Average Error: 20.8 → 17.8
Time: 13.1s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -7.67127596277169436 \cdot 10^{304} \lor \neg \left(z \cdot t \le 1.3317928530781586 \cdot 10^{290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \log \left(e^{\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y}}\right)\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r745941 = 2.0;
        double r745942 = x;
        double r745943 = sqrt(r745942);
        double r745944 = r745941 * r745943;
        double r745945 = y;
        double r745946 = z;
        double r745947 = t;
        double r745948 = r745946 * r745947;
        double r745949 = 3.0;
        double r745950 = r745948 / r745949;
        double r745951 = r745945 - r745950;
        double r745952 = cos(r745951);
        double r745953 = r745944 * r745952;
        double r745954 = a;
        double r745955 = b;
        double r745956 = r745955 * r745949;
        double r745957 = r745954 / r745956;
        double r745958 = r745953 - r745957;
        return r745958;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r745959 = z;
        double r745960 = t;
        double r745961 = r745959 * r745960;
        double r745962 = -7.671275962771694e+304;
        bool r745963 = r745961 <= r745962;
        double r745964 = 1.3317928530781586e+290;
        bool r745965 = r745961 <= r745964;
        double r745966 = !r745965;
        bool r745967 = r745963 || r745966;
        double r745968 = 2.0;
        double r745969 = x;
        double r745970 = sqrt(r745969);
        double r745971 = r745968 * r745970;
        double r745972 = 1.0;
        double r745973 = 0.5;
        double r745974 = y;
        double r745975 = 2.0;
        double r745976 = pow(r745974, r745975);
        double r745977 = r745973 * r745976;
        double r745978 = r745972 - r745977;
        double r745979 = r745971 * r745978;
        double r745980 = a;
        double r745981 = b;
        double r745982 = 3.0;
        double r745983 = r745981 * r745982;
        double r745984 = r745980 / r745983;
        double r745985 = r745979 - r745984;
        double r745986 = cos(r745974);
        double r745987 = r745961 / r745982;
        double r745988 = cos(r745987);
        double r745989 = r745986 * r745988;
        double r745990 = 0.3333333333333333;
        double r745991 = r745960 * r745959;
        double r745992 = r745990 * r745991;
        double r745993 = sin(r745992);
        double r745994 = sin(r745974);
        double r745995 = r745993 * r745994;
        double r745996 = cbrt(r745995);
        double r745997 = r745996 * r745996;
        double r745998 = r745997 * r745996;
        double r745999 = exp(r745998);
        double r746000 = log(r745999);
        double r746001 = r745989 + r746000;
        double r746002 = r745971 * r746001;
        double r746003 = r746002 - r745984;
        double r746004 = r745967 ? r745985 : r746003;
        return r746004;
}

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

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

    6. Applied add-log-exp13.7

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

    8. Applied add-cube-cbrt13.7

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

  4. Final simplification17.8

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

Reproduce

herbie shell --seed 2020057 +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))))