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

Average Error: 16.6 → 12.5

Time: 5.8s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} = -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + {\left(\frac{y \cdot b}{t}\right)}^{1}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -4.28609946082964957 \cdot 10^{-261}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y \cdot z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -0.0:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{y \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 7.68622449138201355 \cdot 10^{181}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{y \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r912315 = x;
        double r912316 = y;
        double r912317 = z;
        double r912318 = r912316 * r912317;
        double r912319 = t;
        double r912320 = r912318 / r912319;
        double r912321 = r912315 + r912320;
        double r912322 = a;
        double r912323 = 1.0;
        double r912324 = r912322 + r912323;
        double r912325 = b;
        double r912326 = r912316 * r912325;
        double r912327 = r912326 / r912319;
        double r912328 = r912324 + r912327;
        double r912329 = r912321 / r912328;
        return r912329;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r912330 = x;
        double r912331 = y;
        double r912332 = z;
        double r912333 = r912331 * r912332;
        double r912334 = t;
        double r912335 = r912333 / r912334;
        double r912336 = r912330 + r912335;
        double r912337 = a;
        double r912338 = 1.0;
        double r912339 = r912337 + r912338;
        double r912340 = b;
        double r912341 = r912331 * r912340;
        double r912342 = r912341 / r912334;
        double r912343 = r912339 + r912342;
        double r912344 = r912336 / r912343;
        double r912345 = -inf.0;
        bool r912346 = r912344 <= r912345;
        double r912347 = r912332 / r912334;
        double r912348 = r912331 * r912347;
        double r912349 = r912330 + r912348;
        double r912350 = 1.0;
        double r912351 = pow(r912342, r912350);
        double r912352 = r912339 + r912351;
        double r912353 = r912349 / r912352;
        double r912354 = -4.28609946082965e-261;
        bool r912355 = r912344 <= r912354;
        double r912356 = r912334 / r912333;
        double r912357 = r912350 / r912356;
        double r912358 = r912330 + r912357;
        double r912359 = r912358 / r912343;
        double r912360 = -0.0;
        bool r912361 = r912344 <= r912360;
        double r912362 = cbrt(r912348);
        double r912363 = r912362 * r912362;
        double r912364 = r912363 * r912362;
        double r912365 = r912330 + r912364;
        double r912366 = cbrt(r912334);
        double r912367 = r912366 * r912366;
        double r912368 = r912331 / r912367;
        double r912369 = r912340 / r912366;
        double r912370 = r912368 * r912369;
        double r912371 = r912339 + r912370;
        double r912372 = r912365 / r912371;
        double r912373 = 7.686224491382014e+181;
        bool r912374 = r912344 <= r912373;
        double r912375 = r912334 / r912341;
        double r912376 = r912350 / r912375;
        double r912377 = r912339 + r912376;
        double r912378 = r912336 / r912377;
        double r912379 = r912374 ? r912378 : r912372;
        double r912380 = r912361 ? r912372 : r912379;
        double r912381 = r912355 ? r912359 : r912380;
        double r912382 = r912346 ? r912353 : r912381;
        return r912382;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	16.6
Target	13.0
Herbie	12.5

\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -inf.0

   1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 64.0

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

   4. Applied times-frac 64.0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 64.0

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

   7. Applied times-frac 37.9

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

   8. Simplified 37.9

      \[\leadsto \frac{x + \color{blue}{y} \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]
   9. Using strategy rm

   10. Applied pow1 37.9

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

   11. Applied pow1 37.9

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

   12. Applied pow-prod-down 37.9

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

   13. Simplified 37.9

       \[\leadsto \frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + {\color{blue}{\left(\frac{y \cdot b}{t}\right)}}^{1}}\]

   if -inf.0 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -4.28609946082965e-261

   1. Initial program 0.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied clear-num 0.4

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t}{y \cdot z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

   if -4.28609946082965e-261 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -0.0 or 7.686224491382014e+181 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

   1. Initial program 38.2

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 38.2

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

   4. Applied times-frac 33.7

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 33.7

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

   7. Applied times-frac 29.6

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

   8. Simplified 29.6

      \[\leadsto \frac{x + \color{blue}{y} \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 29.7

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

   if -0.0 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 7.686224491382014e+181

   1. Initial program 0.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied clear-num 0.3

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{1}{\frac{t}{y \cdot b}}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} = -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + {\left(\frac{y \cdot b}{t}\right)}^{1}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -4.28609946082964957 \cdot 10^{-261}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y \cdot z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -0.0:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{y \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 7.68622449138201355 \cdot 10^{181}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{y \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))