Average Error: 16.0 → 12.6
Time: 1.5m
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le -1.726406946855666 \cdot 10^{+238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le -1.7605438012672034 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le 1.3135635701590148 \cdot 10^{-257}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le 7.061941886181377 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le -1.726406946855666 \cdot 10^{+238}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\

\mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le -1.7605438012672034 \cdot 10^{-260}:\\
\;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)}\\

\mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le 1.3135635701590148 \cdot 10^{-257}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\

\mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le 7.061941886181377 \cdot 10^{+145}:\\
\;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r29869505 = x;
        double r29869506 = y;
        double r29869507 = z;
        double r29869508 = r29869506 * r29869507;
        double r29869509 = t;
        double r29869510 = r29869508 / r29869509;
        double r29869511 = r29869505 + r29869510;
        double r29869512 = a;
        double r29869513 = 1.0;
        double r29869514 = r29869512 + r29869513;
        double r29869515 = b;
        double r29869516 = r29869506 * r29869515;
        double r29869517 = r29869516 / r29869509;
        double r29869518 = r29869514 + r29869517;
        double r29869519 = r29869511 / r29869518;
        return r29869519;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r29869520 = z;
        double r29869521 = y;
        double r29869522 = r29869520 * r29869521;
        double r29869523 = t;
        double r29869524 = r29869522 / r29869523;
        double r29869525 = x;
        double r29869526 = r29869524 + r29869525;
        double r29869527 = b;
        double r29869528 = r29869527 * r29869521;
        double r29869529 = r29869528 / r29869523;
        double r29869530 = a;
        double r29869531 = 1.0;
        double r29869532 = r29869530 + r29869531;
        double r29869533 = r29869529 + r29869532;
        double r29869534 = r29869526 / r29869533;
        double r29869535 = -1.726406946855666e+238;
        bool r29869536 = r29869534 <= r29869535;
        double r29869537 = r29869520 / r29869523;
        double r29869538 = fma(r29869537, r29869521, r29869525);
        double r29869539 = r29869521 / r29869523;
        double r29869540 = fma(r29869539, r29869527, r29869532);
        double r29869541 = r29869538 / r29869540;
        double r29869542 = -1.7605438012672034e-260;
        bool r29869543 = r29869534 <= r29869542;
        double r29869544 = 1.3135635701590148e-257;
        bool r29869545 = r29869534 <= r29869544;
        double r29869546 = 7.061941886181377e+145;
        bool r29869547 = r29869534 <= r29869546;
        double r29869548 = r29869547 ? r29869534 : r29869541;
        double r29869549 = r29869545 ? r29869541 : r29869548;
        double r29869550 = r29869543 ? r29869534 : r29869549;
        double r29869551 = r29869536 ? r29869541 : r29869550;
        return r29869551;
}

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

Original16.0
Target13.2
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -1.726406946855666e+238 or -1.7605438012672034e-260 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 1.3135635701590148e-257 or 7.061941886181377e+145 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

    1. Initial program 34.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

    2. Simplified27.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}}\]
    3. Using strategy rm

    4. Applied div-inv27.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}}\]
    5. Using strategy rm

    6. Applied associate-*r/27.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot 1}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}}\]

    7. Simplified27.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\]

    if -1.726406946855666e+238 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -1.7605438012672034e-260 or 1.3135635701590148e-257 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 7.061941886181377e+145

    1. Initial program 0.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le -1.726406946855666 \cdot 10^{+238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le -1.7605438012672034 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le 1.3135635701590148 \cdot 10^{-257}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)} \le 7.061941886181377 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))