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

Average Error: 21.1 → 18.4

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999932276395497865451034158468:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\log \left(e^{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r652708 = 2.0;
        double r652709 = x;
        double r652710 = sqrt(r652709);
        double r652711 = r652708 * r652710;
        double r652712 = y;
        double r652713 = z;
        double r652714 = t;
        double r652715 = r652713 * r652714;
        double r652716 = 3.0;
        double r652717 = r652715 / r652716;
        double r652718 = r652712 - r652717;
        double r652719 = cos(r652718);
        double r652720 = r652711 * r652719;
        double r652721 = a;
        double r652722 = b;
        double r652723 = r652722 * r652716;
        double r652724 = r652721 / r652723;
        double r652725 = r652720 - r652724;
        return r652725;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r652726 = y;
        double r652727 = z;
        double r652728 = t;
        double r652729 = r652727 * r652728;
        double r652730 = 3.0;
        double r652731 = r652729 / r652730;
        double r652732 = r652726 - r652731;
        double r652733 = cos(r652732);
        double r652734 = 0.9999999999999932;
        bool r652735 = r652733 <= r652734;
        double r652736 = 2.0;
        double r652737 = x;
        double r652738 = sqrt(r652737);
        double r652739 = r652736 * r652738;
        double r652740 = r652728 * r652727;
        double r652741 = sqrt(r652730);
        double r652742 = 2.0;
        double r652743 = pow(r652741, r652742);
        double r652744 = r652740 / r652743;
        double r652745 = cos(r652744);
        double r652746 = exp(r652745);
        double r652747 = log(r652746);
        double r652748 = 1.0;
        double r652749 = r652748 * r652726;
        double r652750 = cos(r652749);
        double r652751 = r652747 * r652750;
        double r652752 = sin(r652749);
        double r652753 = -r652744;
        double r652754 = sin(r652753);
        double r652755 = r652752 * r652754;
        double r652756 = r652751 - r652755;
        double r652757 = 3.0;
        double r652758 = pow(r652756, r652757);
        double r652759 = cbrt(r652758);
        double r652760 = r652728 / r652741;
        double r652761 = -r652760;
        double r652762 = r652727 / r652741;
        double r652763 = r652760 * r652762;
        double r652764 = fma(r652761, r652762, r652763);
        double r652765 = cos(r652764);
        double r652766 = r652759 * r652765;
        double r652767 = -r652763;
        double r652768 = fma(r652748, r652726, r652767);
        double r652769 = sin(r652768);
        double r652770 = sin(r652764);
        double r652771 = r652769 * r652770;
        double r652772 = r652766 - r652771;
        double r652773 = r652739 * r652772;
        double r652774 = a;
        double r652775 = b;
        double r652776 = r652775 * r652730;
        double r652777 = r652774 / r652776;
        double r652778 = r652773 - r652777;
        double r652779 = 0.5;
        double r652780 = pow(r652726, r652742);
        double r652781 = r652779 * r652780;
        double r652782 = r652748 - r652781;
        double r652783 = r652739 * r652782;
        double r652784 = r652783 - r652777;
        double r652785 = r652735 ? r652778 : r652784;
        return r652785;
}

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	21.1
Target	19.1
Herbie	18.4

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999932

   1. Initial program 20.3

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

   3. Applied add-sqr-sqrt 20.5

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

   4. Applied times-frac 20.4

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

   5. Applied add-sqr-sqrt 45.3

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

   6. Applied prod-diff 45.3

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

   7. Applied cos-sum 45.2

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

   8. Simplified 42.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)} - \sin \left(\mathsf{fma}\left(\sqrt{y}, \sqrt{y}, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   9. Simplified 20.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \color{blue}{\sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)}\right) - \frac{a}{b \cdot 3}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 20.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   12. Simplified 20.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{\color{blue}{{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)\right)}^{3}}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   13. Using strategy rm

   14. Applied fma-udef 20.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\cos \color{blue}{\left(1 \cdot y + \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   15. Applied cos-sum 19.8

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\color{blue}{\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   16. Simplified 19.8

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\color{blue}{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right) \cdot \cos \left(1 \cdot y\right)} - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   17. Using strategy rm

   18. Applied add-log-exp 19.8

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\color{blue}{\log \left(e^{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)}\right)} \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999999999932 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.5

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

   2. Taylor expanded around 0 15.8

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999932276395497865451034158468:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\log \left(e^{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))