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

Average Error: 10.6 → 7.2

Time: 27.0s

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 r447097 = x;
        double r447098 = y;
        double r447099 = z;
        double r447100 = r447098 * r447099;
        double r447101 = r447097 - r447100;
        double r447102 = t;
        double r447103 = a;
        double r447104 = r447103 * r447099;
        double r447105 = r447102 - r447104;
        double r447106 = r447101 / r447105;
        return r447106;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r447107 = x;
        double r447108 = t;
        double r447109 = a;
        double r447110 = z;
        double r447111 = r447109 * r447110;
        double r447112 = r447108 - r447111;
        double r447113 = r447107 / r447112;
        double r447114 = y;
        double r447115 = cbrt(r447110);
        double r447116 = r447115 * r447115;
        double r447117 = cbrt(r447112);
        double r447118 = r447117 * r447117;
        double r447119 = r447116 / r447118;
        double r447120 = r447114 * r447119;
        double r447121 = r447115 / r447117;
        double r447122 = r447120 * r447121;
        double r447123 = r447113 - r447122;
        return r447123;
}

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))))