Average Error: 10.6 → 1.9
Time: 16.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.893522426434000231966824381347828386842 \cdot 10^{-112} \lor \neg \left(z \le 2.858845591596935509716292622866871578373 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{t - a \cdot z}, -\frac{y}{\frac{t}{z} - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.893522426434000231966824381347828386842 \cdot 10^{-112} \lor \neg \left(z \le 2.858845591596935509716292622866871578373 \cdot 10^{-23}\right):\\
\;\;\;\;\mathsf{fma}\left(x, \frac{1}{t - a \cdot z}, -\frac{y}{\frac{t}{z} - a}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r621241 = x;
        double r621242 = y;
        double r621243 = z;
        double r621244 = r621242 * r621243;
        double r621245 = r621241 - r621244;
        double r621246 = t;
        double r621247 = a;
        double r621248 = r621247 * r621243;
        double r621249 = r621246 - r621248;
        double r621250 = r621245 / r621249;
        return r621250;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r621251 = z;
        double r621252 = -2.8935224264340002e-112;
        bool r621253 = r621251 <= r621252;
        double r621254 = 2.8588455915969355e-23;
        bool r621255 = r621251 <= r621254;
        double r621256 = !r621255;
        bool r621257 = r621253 || r621256;
        double r621258 = x;
        double r621259 = 1.0;
        double r621260 = t;
        double r621261 = a;
        double r621262 = r621261 * r621251;
        double r621263 = r621260 - r621262;
        double r621264 = r621259 / r621263;
        double r621265 = y;
        double r621266 = r621260 / r621251;
        double r621267 = r621266 - r621261;
        double r621268 = r621265 / r621267;
        double r621269 = -r621268;
        double r621270 = fma(r621258, r621264, r621269);
        double r621271 = r621265 * r621251;
        double r621272 = r621258 - r621271;
        double r621273 = r621263 / r621272;
        double r621274 = r621259 / r621273;
        double r621275 = r621257 ? r621270 : r621274;
        return r621275;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.6
Target1.7
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 < -2.8935224264340002e-112 or 2.8588455915969355e-23 < z

    1. Initial program 17.4

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

    3. Applied div-sub17.4

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

    4. Simplified11.6

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

    6. Applied div-inv11.6

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

    7. Applied fma-neg11.6

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

    8. Simplified11.6

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

    10. Applied *-un-lft-identity11.6

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

    11. Applied *-un-lft-identity11.6

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

    12. Applied times-frac11.6

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

    13. Applied associate-*l*11.6

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

    14. Simplified2.7

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

    if -2.8935224264340002e-112 < z < 2.8588455915969355e-23

    1. Initial program 0.1

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

    3. Applied clear-num0.6

      \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.893522426434000231966824381347828386842 \cdot 10^{-112} \lor \neg \left(z \le 2.858845591596935509716292622866871578373 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{t - a \cdot z}, -\frac{y}{\frac{t}{z} - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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