Average Error: 23.6 → 19.7
Time: 9.3s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.01373759864766076 \cdot 10^{106} \lor \neg \left(z \le 9.0608695367454199 \cdot 10^{67}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(b - y, z, y\right)}}{\frac{1}{x \cdot y + z \cdot \left(t - a\right)}}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.01373759864766076 \cdot 10^{106} \lor \neg \left(z \le 9.0608695367454199 \cdot 10^{67}\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r797275 = x;
        double r797276 = y;
        double r797277 = r797275 * r797276;
        double r797278 = z;
        double r797279 = t;
        double r797280 = a;
        double r797281 = r797279 - r797280;
        double r797282 = r797278 * r797281;
        double r797283 = r797277 + r797282;
        double r797284 = b;
        double r797285 = r797284 - r797276;
        double r797286 = r797278 * r797285;
        double r797287 = r797276 + r797286;
        double r797288 = r797283 / r797287;
        return r797288;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r797289 = z;
        double r797290 = -3.0137375986476608e+106;
        bool r797291 = r797289 <= r797290;
        double r797292 = 9.06086953674542e+67;
        bool r797293 = r797289 <= r797292;
        double r797294 = !r797293;
        bool r797295 = r797291 || r797294;
        double r797296 = t;
        double r797297 = b;
        double r797298 = r797296 / r797297;
        double r797299 = a;
        double r797300 = r797299 / r797297;
        double r797301 = r797298 - r797300;
        double r797302 = 1.0;
        double r797303 = y;
        double r797304 = r797297 - r797303;
        double r797305 = fma(r797304, r797289, r797303);
        double r797306 = r797302 / r797305;
        double r797307 = x;
        double r797308 = r797307 * r797303;
        double r797309 = r797296 - r797299;
        double r797310 = r797289 * r797309;
        double r797311 = r797308 + r797310;
        double r797312 = r797302 / r797311;
        double r797313 = r797306 / r797312;
        double r797314 = r797295 ? r797301 : r797313;
        return r797314;
}

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

Original23.6
Target18.0
Herbie19.7
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.0137375986476608e+106 or 9.06086953674542e+67 < z

    1. Initial program 45.6

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num45.7

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

    5. Applied div-inv45.7

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

    6. Applied associate-/r*45.7

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

    7. Simplified45.7

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

    8. Taylor expanded around inf 34.1

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

    if -3.0137375986476608e+106 < z < 9.06086953674542e+67

    1. Initial program 11.5

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num11.6

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

    5. Applied div-inv11.8

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

    6. Applied associate-/r*11.7

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

    7. Simplified11.7

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

  4. Final simplification19.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.01373759864766076 \cdot 10^{106} \lor \neg \left(z \le 9.0608695367454199 \cdot 10^{67}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(b - y, z, y\right)}}{\frac{1}{x \cdot y + z \cdot \left(t - a\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

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

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