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

Average Error: 16.1 → 12.5

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -7.913399165461686155144823271089923942095 \cdot 10^{-52} \lor \neg \left(t \le 7.626406460332817234650012490007590154937 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r704526 = x;
        double r704527 = y;
        double r704528 = z;
        double r704529 = r704527 * r704528;
        double r704530 = t;
        double r704531 = r704529 / r704530;
        double r704532 = r704526 + r704531;
        double r704533 = a;
        double r704534 = 1.0;
        double r704535 = r704533 + r704534;
        double r704536 = b;
        double r704537 = r704527 * r704536;
        double r704538 = r704537 / r704530;
        double r704539 = r704535 + r704538;
        double r704540 = r704532 / r704539;
        return r704540;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r704541 = t;
        double r704542 = -7.913399165461686e-52;
        bool r704543 = r704541 <= r704542;
        double r704544 = 7.626406460332817e-81;
        bool r704545 = r704541 <= r704544;
        double r704546 = !r704545;
        bool r704547 = r704543 || r704546;
        double r704548 = x;
        double r704549 = y;
        double r704550 = z;
        double r704551 = r704550 / r704541;
        double r704552 = r704549 * r704551;
        double r704553 = r704548 + r704552;
        double r704554 = a;
        double r704555 = 1.0;
        double r704556 = r704554 + r704555;
        double r704557 = b;
        double r704558 = r704557 / r704541;
        double r704559 = r704549 * r704558;
        double r704560 = r704556 + r704559;
        double r704561 = r704553 / r704560;
        double r704562 = r704549 * r704550;
        double r704563 = 1.0;
        double r704564 = r704563 / r704541;
        double r704565 = r704562 * r704564;
        double r704566 = r704548 + r704565;
        double r704567 = r704549 * r704557;
        double r704568 = r704567 / r704541;
        double r704569 = r704556 + r704568;
        double r704570 = r704566 / r704569;
        double r704571 = r704547 ? r704561 : r704570;
        return r704571;
}

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	12.9
Herbie	12.5

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

Derivation

1. Split input into 2 regimes

2. if t < -7.913399165461686e-52 or 7.626406460332817e-81 < t

   1. Initial program 11.1

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

   3. Applied *-un-lft-identity 11.1

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

   4. Applied times-frac 8.1

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

   5. Simplified 8.1

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

   7. Applied *-un-lft-identity 8.1

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

   8. Applied times-frac 5.2

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

   9. Simplified 5.2

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

   if -7.913399165461686e-52 < t < 7.626406460332817e-81

   1. Initial program 24.4

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

   3. Applied div-inv 24.4

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.913399165461686155144823271089923942095 \cdot 10^{-52} \lor \neg \left(t \le 7.626406460332817234650012490007590154937 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))