Average Error: 23.5 → 19.8
Time: 22.0s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.165500557925722713276939801225929782816 \cdot 10^{156}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 8.554770822371756506342987216437881364805 \cdot 10^{73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \le 9.130830894406563034137670928590187187137 \cdot 10^{119} \lor \neg \left(z \le 7.562322875695318426226340712787711622372 \cdot 10^{198}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, b - y, y\right)}}{\frac{1}{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.165500557925722713276939801225929782816 \cdot 10^{156}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 8.554770822371756506342987216437881364805 \cdot 10^{73}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\

\mathbf{elif}\;z \le 9.130830894406563034137670928590187187137 \cdot 10^{119} \lor \neg \left(z \le 7.562322875695318426226340712787711622372 \cdot 10^{198}\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r533147 = x;
        double r533148 = y;
        double r533149 = r533147 * r533148;
        double r533150 = z;
        double r533151 = t;
        double r533152 = a;
        double r533153 = r533151 - r533152;
        double r533154 = r533150 * r533153;
        double r533155 = r533149 + r533154;
        double r533156 = b;
        double r533157 = r533156 - r533148;
        double r533158 = r533150 * r533157;
        double r533159 = r533148 + r533158;
        double r533160 = r533155 / r533159;
        return r533160;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r533161 = z;
        double r533162 = -5.165500557925723e+156;
        bool r533163 = r533161 <= r533162;
        double r533164 = t;
        double r533165 = b;
        double r533166 = r533164 / r533165;
        double r533167 = a;
        double r533168 = r533167 / r533165;
        double r533169 = r533166 - r533168;
        double r533170 = 8.554770822371757e+73;
        bool r533171 = r533161 <= r533170;
        double r533172 = x;
        double r533173 = y;
        double r533174 = r533164 - r533167;
        double r533175 = r533161 * r533174;
        double r533176 = fma(r533172, r533173, r533175);
        double r533177 = r533165 - r533173;
        double r533178 = fma(r533161, r533177, r533173);
        double r533179 = r533176 / r533178;
        double r533180 = 9.130830894406563e+119;
        bool r533181 = r533161 <= r533180;
        double r533182 = 7.562322875695318e+198;
        bool r533183 = r533161 <= r533182;
        double r533184 = !r533183;
        bool r533185 = r533181 || r533184;
        double r533186 = 1.0;
        double r533187 = r533186 / r533178;
        double r533188 = r533186 / r533176;
        double r533189 = r533187 / r533188;
        double r533190 = r533185 ? r533169 : r533189;
        double r533191 = r533171 ? r533179 : r533190;
        double r533192 = r533163 ? r533169 : r533191;
        return r533192;
}

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

Original23.5
Target18.2
Herbie19.8
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.165500557925723e+156 or 8.554770822371757e+73 < z < 9.130830894406563e+119 or 7.562322875695318e+198 < z

    1. Initial program 48.6

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

    2. Simplified48.6

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

    4. Applied clear-num48.6

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

    5. Taylor expanded around inf 33.8

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

    if -5.165500557925723e+156 < z < 8.554770822371757e+73

    1. Initial program 13.0

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

    2. Simplified13.0

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

    if 9.130830894406563e+119 < z < 7.562322875695318e+198

    1. Initial program 40.0

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

    2. Simplified40.0

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

    4. Applied clear-num40.1

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

    6. Applied div-inv40.1

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

    7. Applied associate-/r*40.1

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

  4. Final simplification19.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.165500557925722713276939801225929782816 \cdot 10^{156}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 8.554770822371756506342987216437881364805 \cdot 10^{73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \le 9.130830894406563034137670928590187187137 \cdot 10^{119} \lor \neg \left(z \le 7.562322875695318426226340712787711622372 \cdot 10^{198}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, b - y, y\right)}}{\frac{1}{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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)))))