Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Average Error: 10.6 → 1.9

Time: 18.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -469735603.148817598819732666015625 \lor \neg \left(z \le 3.061405469162073408418218211607294509253 \cdot 10^{53}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} + \frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r555830 = x;
        double r555831 = y;
        double r555832 = z;
        double r555833 = r555831 * r555832;
        double r555834 = r555830 - r555833;
        double r555835 = t;
        double r555836 = a;
        double r555837 = r555836 * r555832;
        double r555838 = r555835 - r555837;
        double r555839 = r555834 / r555838;
        return r555839;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r555840 = z;
        double r555841 = -469735603.1488176;
        bool r555842 = r555840 <= r555841;
        double r555843 = 3.0614054691620734e+53;
        bool r555844 = r555840 <= r555843;
        double r555845 = !r555844;
        bool r555846 = r555842 || r555845;
        double r555847 = x;
        double r555848 = t;
        double r555849 = a;
        double r555850 = r555849 * r555840;
        double r555851 = r555848 - r555850;
        double r555852 = r555847 / r555851;
        double r555853 = cbrt(r555852);
        double r555854 = r555853 * r555853;
        double r555855 = r555854 * r555853;
        double r555856 = y;
        double r555857 = -r555856;
        double r555858 = r555848 / r555840;
        double r555859 = r555858 - r555849;
        double r555860 = r555857 / r555859;
        double r555861 = r555855 + r555860;
        double r555862 = 1.0;
        double r555863 = r555856 * r555840;
        double r555864 = r555847 - r555863;
        double r555865 = r555851 / r555864;
        double r555866 = r555862 / r555865;
        double r555867 = r555846 ? r555861 : r555866;
        return r555867;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.6
Target	1.6
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -469735603.1488176 or 3.0614054691620734e+53 < z

   1. Initial program 22.7

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
   2. Using strategy rm

   3. Applied div-sub 22.7

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

   4. Simplified 14.1

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

   6. Applied sub-neg 14.1

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

   7. Simplified 3.0

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

   9. Applied add-cube-cbrt 3.2

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

   if -469735603.1488176 < z < 3.0614054691620734e+53

   1. Initial program 0.4

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
   2. Using strategy rm

   3. Applied clear-num 0.9

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -469735603.148817598819732666015625 \lor \neg \left(z \le 3.061405469162073408418218211607294509253 \cdot 10^{53}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} + \frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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