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

Average Error: 7.7 → 4.8

Time: 26.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 3.053927322628306114212532979821395041866 \cdot 10^{156}:\\ \;\;\;\;\frac{x \cdot y - \sqrt{9} \cdot \left(\sqrt{9} \cdot \left(t \cdot z\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r83724331 = x;
        double r83724332 = y;
        double r83724333 = r83724331 * r83724332;
        double r83724334 = z;
        double r83724335 = 9.0;
        double r83724336 = r83724334 * r83724335;
        double r83724337 = t;
        double r83724338 = r83724336 * r83724337;
        double r83724339 = r83724333 - r83724338;
        double r83724340 = a;
        double r83724341 = 2.0;
        double r83724342 = r83724340 * r83724341;
        double r83724343 = r83724339 / r83724342;
        return r83724343;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r83724344 = z;
        double r83724345 = 9.0;
        double r83724346 = r83724344 * r83724345;
        double r83724347 = t;
        double r83724348 = r83724346 * r83724347;
        double r83724349 = -inf.0;
        bool r83724350 = r83724348 <= r83724349;
        double r83724351 = x;
        double r83724352 = y;
        double r83724353 = r83724351 * r83724352;
        double r83724354 = a;
        double r83724355 = 2.0;
        double r83724356 = r83724354 * r83724355;
        double r83724357 = r83724353 / r83724356;
        double r83724358 = r83724345 * r83724347;
        double r83724359 = r83724358 / r83724354;
        double r83724360 = r83724344 / r83724355;
        double r83724361 = r83724359 * r83724360;
        double r83724362 = r83724357 - r83724361;
        double r83724363 = 3.053927322628306e+156;
        bool r83724364 = r83724348 <= r83724363;
        double r83724365 = sqrt(r83724345);
        double r83724366 = r83724347 * r83724344;
        double r83724367 = r83724365 * r83724366;
        double r83724368 = r83724365 * r83724367;
        double r83724369 = r83724353 - r83724368;
        double r83724370 = r83724369 / r83724356;
        double r83724371 = r83724364 ? r83724370 : r83724362;
        double r83724372 = r83724350 ? r83724362 : r83724371;
        return r83724372;
}

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	5.7
Herbie	4.8

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

Derivation

1. Split input into 2 regimes

2. if (* (* z 9.0) t) < -inf.0 or 3.053927322628306e+156 < (* (* z 9.0) t)

   1. Initial program 33.3

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

   2. Taylor expanded around inf 33.2

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

   4. Applied add-sqr-sqrt 33.2

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

   5. Applied associate-*l* 33.2

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

   7. Applied associate-*r* 33.3

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

   9. Applied div-sub 33.3

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

   10. Simplified 7.4

       \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{9 \cdot t}{a} \cdot \frac{z}{2}}\]

   if -inf.0 < (* (* z 9.0) t) < 3.053927322628306e+156

   1. Initial program 4.4

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

   2. Taylor expanded around inf 4.4

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

   4. Applied add-sqr-sqrt 4.4

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

   5. Applied associate-*l* 4.4

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

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 3.053927322628306114212532979821395041866 \cdot 10^{156}:\\ \;\;\;\;\frac{x \cdot y - \sqrt{9} \cdot \left(\sqrt{9} \cdot \left(t \cdot z\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))