Average Error: 20.8 → 17.8
Time: 10.2s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -7.67127596277169436 \cdot 10^{304} \lor \neg \left(z \cdot t \le 1.3317928530781586 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \log \left(e^{\left(\sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;z \cdot t \le -7.67127596277169436 \cdot 10^{304} \lor \neg \left(z \cdot t \le 1.3317928530781586 \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 \cdot t}{3}\right) + \log \left(e^{\left(\sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right)\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r884901 = 2.0;
        double r884902 = x;
        double r884903 = sqrt(r884902);
        double r884904 = r884901 * r884903;
        double r884905 = y;
        double r884906 = z;
        double r884907 = t;
        double r884908 = r884906 * r884907;
        double r884909 = 3.0;
        double r884910 = r884908 / r884909;
        double r884911 = r884905 - r884910;
        double r884912 = cos(r884911);
        double r884913 = r884904 * r884912;
        double r884914 = a;
        double r884915 = b;
        double r884916 = r884915 * r884909;
        double r884917 = r884914 / r884916;
        double r884918 = r884913 - r884917;
        return r884918;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r884919 = z;
        double r884920 = t;
        double r884921 = r884919 * r884920;
        double r884922 = -7.671275962771694e+304;
        bool r884923 = r884921 <= r884922;
        double r884924 = 1.3317928530781586e+290;
        bool r884925 = r884921 <= r884924;
        double r884926 = !r884925;
        bool r884927 = r884923 || r884926;
        double r884928 = 2.0;
        double r884929 = x;
        double r884930 = sqrt(r884929);
        double r884931 = r884928 * r884930;
        double r884932 = 1.0;
        double r884933 = 0.5;
        double r884934 = y;
        double r884935 = 2.0;
        double r884936 = pow(r884934, r884935);
        double r884937 = r884933 * r884936;
        double r884938 = r884932 - r884937;
        double r884939 = r884931 * r884938;
        double r884940 = a;
        double r884941 = b;
        double r884942 = 3.0;
        double r884943 = r884941 * r884942;
        double r884944 = r884940 / r884943;
        double r884945 = r884939 - r884944;
        double r884946 = cos(r884934);
        double r884947 = r884921 / r884942;
        double r884948 = cos(r884947);
        double r884949 = r884946 * r884948;
        double r884950 = sin(r884934);
        double r884951 = 0.3333333333333333;
        double r884952 = r884920 * r884919;
        double r884953 = r884951 * r884952;
        double r884954 = sin(r884953);
        double r884955 = r884950 * r884954;
        double r884956 = cbrt(r884955);
        double r884957 = r884956 * r884956;
        double r884958 = r884957 * r884956;
        double r884959 = exp(r884958);
        double r884960 = log(r884959);
        double r884961 = r884949 + r884960;
        double r884962 = r884931 * r884961;
        double r884963 = r884962 - r884944;
        double r884964 = r884927 ? r884945 : r884963;
        return r884964;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target18.5
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -7.671275962771694e+304 or 1.3317928530781586e+290 < (* z t)

    1. Initial program 62.2

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

    2. Taylor expanded around 0 43.3

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

    if -7.671275962771694e+304 < (* z t) < 1.3317928530781586e+290

    1. Initial program 14.2

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

    3. Applied cos-diff13.7

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

    4. Taylor expanded around inf 13.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \color{blue}{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) - \frac{a}{b \cdot 3}\]
    5. Using strategy rm

    6. Applied add-log-exp13.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \color{blue}{\log \left(e^{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)}\right) - \frac{a}{b \cdot 3}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt13.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \log \left(e^{\color{blue}{\left(\sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}}\right)\right) - \frac{a}{b \cdot 3}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -7.67127596277169436 \cdot 10^{304} \lor \neg \left(z \cdot t \le 1.3317928530781586 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \log \left(e^{\left(\sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))