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

Average Error: 2.7 → 1.8

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \frac{z}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{t}{\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}}\\ \mathbf{elif}\;z \cdot t \le 3.982214551187210407512533577328782205679 \cdot 10^{305}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{t}{\sqrt[3]{\sqrt[3]{x}}}\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r406614 = x;
        double r406615 = y;
        double r406616 = z;
        double r406617 = t;
        double r406618 = r406616 * r406617;
        double r406619 = r406615 - r406618;
        double r406620 = r406614 / r406619;
        return r406620;
}

↓

double f(double x, double y, double z, double t) {
        double r406621 = z;
        double r406622 = t;
        double r406623 = r406621 * r406622;
        double r406624 = -inf.0;
        bool r406625 = r406623 <= r406624;
        double r406626 = x;
        double r406627 = cbrt(r406626);
        double r406628 = r406627 * r406627;
        double r406629 = y;
        double r406630 = r406629 / r406627;
        double r406631 = cbrt(r406627);
        double r406632 = r406631 * r406631;
        double r406633 = r406621 / r406632;
        double r406634 = r406632 * r406631;
        double r406635 = cbrt(r406634);
        double r406636 = r406622 / r406635;
        double r406637 = r406633 * r406636;
        double r406638 = r406630 - r406637;
        double r406639 = r406628 / r406638;
        double r406640 = 3.9822145511872104e+305;
        bool r406641 = r406623 <= r406640;
        double r406642 = r406629 - r406623;
        double r406643 = r406626 / r406642;
        double r406644 = cbrt(r406621);
        double r406645 = r406644 * r406644;
        double r406646 = r406645 / r406631;
        double r406647 = r406644 / r406631;
        double r406648 = r406622 / r406631;
        double r406649 = r406647 * r406648;
        double r406650 = r406646 * r406649;
        double r406651 = r406630 - r406650;
        double r406652 = r406628 / r406651;
        double r406653 = r406641 ? r406643 : r406652;
        double r406654 = r406625 ? r406639 : r406653;
        return r406654;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.7
Target	1.9
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;x \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x \lt 2.137830643487644440407921345820165445823 \cdot 10^{131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* z t) < -inf.0

   1. Initial program 18.0

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

   3. Applied add-cube-cbrt 18.0

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

   4. Applied associate-/l* 18.0

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

   6. Applied div-sub 19.5

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

   8. Applied add-cube-cbrt 19.5

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

   9. Applied times-frac 11.5

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

   11. Applied add-cube-cbrt 11.5

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

   if -inf.0 < (* z t) < 3.9822145511872104e+305

   1. Initial program 0.1

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

   if 3.9822145511872104e+305 < (* z t)

   1. Initial program 21.7

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

   3. Applied add-cube-cbrt 21.7

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

   4. Applied associate-/l* 21.7

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

   6. Applied div-sub 23.2

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

   8. Applied add-cube-cbrt 23.2

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

   9. Applied times-frac 14.5

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

   11. Applied add-cube-cbrt 14.5

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

   12. Applied times-frac 14.5

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

   13. Applied associate-*l* 14.5

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \frac{z}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{t}{\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}}\\ \mathbf{elif}\;z \cdot t \le 3.982214551187210407512533577328782205679 \cdot 10^{305}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{t}{\sqrt[3]{\sqrt[3]{x}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t)))))

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