Average Error: 20.5 → 18.8
Time: 10.6s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.711112522981125:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \end{array}\]
\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}\;y \le -2.711112522981125:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\

\mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\
\;\;\;\;\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(0.333333333333333315 \cdot \left(t \cdot z\right)\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r862943 = 2.0;
        double r862944 = x;
        double r862945 = sqrt(r862944);
        double r862946 = r862943 * r862945;
        double r862947 = y;
        double r862948 = z;
        double r862949 = t;
        double r862950 = r862948 * r862949;
        double r862951 = 3.0;
        double r862952 = r862950 / r862951;
        double r862953 = r862947 - r862952;
        double r862954 = cos(r862953);
        double r862955 = r862946 * r862954;
        double r862956 = a;
        double r862957 = b;
        double r862958 = r862957 * r862951;
        double r862959 = r862956 / r862958;
        double r862960 = r862955 - r862959;
        return r862960;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r862961 = y;
        double r862962 = -2.711112522981125;
        bool r862963 = r862961 <= r862962;
        double r862964 = 2.0;
        double r862965 = x;
        double r862966 = sqrt(r862965);
        double r862967 = r862964 * r862966;
        double r862968 = z;
        double r862969 = 3.0;
        double r862970 = sqrt(r862969);
        double r862971 = r862968 / r862970;
        double r862972 = t;
        double r862973 = r862972 / r862970;
        double r862974 = r862971 * r862973;
        double r862975 = r862961 - r862974;
        double r862976 = cos(r862975);
        double r862977 = r862967 * r862976;
        double r862978 = a;
        double r862979 = b;
        double r862980 = r862979 * r862969;
        double r862981 = r862978 / r862980;
        double r862982 = r862977 - r862981;
        double r862983 = 2.4348734333899417e-18;
        bool r862984 = r862961 <= r862983;
        double r862985 = 1.0;
        double r862986 = 0.5;
        double r862987 = 2.0;
        double r862988 = pow(r862961, r862987);
        double r862989 = r862986 * r862988;
        double r862990 = r862985 - r862989;
        double r862991 = r862967 * r862990;
        double r862992 = r862991 - r862981;
        double r862993 = cos(r862961);
        double r862994 = 0.3333333333333333;
        double r862995 = r862972 * r862968;
        double r862996 = r862994 * r862995;
        double r862997 = cos(r862996);
        double r862998 = r862993 * r862997;
        double r862999 = sin(r862961);
        double r863000 = r862968 * r862972;
        double r863001 = r863000 / r862969;
        double r863002 = -r863001;
        double r863003 = sin(r863002);
        double r863004 = r862999 * r863003;
        double r863005 = r862998 - r863004;
        double r863006 = r862967 * r863005;
        double r863007 = r862985 / r862979;
        double r863008 = r862978 / r862969;
        double r863009 = r863007 * r863008;
        double r863010 = r863006 - r863009;
        double r863011 = r862984 ? r862992 : r863010;
        double r863012 = r862963 ? r862982 : r863011;
        return r863012;
}

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
Target18.5
Herbie18.8
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.711112522981125

    1. Initial program 21.1

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

    3. Applied add-sqr-sqrt21.0

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

    4. Applied times-frac21.1

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

    if -2.711112522981125 < y < 2.4348734333899417e-18

    1. Initial program 19.6

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

    2. Taylor expanded around 0 16.6

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

    if 2.4348734333899417e-18 < y

    1. Initial program 21.5

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

    3. Applied sub-neg21.5

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

    4. Applied cos-sum20.7

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

    5. Simplified20.7

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

    6. Taylor expanded around inf 20.8

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

    8. Applied *-un-lft-identity20.8

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{\color{blue}{1 \cdot a}}{b \cdot 3}\]

    9. Applied times-frac20.8

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.711112522981125:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))