Average Error: 7.8 → 3.4
Time: 4.7s
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 r782226 = x;
        double r782227 = y;
        double r782228 = z;
        double r782229 = r782227 * r782228;
        double r782230 = r782229 - r782226;
        double r782231 = t;
        double r782232 = r782231 * r782228;
        double r782233 = r782232 - r782226;
        double r782234 = r782230 / r782233;
        double r782235 = r782226 + r782234;
        double r782236 = 1.0;
        double r782237 = r782226 + r782236;
        double r782238 = r782235 / r782237;
        return r782238;
}
↓
double f(double x, double y, double z, double t) {
        double r782239 = z;
        double r782240 = -5.672541625417554e-24;
        bool r782241 = r782239 <= r782240;
        double r782242 = y;
        double r782243 = t;
        double r782244 = r782243 * r782239;
        double r782245 = x;
        double r782246 = r782244 - r782245;
        double r782247 = r782242 / r782246;
        double r782248 = fma(r782247, r782239, r782245);
        double r782249 = 1.0;
        double r782250 = r782245 + r782249;
        double r782251 = 1.0;
        double r782252 = r782250 * r782251;
        double r782253 = r782248 / r782252;
        double r782254 = r782251 / r782246;
        double r782255 = r782245 * r782254;
        double r782256 = r782255 / r782250;
        double r782257 = r782253 - r782256;
        double r782258 = 5.765663730127239e+55;
        bool r782259 = r782239 <= r782258;
        double r782260 = r782242 * r782239;
        double r782261 = r782260 - r782245;
        double r782262 = r782246 / r782261;
        double r782263 = r782251 / r782262;
        double r782264 = r782245 + r782263;
        double r782265 = r782264 / r782250;
        double r782266 = r782242 / r782243;
        double r782267 = r782245 + r782266;
        double r782268 = r782267 / r782250;
        double r782269 = r782259 ? r782265 : r782268;
        double r782270 = r782241 ? r782257 : r782269;
        return r782270;
}

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)))