Average Error: 20.8 → 17.9
Time: 16.2s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 0.999999999481199109:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\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 r767986 = 2.0;
        double r767987 = x;
        double r767988 = sqrt(r767987);
        double r767989 = r767986 * r767988;
        double r767990 = y;
        double r767991 = z;
        double r767992 = t;
        double r767993 = r767991 * r767992;
        double r767994 = 3.0;
        double r767995 = r767993 / r767994;
        double r767996 = r767990 - r767995;
        double r767997 = cos(r767996);
        double r767998 = r767989 * r767997;
        double r767999 = a;
        double r768000 = b;
        double r768001 = r768000 * r767994;
        double r768002 = r767999 / r768001;
        double r768003 = r767998 - r768002;
        return r768003;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r768004 = y;
        double r768005 = z;
        double r768006 = t;
        double r768007 = r768005 * r768006;
        double r768008 = 3.0;
        double r768009 = r768007 / r768008;
        double r768010 = r768004 - r768009;
        double r768011 = cos(r768010);
        double r768012 = 0.9999999994811991;
        bool r768013 = r768011 <= r768012;
        double r768014 = 2.0;
        double r768015 = x;
        double r768016 = sqrt(r768015);
        double r768017 = r768014 * r768016;
        double r768018 = cos(r768004);
        double r768019 = cos(r768009);
        double r768020 = 3.0;
        double r768021 = pow(r768019, r768020);
        double r768022 = cbrt(r768021);
        double r768023 = cbrt(r768022);
        double r768024 = cbrt(r768019);
        double r768025 = r768023 * r768024;
        double r768026 = r768025 * r768023;
        double r768027 = r768018 * r768026;
        double r768028 = sin(r768004);
        double r768029 = sin(r768009);
        double r768030 = r768028 * r768029;
        double r768031 = r768027 + r768030;
        double r768032 = r768017 * r768031;
        double r768033 = a;
        double r768034 = b;
        double r768035 = r768034 * r768008;
        double r768036 = r768033 / r768035;
        double r768037 = r768032 - r768036;
        double r768038 = 1.0;
        double r768039 = 0.5;
        double r768040 = 2.0;
        double r768041 = pow(r768004, r768040);
        double r768042 = r768039 * r768041;
        double r768043 = r768038 - r768042;
        double r768044 = r768017 * r768043;
        double r768045 = r768044 - r768036;
        double r768046 = r768013 ? r768037 : r768045;
        return r768046;
}

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
Target18.6
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999994811991

    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 cos-diff19.5

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

    5. Applied add-cube-cbrt19.5

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

    7. Applied add-cbrt-cube19.5

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

    8. Simplified19.5

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

    10. Applied add-cbrt-cube19.5

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

    11. Simplified19.5

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

    if 0.9999999994811991 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 21.7

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

    2. Taylor expanded around 0 15.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))