Average Error: 16.5 → 14.7
Time: 17.5s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.198654496605673028369019352681810374144 \cdot 10^{-142} \lor \neg \left(y \le 4.588564977498519865787489769956601900271 \cdot 10^{159}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.198654496605673028369019352681810374144 \cdot 10^{-142} \lor \neg \left(y \le 4.588564977498519865787489769956601900271 \cdot 10^{159}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\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 r462947 = x;
        double r462948 = y;
        double r462949 = z;
        double r462950 = r462948 * r462949;
        double r462951 = t;
        double r462952 = r462950 / r462951;
        double r462953 = r462947 + r462952;
        double r462954 = a;
        double r462955 = 1.0;
        double r462956 = r462954 + r462955;
        double r462957 = b;
        double r462958 = r462948 * r462957;
        double r462959 = r462958 / r462951;
        double r462960 = r462956 + r462959;
        double r462961 = r462953 / r462960;
        return r462961;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r462962 = y;
        double r462963 = -3.198654496605673e-142;
        bool r462964 = r462962 <= r462963;
        double r462965 = 4.58856497749852e+159;
        bool r462966 = r462962 <= r462965;
        double r462967 = !r462966;
        bool r462968 = r462964 || r462967;
        double r462969 = t;
        double r462970 = r462962 / r462969;
        double r462971 = z;
        double r462972 = x;
        double r462973 = fma(r462970, r462971, r462972);
        double r462974 = 1.0;
        double r462975 = b;
        double r462976 = a;
        double r462977 = fma(r462970, r462975, r462976);
        double r462978 = 1.0;
        double r462979 = r462977 + r462978;
        double r462980 = r462974 / r462979;
        double r462981 = r462973 * r462980;
        double r462982 = r462962 * r462971;
        double r462983 = r462982 / r462969;
        double r462984 = r462972 + r462983;
        double r462985 = r462976 + r462978;
        double r462986 = r462962 * r462975;
        double r462987 = r462986 / r462969;
        double r462988 = r462985 + r462987;
        double r462989 = r462984 / r462988;
        double r462990 = r462968 ? r462981 : r462989;
        return r462990;
}

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

Original16.5
Target13.3
Herbie14.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.198654496605673e-142 or 4.58856497749852e+159 < y

    1. Initial program 26.4

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

    2. Simplified22.6

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

    4. Applied div-inv22.6

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

    if -3.198654496605673e-142 < y < 4.58856497749852e+159

    1. Initial program 7.6

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

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.198654496605673028369019352681810374144 \cdot 10^{-142} \lor \neg \left(y \le 4.588564977498519865787489769956601900271 \cdot 10^{159}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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 3.0369671037372459e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))