Average Error: 7.5 → 4.0
Time: 4.5s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.9269626952441295 \cdot 10^{-223} \lor \neg \left(x \le 1.19683684475683126 \cdot 10^{-126}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) \cdot \frac{1}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.9269626952441295 \cdot 10^{-223} \lor \neg \left(x \le 1.19683684475683126 \cdot 10^{-126}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) \cdot \frac{1}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r814297 = x;
        double r814298 = y;
        double r814299 = z;
        double r814300 = r814298 * r814299;
        double r814301 = r814300 - r814297;
        double r814302 = t;
        double r814303 = r814302 * r814299;
        double r814304 = r814303 - r814297;
        double r814305 = r814301 / r814304;
        double r814306 = r814297 + r814305;
        double r814307 = 1.0;
        double r814308 = r814297 + r814307;
        double r814309 = r814306 / r814308;
        return r814309;
}
↓
double f(double x, double y, double z, double t) {
        double r814310 = x;
        double r814311 = -5.9269626952441295e-223;
        bool r814312 = r814310 <= r814311;
        double r814313 = 1.1968368447568313e-126;
        bool r814314 = r814310 <= r814313;
        double r814315 = !r814314;
        bool r814316 = r814312 || r814315;
        double r814317 = y;
        double r814318 = t;
        double r814319 = z;
        double r814320 = r814318 * r814319;
        double r814321 = r814320 - r814310;
        double r814322 = r814317 / r814321;
        double r814323 = fma(r814322, r814319, r814310);
        double r814324 = 1.0;
        double r814325 = 1.0;
        double r814326 = r814310 + r814325;
        double r814327 = r814324 / r814326;
        double r814328 = r814323 * r814327;
        double r814329 = r814310 / r814321;
        double r814330 = r814329 / r814326;
        double r814331 = r814328 - r814330;
        double r814332 = r814317 * r814319;
        double r814333 = r814332 - r814310;
        double r814334 = r814324 / r814321;
        double r814335 = r814333 * r814334;
        double r814336 = r814310 + r814335;
        double r814337 = r814336 / r814326;
        double r814338 = r814316 ? r814331 : r814337;
        return r814338;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.5
Target0.4
Herbie4.0
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.9269626952441295e-223 or 1.1968368447568313e-126 < x

    1. Initial program 7.4

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied div-sub7.4

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

    4. Applied associate-+r-7.4

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

    5. Applied div-sub7.4

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

    6. Simplified2.8

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

    8. Applied div-inv2.9

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

    9. Simplified2.9

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

    if -5.9269626952441295e-223 < x < 1.1968368447568313e-126

    1. Initial program 8.1

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied div-inv8.2

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.9269626952441295 \cdot 10^{-223} \lor \neg \left(x \le 1.19683684475683126 \cdot 10^{-126}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) \cdot \frac{1}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))