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

Average Error: 7.7 → 0.9

Time: 9.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.2344983160605906 \cdot 10^{285}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - 9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r413793 = x;
        double r413794 = y;
        double r413795 = r413793 * r413794;
        double r413796 = z;
        double r413797 = 9.0;
        double r413798 = r413796 * r413797;
        double r413799 = t;
        double r413800 = r413798 * r413799;
        double r413801 = r413795 - r413800;
        double r413802 = a;
        double r413803 = 2.0;
        double r413804 = r413802 * r413803;
        double r413805 = r413801 / r413804;
        return r413805;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r413806 = x;
        double r413807 = y;
        double r413808 = r413806 * r413807;
        double r413809 = z;
        double r413810 = 9.0;
        double r413811 = r413809 * r413810;
        double r413812 = t;
        double r413813 = r413811 * r413812;
        double r413814 = r413808 - r413813;
        double r413815 = -inf.0;
        bool r413816 = r413814 <= r413815;
        double r413817 = 1.2344983160605906e+285;
        bool r413818 = r413814 <= r413817;
        double r413819 = !r413818;
        bool r413820 = r413816 || r413819;
        double r413821 = 0.5;
        double r413822 = a;
        double r413823 = r413807 / r413822;
        double r413824 = r413806 * r413823;
        double r413825 = r413821 * r413824;
        double r413826 = 4.5;
        double r413827 = cbrt(r413822);
        double r413828 = r413827 * r413827;
        double r413829 = r413812 / r413828;
        double r413830 = r413809 / r413827;
        double r413831 = r413829 * r413830;
        double r413832 = r413826 * r413831;
        double r413833 = r413825 - r413832;
        double r413834 = r413812 * r413809;
        double r413835 = r413810 * r413834;
        double r413836 = r413808 - r413835;
        double r413837 = 2.0;
        double r413838 = r413822 * r413837;
        double r413839 = r413836 / r413838;
        double r413840 = r413820 ? r413833 : r413839;
        return r413840;
}

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	5.8
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 1.2344983160605906e+285 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 57.1

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

   2. Taylor expanded around 0 57.0

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

   4. Applied add-cube-cbrt 57.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   5. Applied times-frac 31.3

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 31.3

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\]

   8. Applied times-frac 0.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\]

   9. Simplified 0.9

      \[\leadsto 0.5 \cdot \left(\color{blue}{x} \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\]

   if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 1.2344983160605906e+285

   1. Initial program 0.9

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

   2. Taylor expanded around inf 0.9

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.2344983160605906 \cdot 10^{285}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - 9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :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 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

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