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

Average Error: 16.2 → 13.3

Time: 8.9s

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 r268339 = x;
        double r268340 = y;
        double r268341 = z;
        double r268342 = r268340 * r268341;
        double r268343 = t;
        double r268344 = r268342 / r268343;
        double r268345 = r268339 + r268344;
        double r268346 = a;
        double r268347 = 1.0;
        double r268348 = r268346 + r268347;
        double r268349 = b;
        double r268350 = r268340 * r268349;
        double r268351 = r268350 / r268343;
        double r268352 = r268348 + r268351;
        double r268353 = r268345 / r268352;
        return r268353;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r268354 = y;
        double r268355 = -6.3581812405522e-131;
        bool r268356 = r268354 <= r268355;
        double r268357 = 7.593646484466209e-92;
        bool r268358 = r268354 <= r268357;
        double r268359 = !r268358;
        bool r268360 = r268356 || r268359;
        double r268361 = x;
        double r268362 = z;
        double r268363 = t;
        double r268364 = r268362 / r268363;
        double r268365 = r268354 * r268364;
        double r268366 = r268361 + r268365;
        double r268367 = a;
        double r268368 = 1.0;
        double r268369 = r268367 + r268368;
        double r268370 = b;
        double r268371 = r268370 / r268363;
        double r268372 = r268354 * r268371;
        double r268373 = r268369 + r268372;
        double r268374 = r268366 / r268373;
        double r268375 = r268354 * r268362;
        double r268376 = r268375 / r268363;
        double r268377 = r268361 + r268376;
        double r268378 = r268354 * r268370;
        double r268379 = cbrt(r268363);
        double r268380 = r268379 * r268379;
        double r268381 = r268378 / r268380;
        double r268382 = r268381 / r268379;
        double r268383 = r268369 + r268382;
        double r268384 = r268377 / r268383;
        double r268385 = r268360 ? r268374 : r268384;
        return r268385;
}

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))))