Average Error: 20.2 → 15.5
Time: 1.4m
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999996669330926124530378729105:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \left(\cos y \cdot \cos \left(\frac{-t}{3} \cdot z\right) - \sin y \cdot \left(\sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)} \cdot \sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)}\right)\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{-t}{3} \cdot z\right)\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 0.9999999999999996669330926124530378729105:\\
\;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \left(\cos y \cdot \cos \left(\frac{-t}{3} \cdot z\right) - \sin y \cdot \left(\sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)} \cdot \sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)}\right)\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{-t}{3} \cdot z\right)\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r33528436 = 2.0;
        double r33528437 = x;
        double r33528438 = sqrt(r33528437);
        double r33528439 = r33528436 * r33528438;
        double r33528440 = y;
        double r33528441 = z;
        double r33528442 = t;
        double r33528443 = r33528441 * r33528442;
        double r33528444 = 3.0;
        double r33528445 = r33528443 / r33528444;
        double r33528446 = r33528440 - r33528445;
        double r33528447 = cos(r33528446);
        double r33528448 = r33528439 * r33528447;
        double r33528449 = a;
        double r33528450 = b;
        double r33528451 = r33528450 * r33528444;
        double r33528452 = r33528449 / r33528451;
        double r33528453 = r33528448 - r33528452;
        return r33528453;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r33528454 = y;
        double r33528455 = z;
        double r33528456 = t;
        double r33528457 = r33528455 * r33528456;
        double r33528458 = 3.0;
        double r33528459 = r33528457 / r33528458;
        double r33528460 = r33528454 - r33528459;
        double r33528461 = cos(r33528460);
        double r33528462 = 0.9999999999999997;
        bool r33528463 = r33528461 <= r33528462;
        double r33528464 = -r33528456;
        double r33528465 = r33528464 / r33528458;
        double r33528466 = r33528456 / r33528458;
        double r33528467 = r33528466 * r33528455;
        double r33528468 = fma(r33528465, r33528455, r33528467);
        double r33528469 = cos(r33528468);
        double r33528470 = cos(r33528454);
        double r33528471 = r33528465 * r33528455;
        double r33528472 = cos(r33528471);
        double r33528473 = r33528470 * r33528472;
        double r33528474 = sin(r33528454);
        double r33528475 = sin(r33528471);
        double r33528476 = cbrt(r33528475);
        double r33528477 = r33528476 * r33528476;
        double r33528478 = r33528476 * r33528477;
        double r33528479 = r33528474 * r33528478;
        double r33528480 = r33528473 - r33528479;
        double r33528481 = r33528469 * r33528480;
        double r33528482 = sin(r33528468);
        double r33528483 = 1.0;
        double r33528484 = fma(r33528483, r33528454, r33528471);
        double r33528485 = sin(r33528484);
        double r33528486 = r33528482 * r33528485;
        double r33528487 = r33528481 - r33528486;
        double r33528488 = 2.0;
        double r33528489 = x;
        double r33528490 = sqrt(r33528489);
        double r33528491 = r33528488 * r33528490;
        double r33528492 = r33528487 * r33528491;
        double r33528493 = a;
        double r33528494 = b;
        double r33528495 = r33528494 * r33528458;
        double r33528496 = r33528493 / r33528495;
        double r33528497 = r33528492 - r33528496;
        double r33528498 = -0.5;
        double r33528499 = r33528454 * r33528454;
        double r33528500 = fma(r33528498, r33528499, r33528483);
        double r33528501 = r33528500 * r33528491;
        double r33528502 = r33528501 - r33528496;
        double r33528503 = r33528463 ? r33528497 : r33528502;
        return r33528503;
}

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

Original20.2
Target18.0
Herbie15.5
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999997

    1. Initial program 19.2

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

    3. Applied *-un-lft-identity19.2

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

    4. Applied times-frac19.2

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

    5. Applied *-un-lft-identity19.2

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

    6. Applied prod-diff19.2

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

    7. Applied cos-sum16.4

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

    9. Applied fma-udef16.4

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

    10. Applied cos-sum15.6

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

    12. Applied add-cube-cbrt15.6

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

    if 0.9999999999999997 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 22.0

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

    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. Simplified15.4

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999996669330926124530378729105:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \left(\cos y \cdot \cos \left(\frac{-t}{3} \cdot z\right) - \sin y \cdot \left(\sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)} \cdot \sqrt[3]{\sin \left(\frac{-t}{3} \cdot z\right)}\right)\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{-t}{3} \cdot z\right)\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))