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

Average Error: 2.7 → 1.8

Time: 15.0s

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 r457673 = x;
        double r457674 = y;
        double r457675 = z;
        double r457676 = t;
        double r457677 = r457675 * r457676;
        double r457678 = r457674 - r457677;
        double r457679 = r457673 / r457678;
        return r457679;
}

↓

double f(double x, double y, double z, double t) {
        double r457680 = z;
        double r457681 = t;
        double r457682 = r457680 * r457681;
        double r457683 = -inf.0;
        bool r457684 = r457682 <= r457683;
        double r457685 = x;
        double r457686 = cbrt(r457685);
        double r457687 = r457686 * r457686;
        double r457688 = y;
        double r457689 = r457688 / r457686;
        double r457690 = cbrt(r457686);
        double r457691 = r457690 * r457690;
        double r457692 = r457680 / r457691;
        double r457693 = r457691 * r457690;
        double r457694 = cbrt(r457693);
        double r457695 = r457681 / r457694;
        double r457696 = r457692 * r457695;
        double r457697 = r457689 - r457696;
        double r457698 = r457687 / r457697;
        double r457699 = 3.9822145511872104e+305;
        bool r457700 = r457682 <= r457699;
        double r457701 = r457688 - r457682;
        double r457702 = r457685 / r457701;
        double r457703 = cbrt(r457680);
        double r457704 = r457703 * r457703;
        double r457705 = r457704 / r457690;
        double r457706 = r457703 / r457690;
        double r457707 = r457681 / r457690;
        double r457708 = r457706 * r457707;
        double r457709 = r457705 * r457708;
        double r457710 = r457689 - r457709;
        double r457711 = r457687 / r457710;
        double r457712 = r457700 ? r457702 : r457711;
        double r457713 = r457684 ? r457698 : r457712;
        return r457713;
}

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 
(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))))