Average Error: 16.8 → 13.1
Time: 12.0s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.8857581380493086 \cdot 10^{-59} \lor \neg \left(t \le 7.85936216888135273 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.8857581380493086 \cdot 10^{-59} \lor \neg \left(t \le 7.85936216888135273 \cdot 10^{-21}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\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 r473473 = x;
        double r473474 = y;
        double r473475 = z;
        double r473476 = r473474 * r473475;
        double r473477 = t;
        double r473478 = r473476 / r473477;
        double r473479 = r473473 + r473478;
        double r473480 = a;
        double r473481 = 1.0;
        double r473482 = r473480 + r473481;
        double r473483 = b;
        double r473484 = r473474 * r473483;
        double r473485 = r473484 / r473477;
        double r473486 = r473482 + r473485;
        double r473487 = r473479 / r473486;
        return r473487;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r473488 = t;
        double r473489 = -7.885758138049309e-59;
        bool r473490 = r473488 <= r473489;
        double r473491 = 7.859362168881353e-21;
        bool r473492 = r473488 <= r473491;
        double r473493 = !r473492;
        bool r473494 = r473490 || r473493;
        double r473495 = y;
        double r473496 = r473495 / r473488;
        double r473497 = z;
        double r473498 = x;
        double r473499 = fma(r473496, r473497, r473498);
        double r473500 = b;
        double r473501 = a;
        double r473502 = fma(r473496, r473500, r473501);
        double r473503 = 1.0;
        double r473504 = r473502 + r473503;
        double r473505 = r473499 / r473504;
        double r473506 = r473495 * r473497;
        double r473507 = r473506 / r473488;
        double r473508 = r473498 + r473507;
        double r473509 = r473501 + r473503;
        double r473510 = r473495 * r473500;
        double r473511 = r473510 / r473488;
        double r473512 = r473509 + r473511;
        double r473513 = r473508 / r473512;
        double r473514 = r473494 ? r473505 : r473513;
        return r473514;
}

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

Original16.8
Target13.3
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.885758138049309e-59 or 7.859362168881353e-21 < t

    1. Initial program 11.6

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

    2. Simplified5.1

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

    4. Applied clear-num5.5

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

    6. Applied *-un-lft-identity5.5

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

    7. Applied *-un-lft-identity5.5

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

    8. Applied times-frac5.5

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

    9. Applied add-cube-cbrt5.5

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

    10. Applied times-frac5.5

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

    11. Simplified5.5

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

    12. Simplified5.1

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

    if -7.885758138049309e-59 < t < 7.859362168881353e-21

    1. Initial program 23.9

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.8857581380493086 \cdot 10^{-59} \lor \neg \left(t \le 7.85936216888135273 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))))