Average Error: 10.8 → 7.2
Time: 12.9s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.0472382802125399 \cdot 10^{-76} \lor \neg \left(z \le 2.2308287243209396 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.0472382802125399 \cdot 10^{-76} \lor \neg \left(z \le 2.2308287243209396 \cdot 10^{-8}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - z \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r828353 = x;
        double r828354 = y;
        double r828355 = z;
        double r828356 = r828354 * r828355;
        double r828357 = r828353 - r828356;
        double r828358 = t;
        double r828359 = a;
        double r828360 = r828359 * r828355;
        double r828361 = r828358 - r828360;
        double r828362 = r828357 / r828361;
        return r828362;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r828363 = z;
        double r828364 = -4.04723828021254e-76;
        bool r828365 = r828363 <= r828364;
        double r828366 = 2.2308287243209396e-08;
        bool r828367 = r828363 <= r828366;
        double r828368 = !r828367;
        bool r828369 = r828365 || r828368;
        double r828370 = x;
        double r828371 = t;
        double r828372 = a;
        double r828373 = r828372 * r828363;
        double r828374 = r828371 - r828373;
        double r828375 = r828370 / r828374;
        double r828376 = y;
        double r828377 = r828363 * r828372;
        double r828378 = r828371 - r828377;
        double r828379 = r828363 / r828378;
        double r828380 = r828376 * r828379;
        double r828381 = r828375 - r828380;
        double r828382 = 1.0;
        double r828383 = -r828363;
        double r828384 = fma(r828372, r828383, r828371);
        double r828385 = r828363 * r828376;
        double r828386 = r828370 - r828385;
        double r828387 = r828384 / r828386;
        double r828388 = r828382 / r828387;
        double r828389 = r828369 ? r828381 : r828388;
        return r828389;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.8
Target1.8
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.04723828021254e-76 or 2.2308287243209396e-08 < z

    1. Initial program 18.9

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

    3. Applied div-sub18.9

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

    4. Simplified12.3

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

    if -4.04723828021254e-76 < z < 2.2308287243209396e-08

    1. Initial program 0.1

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

    3. Applied clear-num0.5

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

    4. Simplified0.5

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.0472382802125399 \cdot 10^{-76} \lor \neg \left(z \le 2.2308287243209396 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))))