Average Error: 23.1 → 19.1
Time: 19.9s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.812224200595293507689589697171458927979 \cdot 10^{164}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 1.500183368090720918883242645080141003947 \cdot 10^{85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.812224200595293507689589697171458927979 \cdot 10^{164}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 1.500183368090720918883242645080141003947 \cdot 10^{85}:\\
\;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\mathsf{fma}\left(z, b - y, 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 r35479443 = x;
        double r35479444 = y;
        double r35479445 = r35479443 * r35479444;
        double r35479446 = z;
        double r35479447 = t;
        double r35479448 = a;
        double r35479449 = r35479447 - r35479448;
        double r35479450 = r35479446 * r35479449;
        double r35479451 = r35479445 + r35479450;
        double r35479452 = b;
        double r35479453 = r35479452 - r35479444;
        double r35479454 = r35479446 * r35479453;
        double r35479455 = r35479444 + r35479454;
        double r35479456 = r35479451 / r35479455;
        return r35479456;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r35479457 = z;
        double r35479458 = -4.812224200595294e+164;
        bool r35479459 = r35479457 <= r35479458;
        double r35479460 = t;
        double r35479461 = b;
        double r35479462 = r35479460 / r35479461;
        double r35479463 = a;
        double r35479464 = r35479463 / r35479461;
        double r35479465 = r35479462 - r35479464;
        double r35479466 = 1.500183368090721e+85;
        bool r35479467 = r35479457 <= r35479466;
        double r35479468 = y;
        double r35479469 = x;
        double r35479470 = r35479460 - r35479463;
        double r35479471 = r35479470 * r35479457;
        double r35479472 = fma(r35479468, r35479469, r35479471);
        double r35479473 = r35479461 - r35479468;
        double r35479474 = fma(r35479457, r35479473, r35479468);
        double r35479475 = r35479472 / r35479474;
        double r35479476 = r35479467 ? r35479475 : r35479465;
        double r35479477 = r35479459 ? r35479465 : r35479476;
        return r35479477;
}

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

Original23.1
Target17.9
Herbie19.1
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.812224200595294e+164 or 1.500183368090721e+85 < z

    1. Initial program 48.3

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

    2. Simplified48.3

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

    4. Applied *-commutative48.3

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

    6. Applied clear-num48.3

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

    7. Taylor expanded around inf 34.0

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

    if -4.812224200595294e+164 < z < 1.500183368090721e+85

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied *-commutative13.2

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

  4. Final simplification19.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.812224200595293507689589697171458927979 \cdot 10^{164}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 1.500183368090720918883242645080141003947 \cdot 10^{85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))))