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

Average Error: 16.2 → 13.3

Time: 8.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.3581812405522004 \cdot 10^{-131} \lor \neg \left(y \le 7.59364648446620877 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r614306 = x;
        double r614307 = y;
        double r614308 = z;
        double r614309 = r614307 * r614308;
        double r614310 = t;
        double r614311 = r614309 / r614310;
        double r614312 = r614306 + r614311;
        double r614313 = a;
        double r614314 = 1.0;
        double r614315 = r614313 + r614314;
        double r614316 = b;
        double r614317 = r614307 * r614316;
        double r614318 = r614317 / r614310;
        double r614319 = r614315 + r614318;
        double r614320 = r614312 / r614319;
        return r614320;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r614321 = y;
        double r614322 = -6.3581812405522e-131;
        bool r614323 = r614321 <= r614322;
        double r614324 = 7.593646484466209e-92;
        bool r614325 = r614321 <= r614324;
        double r614326 = !r614325;
        bool r614327 = r614323 || r614326;
        double r614328 = x;
        double r614329 = z;
        double r614330 = t;
        double r614331 = r614329 / r614330;
        double r614332 = r614321 * r614331;
        double r614333 = r614328 + r614332;
        double r614334 = a;
        double r614335 = 1.0;
        double r614336 = r614334 + r614335;
        double r614337 = b;
        double r614338 = r614337 / r614330;
        double r614339 = r614321 * r614338;
        double r614340 = r614336 + r614339;
        double r614341 = r614333 / r614340;
        double r614342 = r614321 * r614329;
        double r614343 = r614342 / r614330;
        double r614344 = r614328 + r614343;
        double r614345 = r614321 * r614337;
        double r614346 = cbrt(r614330);
        double r614347 = r614346 * r614346;
        double r614348 = r614345 / r614347;
        double r614349 = r614348 / r614346;
        double r614350 = r614336 + r614349;
        double r614351 = r614344 / r614350;
        double r614352 = r614327 ? r614341 : r614351;
        return r614352;
}

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	16.2
Target	13.2
Herbie	13.3

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

Derivation

1. Split input into 2 regimes

2. if y < -6.3581812405522e-131 or 7.593646484466209e-92 < y

   1. Initial program 23.2

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

   3. Applied *-un-lft-identity 23.2

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

   4. Applied times-frac 21.7

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

   5. Simplified 21.7

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

   7. Applied *-un-lft-identity 21.7

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

   8. Applied times-frac 18.8

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

   9. Simplified 18.8

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

   if -6.3581812405522e-131 < y < 7.593646484466209e-92

   1. Initial program 2.1

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

   3. Applied add-cube-cbrt 2.2

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

   4. Applied associate-/r* 2.2

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

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.3581812405522004 \cdot 10^{-131} \lor \neg \left(y \le 7.59364648446620877 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))