Average Error: 16.8 → 13.2
Time: 5.2s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.70915755118184234 \cdot 10^{32} \lor \neg \left(y \le 6.6772675351438104 \cdot 10^{75} \lor \neg \left(y \le 4.2862283636253177 \cdot 10^{272}\right)\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.70915755118184234 \cdot 10^{32} \lor \neg \left(y \le 6.6772675351438104 \cdot 10^{75} \lor \neg \left(y \le 4.2862283636253177 \cdot 10^{272}\right)\right):\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r662639 = x;
        double r662640 = y;
        double r662641 = z;
        double r662642 = r662640 * r662641;
        double r662643 = t;
        double r662644 = r662642 / r662643;
        double r662645 = r662639 + r662644;
        double r662646 = a;
        double r662647 = 1.0;
        double r662648 = r662646 + r662647;
        double r662649 = b;
        double r662650 = r662640 * r662649;
        double r662651 = r662650 / r662643;
        double r662652 = r662648 + r662651;
        double r662653 = r662645 / r662652;
        return r662653;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r662654 = y;
        double r662655 = -1.7091575511818423e+32;
        bool r662656 = r662654 <= r662655;
        double r662657 = 6.67726753514381e+75;
        bool r662658 = r662654 <= r662657;
        double r662659 = 4.286228363625318e+272;
        bool r662660 = r662654 <= r662659;
        double r662661 = !r662660;
        bool r662662 = r662658 || r662661;
        double r662663 = !r662662;
        bool r662664 = r662656 || r662663;
        double r662665 = x;
        double r662666 = z;
        double r662667 = t;
        double r662668 = r662666 / r662667;
        double r662669 = r662654 * r662668;
        double r662670 = r662665 + r662669;
        double r662671 = a;
        double r662672 = 1.0;
        double r662673 = r662671 + r662672;
        double r662674 = b;
        double r662675 = r662667 / r662674;
        double r662676 = r662654 / r662675;
        double r662677 = r662673 + r662676;
        double r662678 = r662670 / r662677;
        double r662679 = 1.0;
        double r662680 = r662654 / r662667;
        double r662681 = fma(r662680, r662674, r662673);
        double r662682 = fma(r662680, r662666, r662665);
        double r662683 = r662681 / r662682;
        double r662684 = r662679 / r662683;
        double r662685 = r662664 ? r662678 : r662684;
        return r662685;
}

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
Target13.6
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.7091575511818423e+32 or 6.67726753514381e+75 < y < 4.286228363625318e+272

    1. Initial program 32.0

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

    3. Applied associate-/l*28.7

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

    5. Applied *-un-lft-identity28.7

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

    6. Applied times-frac23.0

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

    7. Simplified23.0

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

    if -1.7091575511818423e+32 < y < 6.67726753514381e+75 or 4.286228363625318e+272 < y

    1. Initial program 7.5

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

    3. Applied associate-/l*10.1

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

    5. Applied *-un-lft-identity10.1

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

    6. Applied *-un-lft-identity10.1

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

    7. Applied times-frac10.1

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

    8. Simplified10.1

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

    9. Simplified7.3

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

    11. Applied clear-num7.6

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

    12. Simplified7.2

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.70915755118184234 \cdot 10^{32} \lor \neg \left(y \le 6.6772675351438104 \cdot 10^{75} \lor \neg \left(y \le 4.2862283636253177 \cdot 10^{272}\right)\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))