Average Error: 20.6 → 18.4
Time: 48.5s
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 r30215391 = 2.0;
        double r30215392 = x;
        double r30215393 = sqrt(r30215392);
        double r30215394 = r30215391 * r30215393;
        double r30215395 = y;
        double r30215396 = z;
        double r30215397 = t;
        double r30215398 = r30215396 * r30215397;
        double r30215399 = 3.0;
        double r30215400 = r30215398 / r30215399;
        double r30215401 = r30215395 - r30215400;
        double r30215402 = cos(r30215401);
        double r30215403 = r30215394 * r30215402;
        double r30215404 = a;
        double r30215405 = b;
        double r30215406 = r30215405 * r30215399;
        double r30215407 = r30215404 / r30215406;
        double r30215408 = r30215403 - r30215407;
        return r30215408;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r30215409 = y;
        double r30215410 = -519168038.2262618;
        bool r30215411 = r30215409 <= r30215410;
        double r30215412 = 2.0;
        double r30215413 = x;
        double r30215414 = sqrt(r30215413);
        double r30215415 = r30215412 * r30215414;
        double r30215416 = sin(r30215409);
        double r30215417 = t;
        double r30215418 = 3.0;
        double r30215419 = sqrt(r30215418);
        double r30215420 = r30215417 / r30215419;
        double r30215421 = z;
        double r30215422 = r30215421 / r30215419;
        double r30215423 = r30215420 * r30215422;
        double r30215424 = sin(r30215423);
        double r30215425 = r30215416 * r30215424;
        double r30215426 = cos(r30215409);
        double r30215427 = r30215417 * r30215421;
        double r30215428 = r30215427 / r30215418;
        double r30215429 = cos(r30215428);
        double r30215430 = r30215426 * r30215429;
        double r30215431 = r30215425 + r30215430;
        double r30215432 = r30215415 * r30215431;
        double r30215433 = a;
        double r30215434 = b;
        double r30215435 = r30215433 / r30215434;
        double r30215436 = r30215435 / r30215418;
        double r30215437 = r30215432 - r30215436;
        double r30215438 = 1.1505339337432292e-20;
        bool r30215439 = r30215409 <= r30215438;
        double r30215440 = r30215417 / r30215418;
        double r30215441 = -r30215440;
        double r30215442 = r30215440 * r30215421;
        double r30215443 = fma(r30215441, r30215421, r30215442);
        double r30215444 = cos(r30215443);
        double r30215445 = cbrt(r30215409);
        double r30215446 = r30215445 * r30215445;
        double r30215447 = -r30215421;
        double r30215448 = r30215447 * r30215440;
        double r30215449 = fma(r30215446, r30215445, r30215448);
        double r30215450 = cos(r30215449);
        double r30215451 = r30215444 * r30215450;
        double r30215452 = sin(r30215443);
        double r30215453 = sin(r30215449);
        double r30215454 = r30215452 * r30215453;
        double r30215455 = r30215451 - r30215454;
        double r30215456 = r30215415 * r30215455;
        double r30215457 = r30215418 * r30215434;
        double r30215458 = r30215433 / r30215457;
        double r30215459 = r30215456 - r30215458;
        double r30215460 = r30215430 * r30215430;
        double r30215461 = sin(r30215428);
        double r30215462 = r30215461 * r30215416;
        double r30215463 = r30215462 * r30215462;
        double r30215464 = r30215460 - r30215463;
        double r30215465 = r30215464 * r30215415;
        double r30215466 = r30215430 - r30215462;
        double r30215467 = r30215465 / r30215466;
        double r30215468 = r30215467 - r30215436;
        double r30215469 = r30215439 ? r30215459 : r30215468;
        double r30215470 = r30215411 ? r30215437 : r30215469;
        return r30215470;
}

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))))