Average Error: 7.8 → 3.4
Time: 4.1s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.67254162541755391 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{x \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 5.76566373012723862 \cdot 10^{55}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.67254162541755391 \cdot 10^{-24}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{x \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

\mathbf{elif}\;z \le 5.76566373012723862 \cdot 10^{55}:\\
\;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r752971 = x;
        double r752972 = y;
        double r752973 = z;
        double r752974 = r752972 * r752973;
        double r752975 = r752974 - r752971;
        double r752976 = t;
        double r752977 = r752976 * r752973;
        double r752978 = r752977 - r752971;
        double r752979 = r752975 / r752978;
        double r752980 = r752971 + r752979;
        double r752981 = 1.0;
        double r752982 = r752971 + r752981;
        double r752983 = r752980 / r752982;
        return r752983;
}
↓
double f(double x, double y, double z, double t) {
        double r752984 = z;
        double r752985 = -5.672541625417554e-24;
        bool r752986 = r752984 <= r752985;
        double r752987 = y;
        double r752988 = t;
        double r752989 = r752988 * r752984;
        double r752990 = x;
        double r752991 = r752989 - r752990;
        double r752992 = r752987 / r752991;
        double r752993 = fma(r752992, r752984, r752990);
        double r752994 = 1.0;
        double r752995 = r752990 + r752994;
        double r752996 = 1.0;
        double r752997 = r752995 * r752996;
        double r752998 = r752993 / r752997;
        double r752999 = r752996 / r752991;
        double r753000 = r752990 * r752999;
        double r753001 = r753000 / r752995;
        double r753002 = r752998 - r753001;
        double r753003 = 5.765663730127239e+55;
        bool r753004 = r752984 <= r753003;
        double r753005 = r752987 * r752984;
        double r753006 = r753005 - r752990;
        double r753007 = r752991 / r753006;
        double r753008 = r752996 / r753007;
        double r753009 = r752990 + r753008;
        double r753010 = r753009 / r752995;
        double r753011 = r752987 / r752988;
        double r753012 = r752990 + r753011;
        double r753013 = r753012 / r752995;
        double r753014 = r753004 ? r753010 : r753013;
        double r753015 = r752986 ? r753002 : r753014;
        return r753015;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.8
Target0.5
Herbie3.4
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.672541625417554e-24

    1. Initial program 14.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied div-sub14.9

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

    4. Applied associate-+r-14.9

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

    5. Applied div-sub14.9

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t \cdot z - x}}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}}\]

    6. Simplified5.5

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

    8. Applied div-inv5.5

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

    if -5.672541625417554e-24 < z < 5.765663730127239e+55

    1. Initial program 0.3

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied clear-num0.3

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1}\]

    if 5.765663730127239e+55 < z

    1. Initial program 19.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

    2. Taylor expanded around inf 9.3

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.67254162541755391 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{x \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 5.76566373012723862 \cdot 10^{55}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))