Average Error: 7.5 → 1.6
Time: 4.5s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{1}{\frac{t \cdot z - x}{x}}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.48483285561730493 \cdot 10^{301}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{1}{\frac{t \cdot z - x}{x}}}{x + 1}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.48483285561730493 \cdot 10^{301}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{t \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 r617837 = x;
        double r617838 = y;
        double r617839 = z;
        double r617840 = r617838 * r617839;
        double r617841 = r617840 - r617837;
        double r617842 = t;
        double r617843 = r617842 * r617839;
        double r617844 = r617843 - r617837;
        double r617845 = r617841 / r617844;
        double r617846 = r617837 + r617845;
        double r617847 = 1.0;
        double r617848 = r617837 + r617847;
        double r617849 = r617846 / r617848;
        return r617849;
}
↓
double f(double x, double y, double z, double t) {
        double r617850 = x;
        double r617851 = y;
        double r617852 = z;
        double r617853 = r617851 * r617852;
        double r617854 = r617853 - r617850;
        double r617855 = t;
        double r617856 = r617855 * r617852;
        double r617857 = r617856 - r617850;
        double r617858 = r617854 / r617857;
        double r617859 = r617850 + r617858;
        double r617860 = 1.0;
        double r617861 = r617850 + r617860;
        double r617862 = r617859 / r617861;
        double r617863 = -inf.0;
        bool r617864 = r617862 <= r617863;
        double r617865 = 1.0;
        double r617866 = r617857 / r617851;
        double r617867 = r617865 / r617866;
        double r617868 = fma(r617867, r617852, r617850);
        double r617869 = r617857 / r617850;
        double r617870 = r617865 / r617869;
        double r617871 = r617868 - r617870;
        double r617872 = r617871 / r617861;
        double r617873 = 1.484832855617305e+301;
        bool r617874 = r617862 <= r617873;
        double r617875 = r617851 / r617855;
        double r617876 = r617850 + r617875;
        double r617877 = r617876 / r617861;
        double r617878 = r617874 ? r617862 : r617877;
        double r617879 = r617864 ? r617872 : r617878;
        return r617879;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.5
Target0.4
Herbie1.6
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0

    1. Initial program 64.0

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

    3. Applied div-sub64.0

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

    4. Applied associate-+r-64.0

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

    5. Simplified3.5

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

    7. Applied clear-num3.5

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

    9. Applied clear-num3.5

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

    if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 1.484832855617305e+301

    1. Initial program 0.7

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

    if 1.484832855617305e+301 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program 63.3

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

    2. Taylor expanded around inf 12.0

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{1}{\frac{t \cdot z - x}{x}}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.48483285561730493 \cdot 10^{301}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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)))