Average Error: 23.8 → 17.8
Time: 19.4s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.0453105058496541 \cdot 10^{-301}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.9961735121647787 \cdot 10^{271}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \mathsf{fma}\left(z, t - a, x \cdot y\right)\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.0453105058496541 \cdot 10^{-301}:\\
\;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.9961735121647787 \cdot 10^{271}\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r865857 = x;
        double r865858 = y;
        double r865859 = r865857 * r865858;
        double r865860 = z;
        double r865861 = t;
        double r865862 = a;
        double r865863 = r865861 - r865862;
        double r865864 = r865860 * r865863;
        double r865865 = r865859 + r865864;
        double r865866 = b;
        double r865867 = r865866 - r865858;
        double r865868 = r865860 * r865867;
        double r865869 = r865858 + r865868;
        double r865870 = r865865 / r865869;
        return r865870;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r865871 = x;
        double r865872 = y;
        double r865873 = r865871 * r865872;
        double r865874 = z;
        double r865875 = t;
        double r865876 = a;
        double r865877 = r865875 - r865876;
        double r865878 = r865874 * r865877;
        double r865879 = r865873 + r865878;
        double r865880 = b;
        double r865881 = r865880 - r865872;
        double r865882 = r865874 * r865881;
        double r865883 = r865872 + r865882;
        double r865884 = r865879 / r865883;
        double r865885 = -1.045310505849654e-301;
        bool r865886 = r865884 <= r865885;
        double r865887 = 0.0;
        bool r865888 = r865884 <= r865887;
        double r865889 = 1.9961735121647787e+271;
        bool r865890 = r865884 <= r865889;
        double r865891 = !r865890;
        bool r865892 = r865888 || r865891;
        double r865893 = r865875 / r865880;
        double r865894 = r865876 / r865880;
        double r865895 = r865893 - r865894;
        double r865896 = 1.0;
        double r865897 = fma(r865881, r865874, r865872);
        double r865898 = r865896 / r865897;
        double r865899 = fma(r865874, r865877, r865873);
        double r865900 = r865898 * r865899;
        double r865901 = r865892 ? r865895 : r865900;
        double r865902 = r865886 ? r865884 : r865901;
        return r865902;
}

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

Original23.8
Target18.2
Herbie17.8
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -1.045310505849654e-301

    1. Initial program 13.1

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

    if -1.045310505849654e-301 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 0.0 or 1.9961735121647787e+271 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

    1. Initial program 56.8

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

    3. Applied clear-num56.8

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

    4. Simplified56.8

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

    5. Taylor expanded around inf 38.3

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

    if 0.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 1.9961735121647787e+271

    1. Initial program 0.3

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

    3. Applied clear-num0.5

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

    4. Simplified0.4

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

    6. Applied div-inv0.6

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

    7. Applied add-cube-cbrt0.6

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

    8. Applied times-frac0.6

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

    9. Simplified0.6

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

    10. Simplified0.4

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.0453105058496541 \cdot 10^{-301}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.9961735121647787 \cdot 10^{271}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \mathsf{fma}\left(z, t - a, x \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))))