Average Error: 7.4 → 3.3
Time: 17.7s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.85708464995288014 \cdot 10^{-84}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right) - \frac{x}{t \cdot z - x}}}\\ \mathbf{elif}\;z \le 8.97764022453410417 \cdot 10^{-118}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \frac{1}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.85708464995288014 \cdot 10^{-84}:\\
\;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right) - \frac{x}{t \cdot z - x}}}\\

\mathbf{elif}\;z \le 8.97764022453410417 \cdot 10^{-118}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(y \cdot \frac{1}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r501967 = x;
        double r501968 = y;
        double r501969 = z;
        double r501970 = r501968 * r501969;
        double r501971 = r501970 - r501967;
        double r501972 = t;
        double r501973 = r501972 * r501969;
        double r501974 = r501973 - r501967;
        double r501975 = r501971 / r501974;
        double r501976 = r501967 + r501975;
        double r501977 = 1.0;
        double r501978 = r501967 + r501977;
        double r501979 = r501976 / r501978;
        return r501979;
}
↓
double f(double x, double y, double z, double t) {
        double r501980 = z;
        double r501981 = -2.85708464995288e-84;
        bool r501982 = r501980 <= r501981;
        double r501983 = 1.0;
        double r501984 = x;
        double r501985 = 1.0;
        double r501986 = r501984 + r501985;
        double r501987 = y;
        double r501988 = t;
        double r501989 = -r501984;
        double r501990 = fma(r501980, r501988, r501989);
        double r501991 = r501987 / r501990;
        double r501992 = fma(r501991, r501980, r501984);
        double r501993 = r501988 * r501980;
        double r501994 = r501993 - r501984;
        double r501995 = r501984 / r501994;
        double r501996 = r501992 - r501995;
        double r501997 = r501986 / r501996;
        double r501998 = r501983 / r501997;
        double r501999 = 8.977640224534104e-118;
        bool r502000 = r501980 <= r501999;
        double r502001 = r501987 * r501980;
        double r502002 = r502001 - r501984;
        double r502003 = r501983 / r501994;
        double r502004 = r502002 * r502003;
        double r502005 = r501984 + r502004;
        double r502006 = r502005 / r501986;
        double r502007 = r501987 * r502003;
        double r502008 = fma(r502007, r501980, r501984);
        double r502009 = r502008 - r501995;
        double r502010 = r502009 / r501986;
        double r502011 = r502000 ? r502006 : r502010;
        double r502012 = r501982 ? r501998 : r502011;
        return r502012;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.4
Target0.4
Herbie3.3
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.85708464995288e-84

    1. Initial program 11.2

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

    3. Applied div-sub11.2

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

    4. Applied associate-+r-11.2

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

    5. Simplified4.7

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

    7. Applied clear-num4.7

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

    if -2.85708464995288e-84 < z < 8.977640224534104e-118

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    if 8.977640224534104e-118 < z

    1. Initial program 11.1

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

    3. Applied div-sub11.1

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

    4. Applied associate-+r-11.1

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

    5. Simplified5.0

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

    7. Applied div-inv5.1

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

    8. Simplified5.1

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.85708464995288014 \cdot 10^{-84}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right) - \frac{x}{t \cdot z - x}}}\\ \mathbf{elif}\;z \le 8.97764022453410417 \cdot 10^{-118}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \frac{1}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))