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

Average Error: 7.4 → 7.4

Time: 11.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r494023 = x;
        double r494024 = y;
        double r494025 = r494023 * r494024;
        double r494026 = z;
        double r494027 = 9.0;
        double r494028 = r494026 * r494027;
        double r494029 = t;
        double r494030 = r494028 * r494029;
        double r494031 = r494025 - r494030;
        double r494032 = a;
        double r494033 = 2.0;
        double r494034 = r494032 * r494033;
        double r494035 = r494031 / r494034;
        return r494035;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r494036 = x;
        double r494037 = y;
        double r494038 = r494036 * r494037;
        double r494039 = 9.0;
        double r494040 = cbrt(r494039);
        double r494041 = r494040 * r494040;
        double r494042 = t;
        double r494043 = z;
        double r494044 = r494042 * r494043;
        double r494045 = r494040 * r494044;
        double r494046 = r494041 * r494045;
        double r494047 = r494038 - r494046;
        double r494048 = a;
        double r494049 = 2.0;
        double r494050 = r494048 * r494049;
        double r494051 = r494047 / r494050;
        return r494051;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.4
Target	5.5
Herbie	7.4

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

Derivation

1. Initial program 7.4

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

2. Taylor expanded around inf 7.4

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

4. Applied add-cube-cbrt 7.4

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

5. Applied associate-*l* 7.4

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

6. Final simplification 7.4

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

Reproduce

herbie shell --seed 2019298 
(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)))