Average Error: 22.9 → 19.4
Time: 50.2s
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 r29480242 = x;
        double r29480243 = y;
        double r29480244 = r29480242 * r29480243;
        double r29480245 = z;
        double r29480246 = t;
        double r29480247 = a;
        double r29480248 = r29480246 - r29480247;
        double r29480249 = r29480245 * r29480248;
        double r29480250 = r29480244 + r29480249;
        double r29480251 = b;
        double r29480252 = r29480251 - r29480243;
        double r29480253 = r29480245 * r29480252;
        double r29480254 = r29480243 + r29480253;
        double r29480255 = r29480250 / r29480254;
        return r29480255;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r29480256 = z;
        double r29480257 = -7.711212480300809e+186;
        bool r29480258 = r29480256 <= r29480257;
        double r29480259 = t;
        double r29480260 = b;
        double r29480261 = r29480259 / r29480260;
        double r29480262 = a;
        double r29480263 = r29480262 / r29480260;
        double r29480264 = r29480261 - r29480263;
        double r29480265 = 3.7160992898136775e+157;
        bool r29480266 = r29480256 <= r29480265;
        double r29480267 = r29480259 - r29480262;
        double r29480268 = y;
        double r29480269 = x;
        double r29480270 = r29480268 * r29480269;
        double r29480271 = fma(r29480256, r29480267, r29480270);
        double r29480272 = r29480260 - r29480268;
        double r29480273 = fma(r29480256, r29480272, r29480268);
        double r29480274 = r29480271 / r29480273;
        double r29480275 = r29480266 ? r29480274 : r29480264;
        double r29480276 = r29480258 ? r29480264 : r29480275;
        return r29480276;
}

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)))))