Average Error: 23.6 → 19.3
Time: 58.2s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.304579087607779 \cdot 10^{+161}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 5.613870853735117 \cdot 10^{+120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.304579087607779 \cdot 10^{+161}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 5.613870853735117 \cdot 10^{+120}:\\
\;\;\;\;\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 r34429847 = x;
        double r34429848 = y;
        double r34429849 = r34429847 * r34429848;
        double r34429850 = z;
        double r34429851 = t;
        double r34429852 = a;
        double r34429853 = r34429851 - r34429852;
        double r34429854 = r34429850 * r34429853;
        double r34429855 = r34429849 + r34429854;
        double r34429856 = b;
        double r34429857 = r34429856 - r34429848;
        double r34429858 = r34429850 * r34429857;
        double r34429859 = r34429848 + r34429858;
        double r34429860 = r34429855 / r34429859;
        return r34429860;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r34429861 = z;
        double r34429862 = -1.304579087607779e+161;
        bool r34429863 = r34429861 <= r34429862;
        double r34429864 = t;
        double r34429865 = b;
        double r34429866 = r34429864 / r34429865;
        double r34429867 = a;
        double r34429868 = r34429867 / r34429865;
        double r34429869 = r34429866 - r34429868;
        double r34429870 = 5.613870853735117e+120;
        bool r34429871 = r34429861 <= r34429870;
        double r34429872 = r34429864 - r34429867;
        double r34429873 = y;
        double r34429874 = x;
        double r34429875 = r34429873 * r34429874;
        double r34429876 = fma(r34429861, r34429872, r34429875);
        double r34429877 = r34429865 - r34429873;
        double r34429878 = fma(r34429877, r34429861, r34429873);
        double r34429879 = r34429876 / r34429878;
        double r34429880 = r34429871 ? r34429879 : r34429869;
        double r34429881 = r34429863 ? r34429869 : r34429880;
        return r34429881;
}

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
Target18.6
Herbie19.3
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.304579087607779e+161 or 5.613870853735117e+120 < z

    1. Initial program 49.1

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

    2. Simplified49.1

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

    4. Applied clear-num49.1

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

    5. Taylor expanded around inf 32.7

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

    if -1.304579087607779e+161 < z < 5.613870853735117e+120

    1. Initial program 14.5

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

    2. Simplified14.5

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

  4. Final simplification19.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.304579087607779 \cdot 10^{+161}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 5.613870853735117 \cdot 10^{+120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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)))))