Average Error: 17.0 → 13.5
Time: 17.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.352849990458678853678224640581509873806 \cdot 10^{-21} \lor \neg \left(t \le 7.053224262423602044403334689875017003314 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.352849990458678853678224640581509873806 \cdot 10^{-21} \lor \neg \left(t \le 7.053224262423602044403334689875017003314 \cdot 10^{-122}\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 r460483 = x;
        double r460484 = y;
        double r460485 = z;
        double r460486 = r460484 * r460485;
        double r460487 = t;
        double r460488 = r460486 / r460487;
        double r460489 = r460483 + r460488;
        double r460490 = a;
        double r460491 = 1.0;
        double r460492 = r460490 + r460491;
        double r460493 = b;
        double r460494 = r460484 * r460493;
        double r460495 = r460494 / r460487;
        double r460496 = r460492 + r460495;
        double r460497 = r460489 / r460496;
        return r460497;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r460498 = t;
        double r460499 = -3.352849990458679e-21;
        bool r460500 = r460498 <= r460499;
        double r460501 = 7.053224262423602e-122;
        bool r460502 = r460498 <= r460501;
        double r460503 = !r460502;
        bool r460504 = r460500 || r460503;
        double r460505 = y;
        double r460506 = r460505 / r460498;
        double r460507 = z;
        double r460508 = x;
        double r460509 = fma(r460506, r460507, r460508);
        double r460510 = b;
        double r460511 = a;
        double r460512 = fma(r460506, r460510, r460511);
        double r460513 = 1.0;
        double r460514 = r460512 + r460513;
        double r460515 = r460509 / r460514;
        double r460516 = r460505 * r460507;
        double r460517 = r460516 / r460498;
        double r460518 = r460508 + r460517;
        double r460519 = r460511 + r460513;
        double r460520 = r460505 * r460510;
        double r460521 = r460520 / r460498;
        double r460522 = r460519 + r460521;
        double r460523 = r460518 / r460522;
        double r460524 = r460504 ? r460515 : r460523;
        return r460524;
}

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

Original17.0
Target13.6
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.352849990458679e-21 or 7.053224262423602e-122 < t

    1. Initial program 12.0

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

    2. Simplified6.3

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

    4. Applied div-inv6.4

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

    6. Applied *-un-lft-identity6.4

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

    7. Applied associate-*l*6.4

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

    8. Simplified6.3

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

    if -3.352849990458679e-21 < t < 7.053224262423602e-122

    1. Initial program 25.5

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.352849990458678853678224640581509873806 \cdot 10^{-21} \lor \neg \left(t \le 7.053224262423602044403334689875017003314 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))