Average Error: 7.7 → 3.6
Time: 5.9s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.4707659967563265 \cdot 10^{138} \lor \neg \left(z \le 8.89564883510701823 \cdot 10^{120}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{\frac{1}{t \cdot z - x}}{\frac{1}{y \cdot z - x}}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.4707659967563265 \cdot 10^{138} \lor \neg \left(z \le 8.89564883510701823 \cdot 10^{120}\right):\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r646235 = x;
        double r646236 = y;
        double r646237 = z;
        double r646238 = r646236 * r646237;
        double r646239 = r646238 - r646235;
        double r646240 = t;
        double r646241 = r646240 * r646237;
        double r646242 = r646241 - r646235;
        double r646243 = r646239 / r646242;
        double r646244 = r646235 + r646243;
        double r646245 = 1.0;
        double r646246 = r646235 + r646245;
        double r646247 = r646244 / r646246;
        return r646247;
}
↓
double f(double x, double y, double z, double t) {
        double r646248 = z;
        double r646249 = -4.4707659967563265e+138;
        bool r646250 = r646248 <= r646249;
        double r646251 = 8.895648835107018e+120;
        bool r646252 = r646248 <= r646251;
        double r646253 = !r646252;
        bool r646254 = r646250 || r646253;
        double r646255 = x;
        double r646256 = y;
        double r646257 = t;
        double r646258 = r646256 / r646257;
        double r646259 = r646255 + r646258;
        double r646260 = 1.0;
        double r646261 = r646255 + r646260;
        double r646262 = r646259 / r646261;
        double r646263 = 1.0;
        double r646264 = r646257 * r646248;
        double r646265 = r646264 - r646255;
        double r646266 = r646263 / r646265;
        double r646267 = r646256 * r646248;
        double r646268 = r646267 - r646255;
        double r646269 = r646263 / r646268;
        double r646270 = r646266 / r646269;
        double r646271 = r646255 + r646270;
        double r646272 = r646271 / r646261;
        double r646273 = r646254 ? r646262 : r646272;
        return r646273;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.7
Target0.3
Herbie3.6
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.4707659967563265e+138 or 8.895648835107018e+120 < z

    1. Initial program 21.1

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

    2. Taylor expanded around inf 6.9

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

    if -4.4707659967563265e+138 < z < 8.895648835107018e+120

    1. Initial program 2.2

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

    3. Applied clear-num2.2

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1}\]
    4. Using strategy rm

    5. Applied div-inv2.2

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

    6. Applied associate-/r*2.2

      \[\leadsto \frac{x + \color{blue}{\frac{\frac{1}{t \cdot z - x}}{\frac{1}{y \cdot z - x}}}}{x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.4707659967563265 \cdot 10^{138} \lor \neg \left(z \le 8.89564883510701823 \cdot 10^{120}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{\frac{1}{t \cdot z - x}}{\frac{1}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))