Average Error: 22.6 → 19.0
Time: 25.5s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.613549175387801 \cdot 10^{+193}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 1.0112549105590363 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.613549175387801 \cdot 10^{+193}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;z \le 1.0112549105590363 \cdot 10^{+124}:\\
\;\;\;\;\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 r25649192 = x;
        double r25649193 = y;
        double r25649194 = r25649192 * r25649193;
        double r25649195 = z;
        double r25649196 = t;
        double r25649197 = a;
        double r25649198 = r25649196 - r25649197;
        double r25649199 = r25649195 * r25649198;
        double r25649200 = r25649194 + r25649199;
        double r25649201 = b;
        double r25649202 = r25649201 - r25649193;
        double r25649203 = r25649195 * r25649202;
        double r25649204 = r25649193 + r25649203;
        double r25649205 = r25649200 / r25649204;
        return r25649205;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r25649206 = z;
        double r25649207 = -5.613549175387801e+193;
        bool r25649208 = r25649206 <= r25649207;
        double r25649209 = t;
        double r25649210 = b;
        double r25649211 = r25649209 / r25649210;
        double r25649212 = a;
        double r25649213 = r25649212 / r25649210;
        double r25649214 = r25649211 - r25649213;
        double r25649215 = 1.0112549105590363e+124;
        bool r25649216 = r25649206 <= r25649215;
        double r25649217 = r25649209 - r25649212;
        double r25649218 = y;
        double r25649219 = x;
        double r25649220 = r25649218 * r25649219;
        double r25649221 = fma(r25649206, r25649217, r25649220);
        double r25649222 = r25649210 - r25649218;
        double r25649223 = fma(r25649222, r25649206, r25649218);
        double r25649224 = r25649221 / r25649223;
        double r25649225 = r25649216 ? r25649224 : r25649214;
        double r25649226 = r25649208 ? r25649214 : r25649225;
        return r25649226;
}

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

Original22.6
Target17.4
Herbie19.0
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.613549175387801e+193 or 1.0112549105590363e+124 < z

    1. Initial program 48.1

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

    2. Simplified48.1

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

    4. Applied clear-num48.2

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

    5. Taylor expanded around inf 33.1

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

    if -5.613549175387801e+193 < z < 1.0112549105590363e+124

    1. Initial program 14.6

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

    2. Simplified14.6

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.613549175387801 \cdot 10^{+193}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;z \le 1.0112549105590363 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))))