Average Error: 16.8 → 13.2
Time: 5.3s
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 r757689 = x;
        double r757690 = y;
        double r757691 = z;
        double r757692 = r757690 * r757691;
        double r757693 = t;
        double r757694 = r757692 / r757693;
        double r757695 = r757689 + r757694;
        double r757696 = a;
        double r757697 = 1.0;
        double r757698 = r757696 + r757697;
        double r757699 = b;
        double r757700 = r757690 * r757699;
        double r757701 = r757700 / r757693;
        double r757702 = r757698 + r757701;
        double r757703 = r757695 / r757702;
        return r757703;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r757704 = y;
        double r757705 = -1.7091575511818423e+32;
        bool r757706 = r757704 <= r757705;
        double r757707 = 6.67726753514381e+75;
        bool r757708 = r757704 <= r757707;
        double r757709 = 4.286228363625318e+272;
        bool r757710 = r757704 <= r757709;
        double r757711 = !r757710;
        bool r757712 = r757708 || r757711;
        double r757713 = !r757712;
        bool r757714 = r757706 || r757713;
        double r757715 = x;
        double r757716 = z;
        double r757717 = t;
        double r757718 = r757716 / r757717;
        double r757719 = r757704 * r757718;
        double r757720 = r757715 + r757719;
        double r757721 = a;
        double r757722 = 1.0;
        double r757723 = r757721 + r757722;
        double r757724 = b;
        double r757725 = r757717 / r757724;
        double r757726 = r757704 / r757725;
        double r757727 = r757723 + r757726;
        double r757728 = r757720 / r757727;
        double r757729 = 1.0;
        double r757730 = r757704 / r757717;
        double r757731 = fma(r757730, r757724, r757723);
        double r757732 = fma(r757730, r757716, r757715);
        double r757733 = r757731 / r757732;
        double r757734 = r757729 / r757733;
        double r757735 = r757714 ? r757728 : r757734;
        return r757735;
}

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))))