Average Error: 22.9 → 19.4
Time: 1.2m
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.711212480300808676492071897373033056253 \cdot 10^{186}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 3.716099289813677493622206799661656232681 \cdot 10^{157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.711212480300808676492071897373033056253 \cdot 10^{186}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 3.716099289813677493622206799661656232681 \cdot 10^{157}:\\
\;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(z, b - y, 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 r37115335 = x;
        double r37115336 = y;
        double r37115337 = r37115335 * r37115336;
        double r37115338 = z;
        double r37115339 = t;
        double r37115340 = a;
        double r37115341 = r37115339 - r37115340;
        double r37115342 = r37115338 * r37115341;
        double r37115343 = r37115337 + r37115342;
        double r37115344 = b;
        double r37115345 = r37115344 - r37115336;
        double r37115346 = r37115338 * r37115345;
        double r37115347 = r37115336 + r37115346;
        double r37115348 = r37115343 / r37115347;
        return r37115348;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r37115349 = z;
        double r37115350 = -7.711212480300809e+186;
        bool r37115351 = r37115349 <= r37115350;
        double r37115352 = t;
        double r37115353 = b;
        double r37115354 = r37115352 / r37115353;
        double r37115355 = a;
        double r37115356 = r37115355 / r37115353;
        double r37115357 = r37115354 - r37115356;
        double r37115358 = 3.7160992898136775e+157;
        bool r37115359 = r37115349 <= r37115358;
        double r37115360 = r37115352 - r37115355;
        double r37115361 = y;
        double r37115362 = x;
        double r37115363 = r37115361 * r37115362;
        double r37115364 = fma(r37115349, r37115360, r37115363);
        double r37115365 = r37115353 - r37115361;
        double r37115366 = fma(r37115349, r37115365, r37115361);
        double r37115367 = r37115364 / r37115366;
        double r37115368 = r37115359 ? r37115367 : r37115357;
        double r37115369 = r37115351 ? r37115357 : r37115368;
        return r37115369;
}

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

Original22.9
Target17.7
Herbie19.4
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.711212480300809e+186 or 3.7160992898136775e+157 < z

    1. Initial program 51.2

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

    2. Simplified51.2

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

    4. Applied clear-num51.2

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

    5. Taylor expanded around inf 34.4

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

    if -7.711212480300809e+186 < z < 3.7160992898136775e+157

    1. Initial program 15.5

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

    2. Simplified15.5

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

    4. Applied clear-num15.6

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

    6. Applied *-un-lft-identity15.6

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

    7. Applied *-un-lft-identity15.6

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

    8. Applied times-frac15.6

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

    9. Applied add-cube-cbrt15.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(z, b - y, y\right)}{\mathsf{fma}\left(z, t - a, y \cdot x\right)}}\]

    10. Applied times-frac15.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{\mathsf{fma}\left(z, t - a, y \cdot x\right)}}}\]

    11. Simplified15.6

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{\mathsf{fma}\left(z, t - a, y \cdot x\right)}}\]

    12. Simplified15.5

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

  4. Final simplification19.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.711212480300808676492071897373033056253 \cdot 10^{186}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 3.716099289813677493622206799661656232681 \cdot 10^{157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))))