Average Error: 23.5 → 20.5
Time: 6.8s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.71025579622562754 \cdot 10^{72} \lor \neg \left(z \le 7.73695011696723767 \cdot 10^{43}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}}}\\ \end{array}\]
\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{1}{\frac{1}{\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 r782906 = x;
        double r782907 = y;
        double r782908 = r782906 * r782907;
        double r782909 = z;
        double r782910 = t;
        double r782911 = a;
        double r782912 = r782910 - r782911;
        double r782913 = r782909 * r782912;
        double r782914 = r782908 + r782913;
        double r782915 = b;
        double r782916 = r782915 - r782907;
        double r782917 = r782909 * r782916;
        double r782918 = r782907 + r782917;
        double r782919 = r782914 / r782918;
        return r782919;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r782920 = z;
        double r782921 = -7.710255796225628e+72;
        bool r782922 = r782920 <= r782921;
        double r782923 = 7.736950116967238e+43;
        bool r782924 = r782920 <= r782923;
        double r782925 = !r782924;
        bool r782926 = r782922 || r782925;
        double r782927 = t;
        double r782928 = b;
        double r782929 = r782927 / r782928;
        double r782930 = a;
        double r782931 = r782930 / r782928;
        double r782932 = r782929 - r782931;
        double r782933 = 1.0;
        double r782934 = x;
        double r782935 = y;
        double r782936 = r782927 - r782930;
        double r782937 = r782920 * r782936;
        double r782938 = fma(r782934, r782935, r782937);
        double r782939 = r782928 - r782935;
        double r782940 = fma(r782920, r782939, r782935);
        double r782941 = r782938 / r782940;
        double r782942 = r782933 / r782941;
        double r782943 = r782933 / r782942;
        double r782944 = r782926 ? r782932 : r782943;
        return r782944;
}

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
Herbie20.5
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

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. Using strategy rm

    3. Applied clear-num42.6

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

    4. Simplified42.6

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

    5. 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. Using strategy rm

    3. Applied clear-num10.7

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

    4. Simplified10.7

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

    6. Applied clear-num10.7

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

  4. Final simplification20.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.71025579622562754 \cdot 10^{72} \lor \neg \left(z \le 7.73695011696723767 \cdot 10^{43}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}}}\\ \end{array}\]

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)))))