Average Error: 16.7 → 13.0
Time: 5.6s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.6145555044474293 \cdot 10^{-55}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le 5.30891579653489513 \cdot 10^{72}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.6145555044474293 \cdot 10^{-55}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{elif}\;t \le 5.30891579653489513 \cdot 10^{72}:\\
\;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r697495 = x;
        double r697496 = y;
        double r697497 = z;
        double r697498 = r697496 * r697497;
        double r697499 = t;
        double r697500 = r697498 / r697499;
        double r697501 = r697495 + r697500;
        double r697502 = a;
        double r697503 = 1.0;
        double r697504 = r697502 + r697503;
        double r697505 = b;
        double r697506 = r697496 * r697505;
        double r697507 = r697506 / r697499;
        double r697508 = r697504 + r697507;
        double r697509 = r697501 / r697508;
        return r697509;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r697510 = t;
        double r697511 = -2.6145555044474293e-55;
        bool r697512 = r697510 <= r697511;
        double r697513 = x;
        double r697514 = y;
        double r697515 = z;
        double r697516 = r697510 / r697515;
        double r697517 = r697514 / r697516;
        double r697518 = r697513 + r697517;
        double r697519 = a;
        double r697520 = 1.0;
        double r697521 = r697519 + r697520;
        double r697522 = b;
        double r697523 = r697522 / r697510;
        double r697524 = r697514 * r697523;
        double r697525 = r697521 + r697524;
        double r697526 = r697518 / r697525;
        double r697527 = 5.308915796534895e+72;
        bool r697528 = r697510 <= r697527;
        double r697529 = r697514 * r697515;
        double r697530 = 1.0;
        double r697531 = r697530 / r697510;
        double r697532 = r697529 * r697531;
        double r697533 = r697513 + r697532;
        double r697534 = r697514 * r697522;
        double r697535 = r697534 / r697510;
        double r697536 = r697521 + r697535;
        double r697537 = r697533 / r697536;
        double r697538 = r697514 / r697510;
        double r697539 = fma(r697538, r697515, r697513);
        double r697540 = r697530 * r697539;
        double r697541 = r697540 / r697525;
        double r697542 = r697528 ? r697537 : r697541;
        double r697543 = r697512 ? r697526 : r697542;
        return r697543;
}

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
Target13.4
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -2.6145555044474293e-55

    1. Initial program 12.2

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

    3. Applied *-un-lft-identity12.2

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

    4. Applied times-frac9.8

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

    5. Simplified9.8

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

    7. Applied associate-/l*5.6

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

    if -2.6145555044474293e-55 < t < 5.308915796534895e+72

    1. Initial program 21.0

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

    3. Applied div-inv21.0

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

    if 5.308915796534895e+72 < t

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied times-frac8.4

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

    5. Simplified8.4

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

    7. Applied associate-/l*3.0

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

    9. Applied *-un-lft-identity3.0

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

    10. Applied *-un-lft-identity3.0

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

    11. Applied distribute-lft-out3.0

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

    12. Simplified3.0

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.6145555044474293 \cdot 10^{-55}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le 5.30891579653489513 \cdot 10^{72}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))))