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

Average Error: 16.4 → 12.8

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -462155825864.207 \lor \neg \left(t \le 4.0158354954516018 \cdot 10^{-7}\right):\\ \;\;\;\;\left(x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1543353 = x;
        double r1543354 = y;
        double r1543355 = z;
        double r1543356 = r1543354 * r1543355;
        double r1543357 = t;
        double r1543358 = r1543356 / r1543357;
        double r1543359 = r1543353 + r1543358;
        double r1543360 = a;
        double r1543361 = 1.0;
        double r1543362 = r1543360 + r1543361;
        double r1543363 = b;
        double r1543364 = r1543354 * r1543363;
        double r1543365 = r1543364 / r1543357;
        double r1543366 = r1543362 + r1543365;
        double r1543367 = r1543359 / r1543366;
        return r1543367;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1543368 = t;
        double r1543369 = -462155825864.20703;
        bool r1543370 = r1543368 <= r1543369;
        double r1543371 = 4.015835495451602e-07;
        bool r1543372 = r1543368 <= r1543371;
        double r1543373 = !r1543372;
        bool r1543374 = r1543370 || r1543373;
        double r1543375 = x;
        double r1543376 = y;
        double r1543377 = cbrt(r1543368);
        double r1543378 = r1543377 * r1543377;
        double r1543379 = r1543376 / r1543378;
        double r1543380 = z;
        double r1543381 = r1543380 / r1543377;
        double r1543382 = r1543379 * r1543381;
        double r1543383 = r1543375 + r1543382;
        double r1543384 = 1.0;
        double r1543385 = a;
        double r1543386 = 1.0;
        double r1543387 = r1543385 + r1543386;
        double r1543388 = b;
        double r1543389 = r1543368 / r1543388;
        double r1543390 = r1543376 / r1543389;
        double r1543391 = r1543387 + r1543390;
        double r1543392 = r1543384 / r1543391;
        double r1543393 = r1543383 * r1543392;
        double r1543394 = r1543376 * r1543388;
        double r1543395 = r1543394 / r1543368;
        double r1543396 = r1543387 + r1543395;
        double r1543397 = r1543376 * r1543380;
        double r1543398 = r1543397 / r1543368;
        double r1543399 = r1543375 + r1543398;
        double r1543400 = r1543396 / r1543399;
        double r1543401 = r1543384 / r1543400;
        double r1543402 = r1543374 ? r1543393 : r1543401;
        return r1543402;
}

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	16.4
Target	13.2
Herbie	12.8

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

Derivation

1. Split input into 2 regimes

2. if t < -462155825864.20703 or 4.015835495451602e-07 < t

   1. Initial program 11.2

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

   3. Applied add-cube-cbrt 11.3

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

   4. Applied times-frac 7.9

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

   6. Applied associate-/l* 4.1

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

   8. Applied div-inv 4.2

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

   if -462155825864.20703 < t < 4.015835495451602e-07

   1. Initial program 21.8

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

   3. Applied clear-num 22.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -462155825864.207 \lor \neg \left(t \le 4.0158354954516018 \cdot 10^{-7}\right):\\ \;\;\;\;\left(x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))