Average Error: 16.7 → 13.0
Time: 5.5s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.6145555044474293 \cdot 10^{-55}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le 5.30891579653489513 \cdot 10^{72}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.6145555044474293 \cdot 10^{-55}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{elif}\;t \le 5.30891579653489513 \cdot 10^{72}:\\
\;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r663822 = x;
        double r663823 = y;
        double r663824 = z;
        double r663825 = r663823 * r663824;
        double r663826 = t;
        double r663827 = r663825 / r663826;
        double r663828 = r663822 + r663827;
        double r663829 = a;
        double r663830 = 1.0;
        double r663831 = r663829 + r663830;
        double r663832 = b;
        double r663833 = r663823 * r663832;
        double r663834 = r663833 / r663826;
        double r663835 = r663831 + r663834;
        double r663836 = r663828 / r663835;
        return r663836;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r663837 = t;
        double r663838 = -2.6145555044474293e-55;
        bool r663839 = r663837 <= r663838;
        double r663840 = x;
        double r663841 = y;
        double r663842 = z;
        double r663843 = r663837 / r663842;
        double r663844 = r663841 / r663843;
        double r663845 = r663840 + r663844;
        double r663846 = a;
        double r663847 = 1.0;
        double r663848 = r663846 + r663847;
        double r663849 = b;
        double r663850 = r663849 / r663837;
        double r663851 = r663841 * r663850;
        double r663852 = r663848 + r663851;
        double r663853 = r663845 / r663852;
        double r663854 = 5.308915796534895e+72;
        bool r663855 = r663837 <= r663854;
        double r663856 = r663841 * r663842;
        double r663857 = 1.0;
        double r663858 = r663857 / r663837;
        double r663859 = r663856 * r663858;
        double r663860 = r663840 + r663859;
        double r663861 = r663841 * r663849;
        double r663862 = r663861 / r663837;
        double r663863 = r663848 + r663862;
        double r663864 = r663860 / r663863;
        double r663865 = r663841 / r663837;
        double r663866 = fma(r663865, r663842, r663840);
        double r663867 = r663857 * r663866;
        double r663868 = r663867 / r663852;
        double r663869 = r663855 ? r663864 : r663868;
        double r663870 = r663839 ? r663853 : r663869;
        return r663870;
}

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

Original16.7
Target13.4
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -2.6145555044474293e-55

    1. Initial program 12.2

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

    3. Applied *-un-lft-identity12.2

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

    4. Applied times-frac9.8

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

    5. Simplified9.8

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

    7. Applied associate-/l*5.6

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

    if -2.6145555044474293e-55 < t < 5.308915796534895e+72

    1. Initial program 21.0

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

    3. Applied div-inv21.0

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

    if 5.308915796534895e+72 < t

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied times-frac8.4

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

    5. Simplified8.4

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

    7. Applied associate-/l*3.0

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

    9. Applied *-un-lft-identity3.0

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

    10. Applied *-un-lft-identity3.0

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

    11. Applied distribute-lft-out3.0

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

    12. Simplified3.0

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.6145555044474293 \cdot 10^{-55}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le 5.30891579653489513 \cdot 10^{72}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))))