Average Error: 23.5 → 20.4
Time: 14.4s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.71025579622562754 \cdot 10^{72} \lor \neg \left(z \le 7.73695011696723767 \cdot 10^{43}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array}\]
\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 r800752 = x;
        double r800753 = y;
        double r800754 = r800752 * r800753;
        double r800755 = z;
        double r800756 = t;
        double r800757 = a;
        double r800758 = r800756 - r800757;
        double r800759 = r800755 * r800758;
        double r800760 = r800754 + r800759;
        double r800761 = b;
        double r800762 = r800761 - r800753;
        double r800763 = r800755 * r800762;
        double r800764 = r800753 + r800763;
        double r800765 = r800760 / r800764;
        return r800765;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r800766 = z;
        double r800767 = -7.710255796225628e+72;
        bool r800768 = r800766 <= r800767;
        double r800769 = 7.736950116967238e+43;
        bool r800770 = r800766 <= r800769;
        double r800771 = !r800770;
        bool r800772 = r800768 || r800771;
        double r800773 = t;
        double r800774 = b;
        double r800775 = r800773 / r800774;
        double r800776 = a;
        double r800777 = r800776 / r800774;
        double r800778 = r800775 - r800777;
        double r800779 = x;
        double r800780 = y;
        double r800781 = r800773 - r800776;
        double r800782 = r800766 * r800781;
        double r800783 = fma(r800779, r800780, r800782);
        double r800784 = r800774 - r800780;
        double r800785 = fma(r800766, r800784, r800780);
        double r800786 = r800783 / r800785;
        double r800787 = r800772 ? r800778 : r800786;
        return r800787;
}

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)))))