Average Error: 20.6 → 18.4
Time: 45.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -519168038.226261794567108154296875:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) + \cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;y \le 1.150533933743229244197856592153169093091 \cdot 10^{-20}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)}{\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) - \sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y} - \frac{\frac{a}{b}}{3}\\ \end{array}\]
\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 -519168038.226261794567108154296875:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) + \cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \frac{\frac{a}{b}}{3}\\

\mathbf{elif}\;y \le 1.150533933743229244197856592153169093091 \cdot 10^{-20}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r30386969 = 2.0;
        double r30386970 = x;
        double r30386971 = sqrt(r30386970);
        double r30386972 = r30386969 * r30386971;
        double r30386973 = y;
        double r30386974 = z;
        double r30386975 = t;
        double r30386976 = r30386974 * r30386975;
        double r30386977 = 3.0;
        double r30386978 = r30386976 / r30386977;
        double r30386979 = r30386973 - r30386978;
        double r30386980 = cos(r30386979);
        double r30386981 = r30386972 * r30386980;
        double r30386982 = a;
        double r30386983 = b;
        double r30386984 = r30386983 * r30386977;
        double r30386985 = r30386982 / r30386984;
        double r30386986 = r30386981 - r30386985;
        return r30386986;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r30386987 = y;
        double r30386988 = -519168038.2262618;
        bool r30386989 = r30386987 <= r30386988;
        double r30386990 = 2.0;
        double r30386991 = x;
        double r30386992 = sqrt(r30386991);
        double r30386993 = r30386990 * r30386992;
        double r30386994 = sin(r30386987);
        double r30386995 = t;
        double r30386996 = 3.0;
        double r30386997 = sqrt(r30386996);
        double r30386998 = r30386995 / r30386997;
        double r30386999 = z;
        double r30387000 = r30386999 / r30386997;
        double r30387001 = r30386998 * r30387000;
        double r30387002 = sin(r30387001);
        double r30387003 = r30386994 * r30387002;
        double r30387004 = cos(r30386987);
        double r30387005 = r30386995 * r30386999;
        double r30387006 = r30387005 / r30386996;
        double r30387007 = cos(r30387006);
        double r30387008 = r30387004 * r30387007;
        double r30387009 = r30387003 + r30387008;
        double r30387010 = r30386993 * r30387009;
        double r30387011 = a;
        double r30387012 = b;
        double r30387013 = r30387011 / r30387012;
        double r30387014 = r30387013 / r30386996;
        double r30387015 = r30387010 - r30387014;
        double r30387016 = 1.1505339337432292e-20;
        bool r30387017 = r30386987 <= r30387016;
        double r30387018 = r30386995 / r30386996;
        double r30387019 = -r30387018;
        double r30387020 = r30386999 * r30387018;
        double r30387021 = fma(r30387019, r30386999, r30387020);
        double r30387022 = cos(r30387021);
        double r30387023 = cbrt(r30386987);
        double r30387024 = r30387023 * r30387023;
        double r30387025 = -r30386999;
        double r30387026 = r30387018 * r30387025;
        double r30387027 = fma(r30387024, r30387023, r30387026);
        double r30387028 = cos(r30387027);
        double r30387029 = r30387022 * r30387028;
        double r30387030 = sin(r30387021);
        double r30387031 = sin(r30387027);
        double r30387032 = r30387030 * r30387031;
        double r30387033 = r30387029 - r30387032;
        double r30387034 = r30386993 * r30387033;
        double r30387035 = r30386996 * r30387012;
        double r30387036 = r30387011 / r30387035;
        double r30387037 = r30387034 - r30387036;
        double r30387038 = r30387008 * r30387008;
        double r30387039 = sin(r30387006);
        double r30387040 = r30387039 * r30386994;
        double r30387041 = r30387040 * r30387040;
        double r30387042 = r30387038 - r30387041;
        double r30387043 = r30387042 * r30386993;
        double r30387044 = r30387008 - r30387040;
        double r30387045 = r30387043 / r30387044;
        double r30387046 = r30387045 - r30387014;
        double r30387047 = r30387017 ? r30387037 : r30387046;
        double r30387048 = r30386989 ? r30387015 : r30387047;
        return r30387048;
}

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

Target

Original20.6
Target18.3
Herbie18.4
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -519168038.2262618

    1. Initial program 21.8

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

    3. Applied cos-diff20.8

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

    5. Applied associate-/r*20.8

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

    7. Applied add-sqr-sqrt20.8

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

    8. Applied times-frac20.8

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

    if -519168038.2262618 < y < 1.1505339337432292e-20

    1. Initial program 19.7

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

    3. Applied *-un-lft-identity19.7

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

    4. Applied times-frac19.7

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

    5. Applied add-cube-cbrt19.8

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

    6. Applied prod-diff19.8

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

    7. Applied cos-sum16.2

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

    if 1.1505339337432292e-20 < y

    1. Initial program 20.9

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

    3. Applied cos-diff20.2

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

    5. Applied associate-/r*20.3

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

    7. Applied flip-+20.3

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

    8. Applied associate-*r/20.3

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -519168038.226261794567108154296875:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) + \cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;y \le 1.150533933743229244197856592153169093091 \cdot 10^{-20}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)}{\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) - \sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y} - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))))