Average Error: 21.0 → 18.8
Time: 14.8s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.995537973235021245:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\left(\sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 0.995537973235021245:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\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(\left(\sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)}\right) \cdot \sqrt[3]{\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}
double f(double x, double y, double z, double t, double a, double b) {
        double r848705 = 2.0;
        double r848706 = x;
        double r848707 = sqrt(r848706);
        double r848708 = r848705 * r848707;
        double r848709 = y;
        double r848710 = z;
        double r848711 = t;
        double r848712 = r848710 * r848711;
        double r848713 = 3.0;
        double r848714 = r848712 / r848713;
        double r848715 = r848709 - r848714;
        double r848716 = cos(r848715);
        double r848717 = r848708 * r848716;
        double r848718 = a;
        double r848719 = b;
        double r848720 = r848719 * r848713;
        double r848721 = r848718 / r848720;
        double r848722 = r848717 - r848721;
        return r848722;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r848723 = y;
        double r848724 = z;
        double r848725 = t;
        double r848726 = r848724 * r848725;
        double r848727 = 3.0;
        double r848728 = r848726 / r848727;
        double r848729 = r848723 - r848728;
        double r848730 = cos(r848729);
        double r848731 = 0.9955379732350212;
        bool r848732 = r848730 <= r848731;
        double r848733 = 2.0;
        double r848734 = x;
        double r848735 = sqrt(r848734);
        double r848736 = r848733 * r848735;
        double r848737 = sqrt(r848727);
        double r848738 = r848725 / r848737;
        double r848739 = r848724 / r848737;
        double r848740 = r848738 * r848739;
        double r848741 = cos(r848740);
        double r848742 = 1.0;
        double r848743 = r848742 * r848723;
        double r848744 = cos(r848743);
        double r848745 = r848741 * r848744;
        double r848746 = sin(r848743);
        double r848747 = -r848740;
        double r848748 = sin(r848747);
        double r848749 = r848746 * r848748;
        double r848750 = r848745 - r848749;
        double r848751 = -r848738;
        double r848752 = fma(r848751, r848739, r848740);
        double r848753 = cos(r848752);
        double r848754 = r848750 * r848753;
        double r848755 = fma(r848742, r848723, r848747);
        double r848756 = sin(r848755);
        double r848757 = cbrt(r848752);
        double r848758 = r848757 * r848757;
        double r848759 = r848758 * r848757;
        double r848760 = sin(r848759);
        double r848761 = r848756 * r848760;
        double r848762 = r848754 - r848761;
        double r848763 = r848736 * r848762;
        double r848764 = a;
        double r848765 = b;
        double r848766 = r848765 * r848727;
        double r848767 = r848764 / r848766;
        double r848768 = r848763 - r848767;
        double r848769 = 0.5;
        double r848770 = 2.0;
        double r848771 = pow(r848723, r848770);
        double r848772 = r848769 * r848771;
        double r848773 = r848742 - r848772;
        double r848774 = r848736 * r848773;
        double r848775 = r848774 - r848767;
        double r848776 = r848732 ? r848768 : r848775;
        return r848776;
}

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

Original21.0
Target19.0
Herbie18.8
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9955379732350212

    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-sqrt20.3

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

    4. Applied times-frac20.3

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

    5. Applied add-sqr-sqrt45.2

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

    6. Applied prod-diff45.2

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

    7. Applied cos-sum45.2

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

    8. Simplified42.1

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

    9. Simplified20.2

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

    11. Applied fma-udef20.2

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

    12. Applied cos-sum19.7

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

    13. Simplified19.7

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

    15. Applied add-cube-cbrt19.7

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

    if 0.9955379732350212 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 22.1

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

    2. Taylor expanded around 0 17.5

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.995537973235021245:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\left(\sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))))