Average Error: 16.3 → 15.5
Time: 16.6s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.065477972910828548290607099195452155009 \cdot 10^{-76}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{y \cdot \frac{b}{t} + \left(a + 1\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.065477972910828548290607099195452155009 \cdot 10^{-76}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r27068140 = x;
        double r27068141 = y;
        double r27068142 = z;
        double r27068143 = r27068141 * r27068142;
        double r27068144 = t;
        double r27068145 = r27068143 / r27068144;
        double r27068146 = r27068140 + r27068145;
        double r27068147 = a;
        double r27068148 = 1.0;
        double r27068149 = r27068147 + r27068148;
        double r27068150 = b;
        double r27068151 = r27068141 * r27068150;
        double r27068152 = r27068151 / r27068144;
        double r27068153 = r27068149 + r27068152;
        double r27068154 = r27068146 / r27068153;
        return r27068154;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r27068155 = z;
        double r27068156 = -1.0654779729108285e-76;
        bool r27068157 = r27068155 <= r27068156;
        double r27068158 = 1.0;
        double r27068159 = y;
        double r27068160 = t;
        double r27068161 = r27068159 / r27068160;
        double r27068162 = b;
        double r27068163 = a;
        double r27068164 = 1.0;
        double r27068165 = r27068163 + r27068164;
        double r27068166 = fma(r27068161, r27068162, r27068165);
        double r27068167 = x;
        double r27068168 = fma(r27068161, r27068155, r27068167);
        double r27068169 = r27068166 / r27068168;
        double r27068170 = r27068158 / r27068169;
        double r27068171 = r27068159 * r27068155;
        double r27068172 = r27068171 / r27068160;
        double r27068173 = r27068167 + r27068172;
        double r27068174 = r27068162 / r27068160;
        double r27068175 = r27068159 * r27068174;
        double r27068176 = r27068175 + r27068165;
        double r27068177 = r27068173 / r27068176;
        double r27068178 = r27068157 ? r27068170 : r27068177;
        return r27068178;
}

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

Original16.3
Target12.8
Herbie15.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0654779729108285e-76

    1. Initial program 21.8

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

    3. Applied clear-num22.0

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

    4. Simplified16.9

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

    if -1.0654779729108285e-76 < z

    1. Initial program 13.8

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

    3. Applied *-un-lft-identity13.8

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

    4. Applied times-frac14.9

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

    5. Simplified14.9

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.065477972910828548290607099195452155009 \cdot 10^{-76}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{y \cdot \frac{b}{t} + \left(a + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))