Average Error: 21.9 → 18.4
Time: 21.3s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.7361233842143253 \cdot 10^{+67}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 8.615954677581334 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.7361233842143253 \cdot 10^{+67}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 8.615954677581334 \cdot 10^{+121}:\\
\;\;\;\;\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 r38390480 = x;
        double r38390481 = y;
        double r38390482 = r38390480 * r38390481;
        double r38390483 = z;
        double r38390484 = t;
        double r38390485 = a;
        double r38390486 = r38390484 - r38390485;
        double r38390487 = r38390483 * r38390486;
        double r38390488 = r38390482 + r38390487;
        double r38390489 = b;
        double r38390490 = r38390489 - r38390481;
        double r38390491 = r38390483 * r38390490;
        double r38390492 = r38390481 + r38390491;
        double r38390493 = r38390488 / r38390492;
        return r38390493;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r38390494 = z;
        double r38390495 = -2.7361233842143253e+67;
        bool r38390496 = r38390494 <= r38390495;
        double r38390497 = t;
        double r38390498 = b;
        double r38390499 = r38390497 / r38390498;
        double r38390500 = a;
        double r38390501 = r38390500 / r38390498;
        double r38390502 = r38390499 - r38390501;
        double r38390503 = 8.615954677581334e+121;
        bool r38390504 = r38390494 <= r38390503;
        double r38390505 = r38390497 - r38390500;
        double r38390506 = y;
        double r38390507 = x;
        double r38390508 = r38390506 * r38390507;
        double r38390509 = fma(r38390494, r38390505, r38390508);
        double r38390510 = r38390498 - r38390506;
        double r38390511 = fma(r38390510, r38390494, r38390506);
        double r38390512 = r38390509 / r38390511;
        double r38390513 = r38390504 ? r38390512 : r38390502;
        double r38390514 = r38390496 ? r38390502 : r38390513;
        return r38390514;
}

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

Original21.9
Target16.7
Herbie18.4
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.7361233842143253e+67 or 8.615954677581334e+121 < z

    1. Initial program 43.7

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

    2. Simplified43.7

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

    4. Applied clear-num43.7

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

    5. Taylor expanded around inf 33.3

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

    if -2.7361233842143253e+67 < z < 8.615954677581334e+121

    1. Initial program 10.5

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

    2. Simplified10.5

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.7361233842143253 \cdot 10^{+67}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 8.615954677581334 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))))