Average Error: 20.3 → 18.3
Time: 31.2s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -4.557684899971343275948588241655457690987 \cdot 10^{175}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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 -4.557684899971343275948588241655457690987 \cdot 10^{175}:\\
\;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\

\mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\
\;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r38486959 = 2.0;
        double r38486960 = x;
        double r38486961 = sqrt(r38486960);
        double r38486962 = r38486959 * r38486961;
        double r38486963 = y;
        double r38486964 = z;
        double r38486965 = t;
        double r38486966 = r38486964 * r38486965;
        double r38486967 = 3.0;
        double r38486968 = r38486966 / r38486967;
        double r38486969 = r38486963 - r38486968;
        double r38486970 = cos(r38486969);
        double r38486971 = r38486962 * r38486970;
        double r38486972 = a;
        double r38486973 = b;
        double r38486974 = r38486973 * r38486967;
        double r38486975 = r38486972 / r38486974;
        double r38486976 = r38486971 - r38486975;
        return r38486976;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r38486977 = z;
        double r38486978 = t;
        double r38486979 = r38486977 * r38486978;
        double r38486980 = -4.557684899971343e+175;
        bool r38486981 = r38486979 <= r38486980;
        double r38486982 = 1.0;
        double r38486983 = y;
        double r38486984 = r38486983 * r38486983;
        double r38486985 = 0.5;
        double r38486986 = r38486984 * r38486985;
        double r38486987 = r38486982 - r38486986;
        double r38486988 = x;
        double r38486989 = sqrt(r38486988);
        double r38486990 = 2.0;
        double r38486991 = r38486989 * r38486990;
        double r38486992 = r38486987 * r38486991;
        double r38486993 = a;
        double r38486994 = 3.0;
        double r38486995 = b;
        double r38486996 = r38486994 * r38486995;
        double r38486997 = r38486993 / r38486996;
        double r38486998 = r38486992 - r38486997;
        double r38486999 = 2.650669621479893e+306;
        bool r38487000 = r38486979 <= r38486999;
        double r38487001 = r38486979 / r38486994;
        double r38487002 = cos(r38487001);
        double r38487003 = cbrt(r38487002);
        double r38487004 = r38487003 * r38487003;
        double r38487005 = r38487003 * r38487004;
        double r38487006 = cos(r38486983);
        double r38487007 = r38487005 * r38487006;
        double r38487008 = sin(r38486983);
        double r38487009 = sin(r38487001);
        double r38487010 = r38487008 * r38487009;
        double r38487011 = r38487007 + r38487010;
        double r38487012 = r38487011 * r38486991;
        double r38487013 = r38487012 - r38486997;
        double r38487014 = r38487000 ? r38487013 : r38486998;
        double r38487015 = r38486981 ? r38486998 : r38487014;
        return r38487015;
}

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

Original20.3
Target18.4
Herbie18.3
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -4.557684899971343e+175 or 2.650669621479893e+306 < (* z t)

    1. Initial program 52.6

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

    2. Taylor expanded around 0 44.3

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

    3. Simplified44.3

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

    if -4.557684899971343e+175 < (* z t) < 2.650669621479893e+306

    1. Initial program 12.8

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

    3. Applied cos-diff12.3

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

    5. Applied add-cube-cbrt12.3

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -4.557684899971343275948588241655457690987 \cdot 10^{175}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))