Average Error: 22.1 → 19.6
Time: 48.2s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.0995968184298553 \cdot 10^{+73}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le -9.689336201481151 \cdot 10^{-225}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \le -1.1807804438070255 \cdot 10^{-295}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \le 1.122440542385563 \cdot 10^{+206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.0995968184298553 \cdot 10^{+73}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le -9.689336201481151 \cdot 10^{-225}:\\
\;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\

\mathbf{elif}\;z \le -1.1807804438070255 \cdot 10^{-295}:\\
\;\;\;\;x\\

\mathbf{elif}\;z \le 1.122440542385563 \cdot 10^{+206}:\\
\;\;\;\;\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 r37793926 = x;
        double r37793927 = y;
        double r37793928 = r37793926 * r37793927;
        double r37793929 = z;
        double r37793930 = t;
        double r37793931 = a;
        double r37793932 = r37793930 - r37793931;
        double r37793933 = r37793929 * r37793932;
        double r37793934 = r37793928 + r37793933;
        double r37793935 = b;
        double r37793936 = r37793935 - r37793927;
        double r37793937 = r37793929 * r37793936;
        double r37793938 = r37793927 + r37793937;
        double r37793939 = r37793934 / r37793938;
        return r37793939;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r37793940 = z;
        double r37793941 = -2.0995968184298553e+73;
        bool r37793942 = r37793940 <= r37793941;
        double r37793943 = t;
        double r37793944 = b;
        double r37793945 = r37793943 / r37793944;
        double r37793946 = a;
        double r37793947 = r37793946 / r37793944;
        double r37793948 = r37793945 - r37793947;
        double r37793949 = -9.689336201481151e-225;
        bool r37793950 = r37793940 <= r37793949;
        double r37793951 = r37793943 - r37793946;
        double r37793952 = y;
        double r37793953 = x;
        double r37793954 = r37793952 * r37793953;
        double r37793955 = fma(r37793940, r37793951, r37793954);
        double r37793956 = r37793944 - r37793952;
        double r37793957 = fma(r37793956, r37793940, r37793952);
        double r37793958 = r37793955 / r37793957;
        double r37793959 = -1.1807804438070255e-295;
        bool r37793960 = r37793940 <= r37793959;
        double r37793961 = 1.122440542385563e+206;
        bool r37793962 = r37793940 <= r37793961;
        double r37793963 = r37793962 ? r37793958 : r37793948;
        double r37793964 = r37793960 ? r37793953 : r37793963;
        double r37793965 = r37793950 ? r37793958 : r37793964;
        double r37793966 = r37793942 ? r37793948 : r37793965;
        return r37793966;
}

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

Original22.1
Target16.7
Herbie19.6
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.0995968184298553e+73 or 1.122440542385563e+206 < z

    1. Initial program 45.3

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

    2. Simplified45.3

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

    4. Applied clear-num45.4

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

    5. Taylor expanded around inf 33.1

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

    if -2.0995968184298553e+73 < z < -9.689336201481151e-225 or -1.1807804438070255e-295 < z < 1.122440542385563e+206

    1. Initial program 14.0

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

    2. Simplified14.0

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

    if -9.689336201481151e-225 < z < -1.1807804438070255e-295

    1. Initial program 8.6

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

    2. Simplified8.6

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

    3. Taylor expanded around 0 22.0

      \[\leadsto \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.0995968184298553 \cdot 10^{+73}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le -9.689336201481151 \cdot 10^{-225}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \le -1.1807804438070255 \cdot 10^{-295}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \le 1.122440542385563 \cdot 10^{+206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))))