Average Error: 20.5 → 17.9
Time: 29.2s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999972455529012593800075592298526316881:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 0.9999972455529012593800075592298526316881:\\
\;\;\;\;\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(\left(\sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\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 r539975 = 2.0;
        double r539976 = x;
        double r539977 = sqrt(r539976);
        double r539978 = r539975 * r539977;
        double r539979 = y;
        double r539980 = z;
        double r539981 = t;
        double r539982 = r539980 * r539981;
        double r539983 = 3.0;
        double r539984 = r539982 / r539983;
        double r539985 = r539979 - r539984;
        double r539986 = cos(r539985);
        double r539987 = r539978 * r539986;
        double r539988 = a;
        double r539989 = b;
        double r539990 = r539989 * r539983;
        double r539991 = r539988 / r539990;
        double r539992 = r539987 - r539991;
        return r539992;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r539993 = y;
        double r539994 = z;
        double r539995 = t;
        double r539996 = r539994 * r539995;
        double r539997 = 3.0;
        double r539998 = r539996 / r539997;
        double r539999 = r539993 - r539998;
        double r540000 = cos(r539999);
        double r540001 = 0.9999972455529013;
        bool r540002 = r540000 <= r540001;
        double r540003 = 2.0;
        double r540004 = x;
        double r540005 = sqrt(r540004);
        double r540006 = r540003 * r540005;
        double r540007 = cos(r539993);
        double r540008 = cos(r539998);
        double r540009 = r540007 * r540008;
        double r540010 = r540006 * r540009;
        double r540011 = sin(r539993);
        double r540012 = sin(r539998);
        double r540013 = r540011 * r540012;
        double r540014 = cbrt(r540013);
        double r540015 = r540014 * r540014;
        double r540016 = r540015 * r540014;
        double r540017 = r540006 * r540016;
        double r540018 = r540010 + r540017;
        double r540019 = a;
        double r540020 = b;
        double r540021 = r540020 * r539997;
        double r540022 = r540019 / r540021;
        double r540023 = r540018 - r540022;
        double r540024 = 1.0;
        double r540025 = 0.5;
        double r540026 = 2.0;
        double r540027 = pow(r539993, r540026);
        double r540028 = r540025 * r540027;
        double r540029 = r540024 - r540028;
        double r540030 = r540006 * r540029;
        double r540031 = r540030 - r540022;
        double r540032 = r540002 ? r540023 : r540031;
        return r540032;
}

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

Original20.5
Target18.8
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999972455529013

    1. Initial program 20.1

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
    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. Applied distribute-lft-in19.5

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

    6. Applied add-cube-cbrt19.5

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

    if 0.9999972455529013 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 21.3

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

    2. Taylor expanded around 0 15.4

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999972455529012593800075592298526316881:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))