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

Average Error: 2.7 → 1.8

Time: 14.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r430211 = x;
        double r430212 = y;
        double r430213 = z;
        double r430214 = t;
        double r430215 = r430213 * r430214;
        double r430216 = r430212 - r430215;
        double r430217 = r430211 / r430216;
        return r430217;
}

↓

double f(double x, double y, double z, double t) {
        double r430218 = z;
        double r430219 = t;
        double r430220 = r430218 * r430219;
        double r430221 = -inf.0;
        bool r430222 = r430220 <= r430221;
        double r430223 = x;
        double r430224 = cbrt(r430223);
        double r430225 = r430224 * r430224;
        double r430226 = y;
        double r430227 = r430226 / r430224;
        double r430228 = cbrt(r430224);
        double r430229 = r430218 / r430228;
        double r430230 = r430229 / r430228;
        double r430231 = r430228 * r430228;
        double r430232 = cbrt(r430231);
        double r430233 = r430230 / r430232;
        double r430234 = cbrt(r430228);
        double r430235 = r430219 / r430234;
        double r430236 = r430233 * r430235;
        double r430237 = r430227 - r430236;
        double r430238 = r430225 / r430237;
        double r430239 = 3.9822145511872104e+305;
        bool r430240 = r430220 <= r430239;
        double r430241 = r430226 - r430220;
        double r430242 = r430223 / r430241;
        double r430243 = cbrt(r430218);
        double r430244 = r430243 * r430243;
        double r430245 = r430244 / r430228;
        double r430246 = r430243 / r430228;
        double r430247 = r430219 / r430228;
        double r430248 = r430246 * r430247;
        double r430249 = r430245 * r430248;
        double r430250 = r430227 - r430249;
        double r430251 = r430225 / r430250;
        double r430252 = r430240 ? r430242 : r430251;
        double r430253 = r430222 ? r430238 : r430252;
        return r430253;
}

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 
(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))))