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

Average Error: 7.8 → 4.0

Time: 12.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -1.508242470936290470296324392538488250699 \cdot 10^{83} \lor \neg \left(\left(z \cdot 9\right) \cdot t \le 6.940538148906954006462762148490763180159 \cdot 10^{247}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r527821 = x;
        double r527822 = y;
        double r527823 = r527821 * r527822;
        double r527824 = z;
        double r527825 = 9.0;
        double r527826 = r527824 * r527825;
        double r527827 = t;
        double r527828 = r527826 * r527827;
        double r527829 = r527823 - r527828;
        double r527830 = a;
        double r527831 = 2.0;
        double r527832 = r527830 * r527831;
        double r527833 = r527829 / r527832;
        return r527833;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r527834 = z;
        double r527835 = 9.0;
        double r527836 = r527834 * r527835;
        double r527837 = t;
        double r527838 = r527836 * r527837;
        double r527839 = -1.5082424709362905e+83;
        bool r527840 = r527838 <= r527839;
        double r527841 = 6.940538148906954e+247;
        bool r527842 = r527838 <= r527841;
        double r527843 = !r527842;
        bool r527844 = r527840 || r527843;
        double r527845 = 0.5;
        double r527846 = x;
        double r527847 = a;
        double r527848 = cbrt(r527847);
        double r527849 = r527848 * r527848;
        double r527850 = r527846 / r527849;
        double r527851 = y;
        double r527852 = r527851 / r527848;
        double r527853 = r527850 * r527852;
        double r527854 = r527845 * r527853;
        double r527855 = 4.5;
        double r527856 = r527834 / r527847;
        double r527857 = r527837 * r527856;
        double r527858 = r527855 * r527857;
        double r527859 = r527854 - r527858;
        double r527860 = r527846 * r527851;
        double r527861 = r527835 * r527837;
        double r527862 = r527834 * r527861;
        double r527863 = r527860 - r527862;
        double r527864 = 2.0;
        double r527865 = r527847 * r527864;
        double r527866 = r527863 / r527865;
        double r527867 = r527844 ? r527859 : r527866;
        return r527867;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.8
Target	5.8
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if (* (* z 9.0) t) < -1.5082424709362905e+83 or 6.940538148906954e+247 < (* (* z 9.0) t)

   1. Initial program 23.8

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

   2. Taylor expanded around 0 23.5

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

   4. Applied add-cube-cbrt 23.6

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

   5. Applied times-frac 21.6

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

   7. Applied *-un-lft-identity 21.6

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

   8. Applied times-frac 3.4

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

   9. Simplified 3.4

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

   if -1.5082424709362905e+83 < (* (* z 9.0) t) < 6.940538148906954e+247

   1. Initial program 4.1

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

   3. Applied associate-*l* 4.1

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -1.508242470936290470296324392538488250699 \cdot 10^{83} \lor \neg \left(\left(z \cdot 9\right) \cdot t \le 6.940538148906954006462762148490763180159 \cdot 10^{247}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))