Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Average Error: 7.7 → 0.9

Time: 21.1s

Precision: 64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.621903211914449632743630620835126462008 \cdot 10^{239}\right):\\ \;\;\;\;\left(\frac{x}{a} \cdot \frac{y}{2} - \frac{z \cdot 9}{a} \cdot \frac{t}{2}\right) + \frac{z \cdot 9}{a} \cdot \left(\left(-\frac{t}{2}\right) + \frac{t}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r552506 = x;
        double r552507 = y;
        double r552508 = r552506 * r552507;
        double r552509 = z;
        double r552510 = 9.0;
        double r552511 = r552509 * r552510;
        double r552512 = t;
        double r552513 = r552511 * r552512;
        double r552514 = r552508 - r552513;
        double r552515 = a;
        double r552516 = 2.0;
        double r552517 = r552515 * r552516;
        double r552518 = r552514 / r552517;
        return r552518;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r552519 = x;
        double r552520 = y;
        double r552521 = r552519 * r552520;
        double r552522 = z;
        double r552523 = 9.0;
        double r552524 = r552522 * r552523;
        double r552525 = t;
        double r552526 = r552524 * r552525;
        double r552527 = r552521 - r552526;
        double r552528 = -inf.0;
        bool r552529 = r552527 <= r552528;
        double r552530 = 6.62190321191445e+239;
        bool r552531 = r552527 <= r552530;
        double r552532 = !r552531;
        bool r552533 = r552529 || r552532;
        double r552534 = a;
        double r552535 = r552519 / r552534;
        double r552536 = 2.0;
        double r552537 = r552520 / r552536;
        double r552538 = r552535 * r552537;
        double r552539 = r552524 / r552534;
        double r552540 = r552525 / r552536;
        double r552541 = r552539 * r552540;
        double r552542 = r552538 - r552541;
        double r552543 = -r552540;
        double r552544 = r552543 + r552540;
        double r552545 = r552539 * r552544;
        double r552546 = r552542 + r552545;
        double r552547 = 0.5;
        double r552548 = r552521 / r552534;
        double r552549 = r552547 * r552548;
        double r552550 = 4.5;
        double r552551 = r552525 * r552522;
        double r552552 = r552550 * r552551;
        double r552553 = 1.0;
        double r552554 = r552553 / r552534;
        double r552555 = r552552 * r552554;
        double r552556 = r552549 - r552555;
        double r552557 = r552533 ? r552546 : r552556;
        return r552557;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.7
Target	5.5
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 6.62190321191445e+239 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 46.2

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
   2. Using strategy rm

   3. Applied div-sub 46.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 46.4

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\left(\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}}\]

   6. Applied add-sqr-sqrt 53.4

      \[\leadsto \color{blue}{\sqrt{\frac{x \cdot y}{a \cdot 2}} \cdot \sqrt{\frac{x \cdot y}{a \cdot 2}}} - \left(\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\]

   7. Applied prod-diff 53.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x \cdot y}{a \cdot 2}}, \sqrt{\frac{x \cdot y}{a \cdot 2}}, -\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \left(\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}, \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}, \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \left(\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right)\right)}\]

   8. Simplified 25.5

      \[\leadsto \color{blue}{\left(\frac{x}{a} \cdot \frac{y}{2} - \frac{z \cdot 9}{a} \cdot \frac{t}{2}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}, \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}, \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \left(\sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right)\right)\]

   9. Simplified 0.7

      \[\leadsto \left(\frac{x}{a} \cdot \frac{y}{2} - \frac{z \cdot 9}{a} \cdot \frac{t}{2}\right) + \color{blue}{\frac{z \cdot 9}{a} \cdot \left(\left(-\frac{t}{2}\right) + \frac{t}{2}\right)}\]

   if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 6.62190321191445e+239

   1. Initial program 1.0

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied div-inv 1.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{a}\right)}\]

   5. Applied associate-*r* 1.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.621903211914449632743630620835126462008 \cdot 10^{239}\right):\\ \;\;\;\;\left(\frac{x}{a} \cdot \frac{y}{2} - \frac{z \cdot 9}{a} \cdot \frac{t}{2}\right) + \frac{z \cdot 9}{a} \cdot \left(\left(-\frac{t}{2}\right) + \frac{t}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.14403070783397609e99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))