Average Error: 16.7 → 14.9
Time: 6.8s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.1627483557735157 \cdot 10^{-52} \lor \neg \left(b \le 1502069550.1706839\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;b \le -6.1627483557735157 \cdot 10^{-52} \lor \neg \left(b \le 1502069550.1706839\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r730403 = x;
        double r730404 = y;
        double r730405 = z;
        double r730406 = r730404 * r730405;
        double r730407 = t;
        double r730408 = r730406 / r730407;
        double r730409 = r730403 + r730408;
        double r730410 = a;
        double r730411 = 1.0;
        double r730412 = r730410 + r730411;
        double r730413 = b;
        double r730414 = r730404 * r730413;
        double r730415 = r730414 / r730407;
        double r730416 = r730412 + r730415;
        double r730417 = r730409 / r730416;
        return r730417;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r730418 = b;
        double r730419 = -6.162748355773516e-52;
        bool r730420 = r730418 <= r730419;
        double r730421 = 1502069550.1706839;
        bool r730422 = r730418 <= r730421;
        double r730423 = !r730422;
        bool r730424 = r730420 || r730423;
        double r730425 = y;
        double r730426 = t;
        double r730427 = r730425 / r730426;
        double r730428 = z;
        double r730429 = x;
        double r730430 = fma(r730427, r730428, r730429);
        double r730431 = 1.0;
        double r730432 = a;
        double r730433 = 1.0;
        double r730434 = r730432 + r730433;
        double r730435 = fma(r730427, r730418, r730434);
        double r730436 = r730431 / r730435;
        double r730437 = r730430 * r730436;
        double r730438 = r730426 / r730428;
        double r730439 = r730425 / r730438;
        double r730440 = r730429 + r730439;
        double r730441 = r730425 * r730418;
        double r730442 = r730441 / r730426;
        double r730443 = r730434 + r730442;
        double r730444 = r730440 / r730443;
        double r730445 = r730424 ? r730437 : r730444;
        return r730445;
}

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

Original16.7
Target13.8
Herbie14.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -6.162748355773516e-52 or 1502069550.1706839 < b

    1. Initial program 21.8

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

    3. Applied div-inv21.9

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

    4. Simplified19.0

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

    6. Applied *-un-lft-identity19.0

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

    7. Applied *-un-lft-identity19.0

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

    8. Applied times-frac19.0

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

    9. Applied associate-*r*19.0

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

    10. Simplified18.5

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

    if -6.162748355773516e-52 < b < 1502069550.1706839

    1. Initial program 10.7

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

    3. Applied associate-/l*10.6

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

  4. Final simplification14.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.1627483557735157 \cdot 10^{-52} \lor \neg \left(b \le 1502069550.1706839\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))