Average Error: 16.5 → 13.3
Time: 7.2s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.26604793545714451 \cdot 10^{90} \lor \neg \left(y \le 2.039473926406 \cdot 10^{-197}\right):\\ \;\;\;\;\left(x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.26604793545714451 \cdot 10^{90} \lor \neg \left(y \le 2.039473926406 \cdot 10^{-197}\right):\\
\;\;\;\;\left(x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\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 r817327 = x;
        double r817328 = y;
        double r817329 = z;
        double r817330 = r817328 * r817329;
        double r817331 = t;
        double r817332 = r817330 / r817331;
        double r817333 = r817327 + r817332;
        double r817334 = a;
        double r817335 = 1.0;
        double r817336 = r817334 + r817335;
        double r817337 = b;
        double r817338 = r817328 * r817337;
        double r817339 = r817338 / r817331;
        double r817340 = r817336 + r817339;
        double r817341 = r817333 / r817340;
        return r817341;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r817342 = y;
        double r817343 = -1.2660479354571445e+90;
        bool r817344 = r817342 <= r817343;
        double r817345 = 2.039473926405995e-197;
        bool r817346 = r817342 <= r817345;
        double r817347 = !r817346;
        bool r817348 = r817344 || r817347;
        double r817349 = x;
        double r817350 = cbrt(r817342);
        double r817351 = r817350 * r817350;
        double r817352 = t;
        double r817353 = cbrt(r817352);
        double r817354 = r817351 / r817353;
        double r817355 = r817350 / r817353;
        double r817356 = z;
        double r817357 = r817356 / r817353;
        double r817358 = r817355 * r817357;
        double r817359 = r817354 * r817358;
        double r817360 = r817349 + r817359;
        double r817361 = 1.0;
        double r817362 = a;
        double r817363 = 1.0;
        double r817364 = r817362 + r817363;
        double r817365 = b;
        double r817366 = r817365 / r817352;
        double r817367 = r817342 * r817366;
        double r817368 = r817364 + r817367;
        double r817369 = r817361 / r817368;
        double r817370 = r817360 * r817369;
        double r817371 = r817342 / r817352;
        double r817372 = fma(r817371, r817356, r817349);
        double r817373 = r817342 * r817365;
        double r817374 = r817373 / r817352;
        double r817375 = r817364 + r817374;
        double r817376 = r817372 / r817375;
        double r817377 = r817348 ? r817370 : r817376;
        return r817377;
}

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

Original16.5
Target13.3
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.2660479354571445e+90 or 2.039473926405995e-197 < y

    1. Initial program 24.3

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

    3. Applied add-cube-cbrt24.5

      \[\leadsto \frac{x + \frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

    4. Applied times-frac23.4

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

    6. Applied *-un-lft-identity23.4

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{\color{blue}{1 \cdot t}}}\]

    7. Applied times-frac20.6

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \color{blue}{\frac{y}{1} \cdot \frac{b}{t}}}\]

    8. Simplified20.6

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

    10. Applied div-inv20.6

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

    12. Applied add-cube-cbrt20.7

      \[\leadsto \left(x + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\]

    13. Applied times-frac20.7

      \[\leadsto \left(x + \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right)} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\]

    14. Applied associate-*l*18.9

      \[\leadsto \left(x + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\]

    if -1.2660479354571445e+90 < y < 2.039473926405995e-197

    1. Initial program 5.7

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

    3. Applied *-un-lft-identity5.7

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

    4. Applied associate-/r*5.7

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

    5. Simplified5.5

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.26604793545714451 \cdot 10^{90} \lor \neg \left(y \le 2.039473926406 \cdot 10^{-197}\right):\\ \;\;\;\;\left(x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))))