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

Average Error: 10.6 → 7.2

Time: 25.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r544780 = x;
        double r544781 = y;
        double r544782 = z;
        double r544783 = r544781 * r544782;
        double r544784 = r544780 - r544783;
        double r544785 = t;
        double r544786 = a;
        double r544787 = r544786 * r544782;
        double r544788 = r544785 - r544787;
        double r544789 = r544784 / r544788;
        return r544789;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r544790 = x;
        double r544791 = t;
        double r544792 = a;
        double r544793 = z;
        double r544794 = r544792 * r544793;
        double r544795 = r544791 - r544794;
        double r544796 = r544790 / r544795;
        double r544797 = y;
        double r544798 = cbrt(r544793);
        double r544799 = r544798 * r544798;
        double r544800 = cbrt(r544795);
        double r544801 = r544800 * r544800;
        double r544802 = r544799 / r544801;
        double r544803 = r544797 * r544802;
        double r544804 = r544798 / r544800;
        double r544805 = r544803 * r544804;
        double r544806 = r544796 - r544805;
        return r544806;
}

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	7.2

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

Derivation

1. Initial program 10.6

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

3. Applied div-sub 10.6

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

4. Simplified 7.9

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

6. Applied add-cube-cbrt 8.2

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

7. Applied add-cube-cbrt 8.3

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

8. Applied times-frac 8.3

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

9. Applied associate-*r* 7.2

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

10. Final simplification 7.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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))))