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

Average Error: 7.9 → 8.1

Time: 11.4s

Precision: 64

Math

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

↓

\[\frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

C

double f(double x, double y, double z, double t, double a) {
        double r465495 = x;
        double r465496 = y;
        double r465497 = r465495 * r465496;
        double r465498 = z;
        double r465499 = 9.0;
        double r465500 = r465498 * r465499;
        double r465501 = t;
        double r465502 = r465500 * r465501;
        double r465503 = r465497 - r465502;
        double r465504 = a;
        double r465505 = 2.0;
        double r465506 = r465504 * r465505;
        double r465507 = r465503 / r465506;
        return r465507;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r465508 = 1.0;
        double r465509 = 2.0;
        double r465510 = r465508 / r465509;
        double r465511 = x;
        double r465512 = a;
        double r465513 = y;
        double r465514 = r465512 / r465513;
        double r465515 = r465511 / r465514;
        double r465516 = r465510 * r465515;
        double r465517 = 9.0;
        double r465518 = r465517 / r465509;
        double r465519 = t;
        double r465520 = z;
        double r465521 = r465512 / r465520;
        double r465522 = r465519 / r465521;
        double r465523 = r465518 * r465522;
        double r465524 = r465516 - r465523;
        return r465524;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.9
Target	5.8
Herbie	8.1

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -3.2487071020495243e+187

   1. Initial program 27.4

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

   2. Taylor expanded around 0 27.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 27.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied associate-/l* 24.6

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]
   6. Using strategy rm

   7. Applied associate-/l* 0.9

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

   if -3.2487071020495243e+187 < (* x y) < 1.1649817974422874e+58

   1. Initial program 4.2

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

   2. Taylor expanded around 0 4.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 4.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied div-inv 4.1

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{1}{a}\right)} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]

   6. Applied associate-*r* 4.1

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{a}} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]

   if 1.1649817974422874e+58 < (* x y)

   1. Initial program 17.0

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

   2. Taylor expanded around 0 16.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 16.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied associate-/l* 14.0

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 14.0

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

   8. Applied times-frac 5.0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

   9. Applied associate-*r* 5.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{1}\right) \cdot \frac{y}{a}} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

   10. Simplified 5.0

       \[\leadsto \color{blue}{\frac{1 \cdot x}{2}} \cdot \frac{y}{a} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 5.7

       \[\leadsto \frac{1 \cdot x}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \frac{y}{a} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

   13. Applied times-frac 5.6

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \frac{x}{\sqrt{2}}\right)} \cdot \frac{y}{a} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

   14. Applied associate-*l* 5.5

       \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{y}{a}\right)} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9}{2} \cdot \frac{t}{\frac{a}{z}}\]

Reproduce

herbie shell --seed 2019303 
(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)))