Average Error: 23.5 → 20.4
Time: 18.2s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.71025579622562754 \cdot 10^{72} \lor \neg \left(z \le 7.73695011696723767 \cdot 10^{43}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.71025579622562754 \cdot 10^{72} \lor \neg \left(z \le 7.73695011696723767 \cdot 10^{43}\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r847794 = x;
        double r847795 = y;
        double r847796 = r847794 * r847795;
        double r847797 = z;
        double r847798 = t;
        double r847799 = a;
        double r847800 = r847798 - r847799;
        double r847801 = r847797 * r847800;
        double r847802 = r847796 + r847801;
        double r847803 = b;
        double r847804 = r847803 - r847795;
        double r847805 = r847797 * r847804;
        double r847806 = r847795 + r847805;
        double r847807 = r847802 / r847806;
        return r847807;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r847808 = z;
        double r847809 = -7.710255796225628e+72;
        bool r847810 = r847808 <= r847809;
        double r847811 = 7.736950116967238e+43;
        bool r847812 = r847808 <= r847811;
        double r847813 = !r847812;
        bool r847814 = r847810 || r847813;
        double r847815 = t;
        double r847816 = b;
        double r847817 = r847815 / r847816;
        double r847818 = a;
        double r847819 = r847818 / r847816;
        double r847820 = r847817 - r847819;
        double r847821 = x;
        double r847822 = y;
        double r847823 = r847815 - r847818;
        double r847824 = r847808 * r847823;
        double r847825 = fma(r847821, r847822, r847824);
        double r847826 = r847816 - r847822;
        double r847827 = fma(r847808, r847826, r847822);
        double r847828 = r847825 / r847827;
        double r847829 = r847814 ? r847820 : r847828;
        return r847829;
}

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
Target18.1
Herbie20.4
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.710255796225628e+72 or 7.736950116967238e+43 < z

    1. Initial program 42.5

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

    2. Simplified42.5

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

    4. Applied clear-num42.6

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

    5. Simplified42.6

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

    6. Taylor expanded around inf 34.9

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

    if -7.710255796225628e+72 < z < 7.736950116967238e+43

    1. Initial program 10.6

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

    2. Simplified10.6

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

  4. Final simplification20.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.71025579622562754 \cdot 10^{72} \lor \neg \left(z \le 7.73695011696723767 \cdot 10^{43}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))))