Average Error: 16.5 → 12.9
Time: 25.0s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.746515370639549540143905675148851571849 \cdot 10^{-50} \lor \neg \left(t \le 5.787492129914299632539133765158045701988 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.746515370639549540143905675148851571849 \cdot 10^{-50} \lor \neg \left(t \le 5.787492129914299632539133765158045701988 \cdot 10^{-52}\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 r632950 = x;
        double r632951 = y;
        double r632952 = z;
        double r632953 = r632951 * r632952;
        double r632954 = t;
        double r632955 = r632953 / r632954;
        double r632956 = r632950 + r632955;
        double r632957 = a;
        double r632958 = 1.0;
        double r632959 = r632957 + r632958;
        double r632960 = b;
        double r632961 = r632951 * r632960;
        double r632962 = r632961 / r632954;
        double r632963 = r632959 + r632962;
        double r632964 = r632956 / r632963;
        return r632964;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r632965 = t;
        double r632966 = -1.7465153706395495e-50;
        bool r632967 = r632965 <= r632966;
        double r632968 = 5.7874921299143e-52;
        bool r632969 = r632965 <= r632968;
        double r632970 = !r632969;
        bool r632971 = r632967 || r632970;
        double r632972 = y;
        double r632973 = r632972 / r632965;
        double r632974 = z;
        double r632975 = x;
        double r632976 = fma(r632973, r632974, r632975);
        double r632977 = b;
        double r632978 = a;
        double r632979 = fma(r632973, r632977, r632978);
        double r632980 = 1.0;
        double r632981 = r632979 + r632980;
        double r632982 = r632976 / r632981;
        double r632983 = r632972 * r632974;
        double r632984 = r632983 / r632965;
        double r632985 = r632975 + r632984;
        double r632986 = r632978 + r632980;
        double r632987 = r632972 * r632977;
        double r632988 = r632987 / r632965;
        double r632989 = r632986 + r632988;
        double r632990 = r632985 / r632989;
        double r632991 = r632971 ? r632982 : r632990;
        return r632991;
}

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
Target13.5
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.7465153706395495e-50 or 5.7874921299143e-52 < t

    1. Initial program 11.4

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

    2. Simplified5.4

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

    if -1.7465153706395495e-50 < t < 5.7874921299143e-52

    1. Initial program 23.8

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.746515370639549540143905675148851571849 \cdot 10^{-50} \lor \neg \left(t \le 5.787492129914299632539133765158045701988 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))