Average Error: 23.4 → 19.5
Time: 59.1s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.61354917538780133180731765986612602828 \cdot 10^{193}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 5.088199112679185338412991581117806833141 \cdot 10^{118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.61354917538780133180731765986612602828 \cdot 10^{193}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 5.088199112679185338412991581117806833141 \cdot 10^{118}:\\
\;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r38427439 = x;
        double r38427440 = y;
        double r38427441 = r38427439 * r38427440;
        double r38427442 = z;
        double r38427443 = t;
        double r38427444 = a;
        double r38427445 = r38427443 - r38427444;
        double r38427446 = r38427442 * r38427445;
        double r38427447 = r38427441 + r38427446;
        double r38427448 = b;
        double r38427449 = r38427448 - r38427440;
        double r38427450 = r38427442 * r38427449;
        double r38427451 = r38427440 + r38427450;
        double r38427452 = r38427447 / r38427451;
        return r38427452;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r38427453 = z;
        double r38427454 = -5.613549175387801e+193;
        bool r38427455 = r38427453 <= r38427454;
        double r38427456 = t;
        double r38427457 = b;
        double r38427458 = r38427456 / r38427457;
        double r38427459 = a;
        double r38427460 = r38427459 / r38427457;
        double r38427461 = r38427458 - r38427460;
        double r38427462 = 5.088199112679185e+118;
        bool r38427463 = r38427453 <= r38427462;
        double r38427464 = r38427456 - r38427459;
        double r38427465 = y;
        double r38427466 = x;
        double r38427467 = r38427465 * r38427466;
        double r38427468 = fma(r38427453, r38427464, r38427467);
        double r38427469 = r38427457 - r38427465;
        double r38427470 = fma(r38427469, r38427453, r38427465);
        double r38427471 = r38427468 / r38427470;
        double r38427472 = r38427463 ? r38427471 : r38427461;
        double r38427473 = r38427455 ? r38427461 : r38427472;
        return r38427473;
}

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

Original23.4
Target18.1
Herbie19.5
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.613549175387801e+193 or 5.088199112679185e+118 < z

    1. Initial program 49.1

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

    2. Simplified49.1

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

    4. Applied clear-num49.1

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

    5. Taylor expanded around inf 33.1

      \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]

    if -5.613549175387801e+193 < z < 5.088199112679185e+118

    1. Initial program 15.1

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

    2. Simplified15.1

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

  4. Final simplification19.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.61354917538780133180731765986612602828 \cdot 10^{193}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 5.088199112679185338412991581117806833141 \cdot 10^{118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"

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

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