Average Error: 16.0 → 13.3
Time: 21.0s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.542078426322413427201724504758608315283 \cdot 10^{-178} \lor \neg \left(z \le 1.580084619156310622194536930557427267653 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.542078426322413427201724504758608315283 \cdot 10^{-178} \lor \neg \left(z \le 1.580084619156310622194536930557427267653 \cdot 10^{-147}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\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 r546812 = x;
        double r546813 = y;
        double r546814 = z;
        double r546815 = r546813 * r546814;
        double r546816 = t;
        double r546817 = r546815 / r546816;
        double r546818 = r546812 + r546817;
        double r546819 = a;
        double r546820 = 1.0;
        double r546821 = r546819 + r546820;
        double r546822 = b;
        double r546823 = r546813 * r546822;
        double r546824 = r546823 / r546816;
        double r546825 = r546821 + r546824;
        double r546826 = r546818 / r546825;
        return r546826;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r546827 = z;
        double r546828 = -2.5420784263224134e-178;
        bool r546829 = r546827 <= r546828;
        double r546830 = 1.5800846191563106e-147;
        bool r546831 = r546827 <= r546830;
        double r546832 = !r546831;
        bool r546833 = r546829 || r546832;
        double r546834 = y;
        double r546835 = t;
        double r546836 = r546834 / r546835;
        double r546837 = x;
        double r546838 = fma(r546836, r546827, r546837);
        double r546839 = b;
        double r546840 = a;
        double r546841 = fma(r546836, r546839, r546840);
        double r546842 = 1.0;
        double r546843 = r546841 + r546842;
        double r546844 = r546838 / r546843;
        double r546845 = r546834 * r546827;
        double r546846 = r546845 / r546835;
        double r546847 = r546837 + r546846;
        double r546848 = r546840 + r546842;
        double r546849 = r546834 * r546839;
        double r546850 = r546849 / r546835;
        double r546851 = r546848 + r546850;
        double r546852 = r546847 / r546851;
        double r546853 = r546833 ? r546844 : r546852;
        return r546853;
}

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

Original16.0
Target13.0
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.5420784263224134e-178 or 1.5800846191563106e-147 < z

    1. Initial program 19.0

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

    2. Simplified15.4

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

    if -2.5420784263224134e-178 < z < 1.5800846191563106e-147

    1. Initial program 6.7

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.542078426322413427201724504758608315283 \cdot 10^{-178} \lor \neg \left(z \le 1.580084619156310622194536930557427267653 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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))))