Average Error: 20.6 → 18.4
Time: 49.1s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -519168038.226261794567108154296875:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) + \cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;y \le 1.150533933743229244197856592153169093091 \cdot 10^{-20}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \left(-z\right) \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \left(-z\right) \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right)\right) - \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)}{\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) - \sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y} - \frac{\frac{a}{b}}{3}\\ \end{array}\]
\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, \frac{t}{3} \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \left(-z\right) \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \left(-z\right) \cdot \frac{t}{3}\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 r26811257 = 2.0;
        double r26811258 = x;
        double r26811259 = sqrt(r26811258);
        double r26811260 = r26811257 * r26811259;
        double r26811261 = y;
        double r26811262 = z;
        double r26811263 = t;
        double r26811264 = r26811262 * r26811263;
        double r26811265 = 3.0;
        double r26811266 = r26811264 / r26811265;
        double r26811267 = r26811261 - r26811266;
        double r26811268 = cos(r26811267);
        double r26811269 = r26811260 * r26811268;
        double r26811270 = a;
        double r26811271 = b;
        double r26811272 = r26811271 * r26811265;
        double r26811273 = r26811270 / r26811272;
        double r26811274 = r26811269 - r26811273;
        return r26811274;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r26811275 = y;
        double r26811276 = -519168038.2262618;
        bool r26811277 = r26811275 <= r26811276;
        double r26811278 = 2.0;
        double r26811279 = x;
        double r26811280 = sqrt(r26811279);
        double r26811281 = r26811278 * r26811280;
        double r26811282 = sin(r26811275);
        double r26811283 = t;
        double r26811284 = 3.0;
        double r26811285 = sqrt(r26811284);
        double r26811286 = r26811283 / r26811285;
        double r26811287 = z;
        double r26811288 = r26811287 / r26811285;
        double r26811289 = r26811286 * r26811288;
        double r26811290 = sin(r26811289);
        double r26811291 = r26811282 * r26811290;
        double r26811292 = cos(r26811275);
        double r26811293 = r26811283 * r26811287;
        double r26811294 = r26811293 / r26811284;
        double r26811295 = cos(r26811294);
        double r26811296 = r26811292 * r26811295;
        double r26811297 = r26811291 + r26811296;
        double r26811298 = r26811281 * r26811297;
        double r26811299 = a;
        double r26811300 = b;
        double r26811301 = r26811299 / r26811300;
        double r26811302 = r26811301 / r26811284;
        double r26811303 = r26811298 - r26811302;
        double r26811304 = 1.1505339337432292e-20;
        bool r26811305 = r26811275 <= r26811304;
        double r26811306 = r26811283 / r26811284;
        double r26811307 = -r26811306;
        double r26811308 = r26811306 * r26811287;
        double r26811309 = fma(r26811307, r26811287, r26811308);
        double r26811310 = cos(r26811309);
        double r26811311 = cbrt(r26811275);
        double r26811312 = r26811311 * r26811311;
        double r26811313 = -r26811287;
        double r26811314 = r26811313 * r26811306;
        double r26811315 = fma(r26811312, r26811311, r26811314);
        double r26811316 = cos(r26811315);
        double r26811317 = r26811310 * r26811316;
        double r26811318 = sin(r26811309);
        double r26811319 = sin(r26811315);
        double r26811320 = r26811318 * r26811319;
        double r26811321 = r26811317 - r26811320;
        double r26811322 = r26811281 * r26811321;
        double r26811323 = r26811284 * r26811300;
        double r26811324 = r26811299 / r26811323;
        double r26811325 = r26811322 - r26811324;
        double r26811326 = r26811296 * r26811296;
        double r26811327 = sin(r26811294);
        double r26811328 = r26811327 * r26811282;
        double r26811329 = r26811328 * r26811328;
        double r26811330 = r26811326 - r26811329;
        double r26811331 = r26811330 * r26811281;
        double r26811332 = r26811296 - r26811328;
        double r26811333 = r26811331 / r26811332;
        double r26811334 = r26811333 - r26811302;
        double r26811335 = r26811305 ? r26811325 : r26811334;
        double r26811336 = r26811277 ? r26811303 : r26811335;
        return r26811336;
}

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

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

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))))