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

Average Error: 7.4 → 0.9

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.86218541740101406 \cdot 10^{182} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.83663500979720747 \cdot 10^{215}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r647772 = x;
        double r647773 = y;
        double r647774 = r647772 * r647773;
        double r647775 = z;
        double r647776 = 9.0;
        double r647777 = r647775 * r647776;
        double r647778 = t;
        double r647779 = r647777 * r647778;
        double r647780 = r647774 - r647779;
        double r647781 = a;
        double r647782 = 2.0;
        double r647783 = r647781 * r647782;
        double r647784 = r647780 / r647783;
        return r647784;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r647785 = x;
        double r647786 = y;
        double r647787 = r647785 * r647786;
        double r647788 = z;
        double r647789 = 9.0;
        double r647790 = r647788 * r647789;
        double r647791 = t;
        double r647792 = r647790 * r647791;
        double r647793 = r647787 - r647792;
        double r647794 = -1.862185417401014e+182;
        bool r647795 = r647793 <= r647794;
        double r647796 = 7.836635009797207e+215;
        bool r647797 = r647793 <= r647796;
        double r647798 = !r647797;
        bool r647799 = r647795 || r647798;
        double r647800 = 0.5;
        double r647801 = a;
        double r647802 = r647786 / r647801;
        double r647803 = r647785 * r647802;
        double r647804 = r647800 * r647803;
        double r647805 = 4.5;
        double r647806 = r647791 * r647805;
        double r647807 = r647788 / r647801;
        double r647808 = r647806 * r647807;
        double r647809 = r647804 - r647808;
        double r647810 = r647787 / r647801;
        double r647811 = r647800 * r647810;
        double r647812 = r647791 * r647788;
        double r647813 = r647812 / r647801;
        double r647814 = r647805 * r647813;
        double r647815 = r647811 - r647814;
        double r647816 = 1.0;
        double r647817 = pow(r647815, r647816);
        double r647818 = r647799 ? r647809 : r647817;
        return r647818;
}

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	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)) < -1.862185417401014e+182 or 7.836635009797207e+215 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 27.4

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

   2. Taylor expanded around 0 26.9

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

   4. Applied *-un-lft-identity 26.9

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

   5. Applied times-frac 14.5

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

   6. Applied associate-*r* 14.5

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

   7. Simplified 14.5

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

   9. Applied *-un-lft-identity 14.5

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

   10. Applied times-frac 1.4

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

   11. Simplified 1.4

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

   if -1.862185417401014e+182 < (- (* x y) (* (* z 9.0) t)) < 7.836635009797207e+215

   1. Initial program 0.7

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

   2. Taylor expanded around 0 0.7

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

   4. Applied pow1 0.7

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.86218541740101406 \cdot 10^{182} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.83663500979720747 \cdot 10^{215}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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 -2.090464557976709e+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)))