Average Error: 23.2 → 19.6
Time: 25.5s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.034080329997317996022135396100574836359 \cdot 10^{49} \lor \neg \left(z \le 3.954445955845695356594268852084466536392 \cdot 10^{76}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(t - a\right) + x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.034080329997317996022135396100574836359 \cdot 10^{49} \lor \neg \left(z \le 3.954445955845695356594268852084466536392 \cdot 10^{76}\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{else}:\\
\;\;\;\;\left(z \cdot \left(t - a\right) + x \cdot y\right) \cdot \frac{1}{\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 r756211 = x;
        double r756212 = y;
        double r756213 = r756211 * r756212;
        double r756214 = z;
        double r756215 = t;
        double r756216 = a;
        double r756217 = r756215 - r756216;
        double r756218 = r756214 * r756217;
        double r756219 = r756213 + r756218;
        double r756220 = b;
        double r756221 = r756220 - r756212;
        double r756222 = r756214 * r756221;
        double r756223 = r756212 + r756222;
        double r756224 = r756219 / r756223;
        return r756224;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r756225 = z;
        double r756226 = -2.034080329997318e+49;
        bool r756227 = r756225 <= r756226;
        double r756228 = 3.9544459558456954e+76;
        bool r756229 = r756225 <= r756228;
        double r756230 = !r756229;
        bool r756231 = r756227 || r756230;
        double r756232 = t;
        double r756233 = b;
        double r756234 = r756232 / r756233;
        double r756235 = a;
        double r756236 = r756235 / r756233;
        double r756237 = r756234 - r756236;
        double r756238 = r756232 - r756235;
        double r756239 = r756225 * r756238;
        double r756240 = x;
        double r756241 = y;
        double r756242 = r756240 * r756241;
        double r756243 = r756239 + r756242;
        double r756244 = 1.0;
        double r756245 = r756233 - r756241;
        double r756246 = fma(r756225, r756245, r756241);
        double r756247 = r756244 / r756246;
        double r756248 = r756243 * r756247;
        double r756249 = r756231 ? r756237 : r756248;
        return r756249;
}

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

Original23.2
Target17.9
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 2 regimes

  2. if z < -2.034080329997318e+49 or 3.9544459558456954e+76 < z

    1. Initial program 43.8

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num43.9

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

    4. Simplified43.9

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

    5. Taylor expanded around inf 34.4

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

    if -2.034080329997318e+49 < z < 3.9544459558456954e+76

    1. Initial program 10.2

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied div-inv10.3

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

    4. Simplified10.3

      \[\leadsto \left(x \cdot y + z \cdot \left(t - a\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(z, b - y, y\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification19.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.034080329997317996022135396100574836359 \cdot 10^{49} \lor \neg \left(z \le 3.954445955845695356594268852084466536392 \cdot 10^{76}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(t - a\right) + x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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)))))