Average Error: 20.7 → 17.8
Time: 30.5s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 8.139896335094742174168253426990729793648 \cdot 10^{149}:\\ \;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 8.139896335094742174168253426990729793648 \cdot 10^{149}:\\
\;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{\frac{a}{b}}{3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r550822 = 2.0;
        double r550823 = x;
        double r550824 = sqrt(r550823);
        double r550825 = r550822 * r550824;
        double r550826 = y;
        double r550827 = z;
        double r550828 = t;
        double r550829 = r550827 * r550828;
        double r550830 = 3.0;
        double r550831 = r550829 / r550830;
        double r550832 = r550826 - r550831;
        double r550833 = cos(r550832);
        double r550834 = r550825 * r550833;
        double r550835 = a;
        double r550836 = b;
        double r550837 = r550836 * r550830;
        double r550838 = r550835 / r550837;
        double r550839 = r550834 - r550838;
        return r550839;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r550840 = 2.0;
        double r550841 = x;
        double r550842 = sqrt(r550841);
        double r550843 = r550840 * r550842;
        double r550844 = y;
        double r550845 = z;
        double r550846 = t;
        double r550847 = r550845 * r550846;
        double r550848 = 3.0;
        double r550849 = r550847 / r550848;
        double r550850 = r550844 - r550849;
        double r550851 = cos(r550850);
        double r550852 = r550843 * r550851;
        double r550853 = 8.139896335094742e+149;
        bool r550854 = r550852 <= r550853;
        double r550855 = cos(r550844);
        double r550856 = cos(r550849);
        double r550857 = cbrt(r550856);
        double r550858 = r550857 * r550857;
        double r550859 = 0.3333333333333333;
        double r550860 = r550846 * r550845;
        double r550861 = r550859 * r550860;
        double r550862 = cos(r550861);
        double r550863 = cbrt(r550862);
        double r550864 = r550858 * r550863;
        double r550865 = r550855 * r550864;
        double r550866 = r550865 * r550843;
        double r550867 = sin(r550844);
        double r550868 = sin(r550849);
        double r550869 = r550867 * r550868;
        double r550870 = r550869 * r550843;
        double r550871 = r550866 + r550870;
        double r550872 = a;
        double r550873 = b;
        double r550874 = r550872 / r550873;
        double r550875 = r550874 / r550848;
        double r550876 = r550871 - r550875;
        double r550877 = 2.0;
        double r550878 = pow(r550844, r550877);
        double r550879 = -0.5;
        double r550880 = 1.0;
        double r550881 = fma(r550878, r550879, r550880);
        double r550882 = r550843 * r550881;
        double r550883 = r550873 * r550848;
        double r550884 = r550872 / r550883;
        double r550885 = r550882 - r550884;
        double r550886 = r550854 ? r550876 : r550885;
        return r550886;
}

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

Original20.7
Target18.7
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 8.139896335094742e+149

    1. Initial program 14.6

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

    3. Applied cos-diff14.0

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

    4. Applied distribute-lft-in14.0

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

    5. Simplified14.0

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

    6. Simplified14.0

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

    8. Applied associate-/r*14.1

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

    10. Applied add-cube-cbrt14.1

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

    11. Taylor expanded around inf 14.1

      \[\leadsto \left(\left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\color{blue}{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{\frac{a}{b}}{3}\]

    if 8.139896335094742e+149 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

    1. Initial program 60.9

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

    2. Taylor expanded around 0 42.3

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

    3. Simplified42.3

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 8.139896335094742174168253426990729793648 \cdot 10^{149}:\\ \;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))