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

Average Error: 16.1 → 13.1

Time: 15.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 7.560625940505470397372999874739945211814 \cdot 10^{-286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 4.353154786484829134540667406486672764698 \cdot 10^{250}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r784284 = x;
        double r784285 = y;
        double r784286 = z;
        double r784287 = r784285 * r784286;
        double r784288 = t;
        double r784289 = r784287 / r784288;
        double r784290 = r784284 + r784289;
        double r784291 = a;
        double r784292 = 1.0;
        double r784293 = r784291 + r784292;
        double r784294 = b;
        double r784295 = r784285 * r784294;
        double r784296 = r784295 / r784288;
        double r784297 = r784293 + r784296;
        double r784298 = r784290 / r784297;
        return r784298;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r784299 = x;
        double r784300 = y;
        double r784301 = z;
        double r784302 = r784300 * r784301;
        double r784303 = t;
        double r784304 = r784302 / r784303;
        double r784305 = r784299 + r784304;
        double r784306 = a;
        double r784307 = 1.0;
        double r784308 = r784306 + r784307;
        double r784309 = b;
        double r784310 = r784300 * r784309;
        double r784311 = r784310 / r784303;
        double r784312 = r784308 + r784311;
        double r784313 = r784305 / r784312;
        double r784314 = 7.56062594050547e-286;
        bool r784315 = r784313 <= r784314;
        double r784316 = r784300 / r784303;
        double r784317 = fma(r784316, r784301, r784299);
        double r784318 = fma(r784316, r784309, r784306);
        double r784319 = r784318 + r784307;
        double r784320 = r784317 / r784319;
        double r784321 = 4.353154786484829e+250;
        bool r784322 = r784313 <= r784321;
        double r784323 = 1.0;
        double r784324 = r784323 / r784319;
        double r784325 = r784317 * r784324;
        double r784326 = r784322 ? r784313 : r784325;
        double r784327 = r784315 ? r784320 : r784326;
        return r784327;
}

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	16.1
Target	12.8
Herbie	13.1

\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 7.56062594050547e-286

   1. Initial program 13.8

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

   2. Simplified 11.1

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

   if 7.56062594050547e-286 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 4.353154786484829e+250

   1. Initial program 0.5

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

   if 4.353154786484829e+250 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

   1. Initial program 57.3

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

   2. Simplified 47.4

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

   4. Applied div-inv 47.4

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 7.560625940505470397372999874739945211814 \cdot 10^{-286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 4.353154786484829134540667406486672764698 \cdot 10^{250}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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))))