Average Error: 16.5 → 12.8
Time: 14.0s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -11426198898.5786876678466796875:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;y \le 6.399239486311505375772837011137183856531 \cdot 10^{-56}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y}{t}}{\frac{1}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -11426198898.5786876678466796875:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{elif}\;y \le 6.399239486311505375772837011137183856531 \cdot 10^{-56}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y}{t}}{\frac{1}{b}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r479465 = x;
        double r479466 = y;
        double r479467 = z;
        double r479468 = r479466 * r479467;
        double r479469 = t;
        double r479470 = r479468 / r479469;
        double r479471 = r479465 + r479470;
        double r479472 = a;
        double r479473 = 1.0;
        double r479474 = r479472 + r479473;
        double r479475 = b;
        double r479476 = r479466 * r479475;
        double r479477 = r479476 / r479469;
        double r479478 = r479474 + r479477;
        double r479479 = r479471 / r479478;
        return r479479;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r479480 = y;
        double r479481 = -11426198898.578688;
        bool r479482 = r479480 <= r479481;
        double r479483 = x;
        double r479484 = z;
        double r479485 = t;
        double r479486 = r479484 / r479485;
        double r479487 = r479480 * r479486;
        double r479488 = r479483 + r479487;
        double r479489 = a;
        double r479490 = 1.0;
        double r479491 = r479489 + r479490;
        double r479492 = b;
        double r479493 = r479492 / r479485;
        double r479494 = r479480 * r479493;
        double r479495 = r479491 + r479494;
        double r479496 = r479488 / r479495;
        double r479497 = 6.399239486311505e-56;
        bool r479498 = r479480 <= r479497;
        double r479499 = r479480 * r479484;
        double r479500 = r479499 / r479485;
        double r479501 = r479483 + r479500;
        double r479502 = r479480 / r479485;
        double r479503 = 1.0;
        double r479504 = r479503 / r479492;
        double r479505 = r479502 / r479504;
        double r479506 = r479491 + r479505;
        double r479507 = r479501 / r479506;
        double r479508 = r479485 / r479484;
        double r479509 = r479480 / r479508;
        double r479510 = r479483 + r479509;
        double r479511 = r479485 / r479492;
        double r479512 = r479480 / r479511;
        double r479513 = r479491 + r479512;
        double r479514 = r479510 / r479513;
        double r479515 = r479498 ? r479507 : r479514;
        double r479516 = r479482 ? r479496 : r479515;
        return r479516;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.5
Target13.5
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -11426198898.578688

    1. Initial program 30.6

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

    3. Applied associate-/l*27.9

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

    5. Applied *-un-lft-identity27.9

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

    6. Applied times-frac23.2

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

    7. Simplified23.2

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

    9. Applied div-inv23.2

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

    10. Simplified23.2

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

    if -11426198898.578688 < y < 6.399239486311505e-56

    1. Initial program 3.7

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

    3. Applied associate-/l*8.0

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

    5. Applied div-inv8.0

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

    6. Applied associate-/r*3.7

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

    if 6.399239486311505e-56 < y

    1. Initial program 26.3

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

    3. Applied associate-/l*23.5

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

    5. Applied associate-/l*19.5

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -11426198898.5786876678466796875:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;y \le 6.399239486311505375772837011137183856531 \cdot 10^{-56}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y}{t}}{\frac{1}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))