999999915457;
        bool r543811 = r543809 <= r543810;
        double r543812 = cos(r543802);
        double r543813 = cos(r543807);
        double r543814 = r543812 * r543813;
        double r543815 = 2.0;
        double r543816 = x;
        double r543817 = sqrt(r543816);
        double r543818 = r543815 * r543817;
        double r543819 = r543814 * r543818;
        double r543820 = sin(r543802);
        double r543821 = cbrt(r543806);
        double r543822 = r543821 * r543821;
        double r543823 = r543803 / r543822;
        double r543824 = cbrt(r543804);
        double r543825 = r543824 * r543824;
        double r543826 = sqrt(r543821);
        double r543827 = r543825 / r543826;
        double r543828 = r543824 / r543826;
        double r543829 = r543827 * r543828;
        double r543830 = r543823 * r543829;
        double r543831 = sin(r543830);
        double r543832 = exp(r543831);
        double r543833 = log(r543832);
        double r543834 = r543820 * r543833;
        double r543835 = r543834 * r543818;
        double r543836 = r543819 + r543835;
        double r543837 = a;
        double r543838 = b;
        double r543839 = r543838 * r543806;
        double r543840 = r543837 / r543839;
        double r543841 = r543836 - r543840;
        double r543842 = -0.5;
        double r543843 = 2.0;
        double r543844 = pow(r543802, r543843);
        double r543845 = 1.0;
        double r543846 = fma(r543842, r543844, r543845);
        double r543847 = r543818 * r543846;
        double r543848 = r543847 - r543840;
        double r543849 = r543811 ? r543841 : r543848;
        return r543849;
}

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	21.3
Target	19.4
Herbie	18.5

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999915457

   1. Initial program 20.5

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

   3. Applied cos-diff 19.8

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

   4. Applied distribute-lft-in 19.8

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

   5. Simplified 19.8

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

   6. Simplified 19.8

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

   8. Applied add-log-exp 19.8

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

   10. Applied add-cube-cbrt 19.8

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

   11. Applied times-frac 19.9

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

   13. Applied add-sqr-sqrt 19.9

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

   14. Applied add-cube-cbrt 19.9

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

   15. Applied times-frac 19.9

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

   if 0.9999999999915457 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.7

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

   2. Taylor expanded around 0 16.1

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

   3. Simplified 16.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999915456516674794329446740448475:\\ \;\;\;\;\left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \log \left(e^{\sin \left(\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt{\sqrt[3]{3}}} \cdot \frac{\sqrt[3]{t}}{\sqrt{\sqrt[3]{3}}}\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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 -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

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