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

Average Error: 2.7 → 1.8

Time: 16.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r383389 = x;
        double r383390 = y;
        double r383391 = z;
        double r383392 = t;
        double r383393 = r383391 * r383392;
        double r383394 = r383390 - r383393;
        double r383395 = r383389 / r383394;
        return r383395;
}

↓

double f(double x, double y, double z, double t) {
        double r383396 = z;
        double r383397 = t;
        double r383398 = r383396 * r383397;
        double r383399 = -inf.0;
        bool r383400 = r383398 <= r383399;
        double r383401 = x;
        double r383402 = cbrt(r383401);
        double r383403 = r383402 * r383402;
        double r383404 = y;
        double r383405 = r383404 / r383402;
        double r383406 = cbrt(r383402);
        double r383407 = r383396 / r383406;
        double r383408 = r383407 / r383406;
        double r383409 = r383406 * r383406;
        double r383410 = cbrt(r383409);
        double r383411 = r383408 / r383410;
        double r383412 = cbrt(r383406);
        double r383413 = r383397 / r383412;
        double r383414 = r383411 * r383413;
        double r383415 = r383405 - r383414;
        double r383416 = r383403 / r383415;
        double r383417 = 3.9822145511872104e+305;
        bool r383418 = r383398 <= r383417;
        double r383419 = r383404 - r383398;
        double r383420 = r383401 / r383419;
        double r383421 = cbrt(r383396);
        double r383422 = r383421 * r383421;
        double r383423 = r383422 / r383406;
        double r383424 = r383421 / r383406;
        double r383425 = r383397 / r383406;
        double r383426 = r383424 * r383425;
        double r383427 = r383423 * r383426;
        double r383428 = r383405 - r383427;
        double r383429 = r383403 / r383428;
        double r383430 = r383418 ? r383420 : r383429;
        double r383431 = r383400 ? r383416 : r383430;
        return r383431;
}

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

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

   12. Applied cbrt-prod 11.5

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

   13. Applied *-un-lft-identity 11.5

       \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \frac{z}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\color{blue}{1 \cdot t}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}}\]

   14. Applied times-frac 11.5

       \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \frac{z}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}} \cdot \frac{t}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}\right)}}\]

   15. Applied associate-*r* 11.5

       \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{y}{\sqrt[3]{x}} - \color{blue}{\left(\frac{z}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}\right) \cdot \frac{t}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}}}\]

   16. Simplified 11.5

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

   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

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

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))))