Average Error: 10.9 → 1.9
Time: 21.9s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.873159407921233748848095007688082806631 \cdot 10^{59} \lor \neg \left(z \le 1.295167332391820426867656083484373540485 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.873159407921233748848095007688082806631 \cdot 10^{59} \lor \neg \left(z \le 1.295167332391820426867656083484373540485 \cdot 10^{-63}\right):\\
\;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r471853 = x;
        double r471854 = y;
        double r471855 = z;
        double r471856 = r471854 * r471855;
        double r471857 = r471853 - r471856;
        double r471858 = t;
        double r471859 = a;
        double r471860 = r471859 * r471855;
        double r471861 = r471858 - r471860;
        double r471862 = r471857 / r471861;
        return r471862;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r471863 = z;
        double r471864 = -1.8731594079212337e+59;
        bool r471865 = r471863 <= r471864;
        double r471866 = 1.2951673323918204e-63;
        bool r471867 = r471863 <= r471866;
        double r471868 = !r471867;
        bool r471869 = r471865 || r471868;
        double r471870 = x;
        double r471871 = 1.0;
        double r471872 = t;
        double r471873 = a;
        double r471874 = r471873 * r471863;
        double r471875 = r471872 - r471874;
        double r471876 = r471871 / r471875;
        double r471877 = r471870 * r471876;
        double r471878 = y;
        double r471879 = r471872 / r471863;
        double r471880 = r471879 - r471873;
        double r471881 = r471878 / r471880;
        double r471882 = r471877 - r471881;
        double r471883 = r471878 * r471863;
        double r471884 = r471870 - r471883;
        double r471885 = r471884 / r471875;
        double r471886 = r471869 ? r471882 : r471885;
        return r471886;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.9
Target1.8
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.8731594079212337e+59 or 1.2951673323918204e-63 < z

    1. Initial program 20.6

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub20.6

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]

    4. Simplified13.0

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{y \cdot \frac{z}{t - a \cdot z}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity13.0

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

    7. Applied associate-*l*13.0

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

    8. Simplified3.0

      \[\leadsto \frac{x}{t - a \cdot z} - 1 \cdot \color{blue}{\frac{y}{\frac{t}{z} - a}}\]
    9. Using strategy rm

    10. Applied div-inv3.0

      \[\leadsto \color{blue}{x \cdot \frac{1}{t - a \cdot z}} - 1 \cdot \frac{y}{\frac{t}{z} - a}\]

    if -1.8731594079212337e+59 < z < 1.2951673323918204e-63

    1. Initial program 0.6

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.873159407921233748848095007688082806631 \cdot 10^{59} \lor \neg \left(z \le 1.295167332391820426867656083484373540485 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))