Average Error: 7.1 → 3.2
Time: 4.4s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.5176478096091269 \cdot 10^{137}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 1.00049457693685056 \cdot 10^{65}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.5176478096091269 \cdot 10^{137}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\mathbf{elif}\;z \le 1.00049457693685056 \cdot 10^{65}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r612329 = x;
        double r612330 = y;
        double r612331 = z;
        double r612332 = r612330 * r612331;
        double r612333 = r612332 - r612329;
        double r612334 = t;
        double r612335 = r612334 * r612331;
        double r612336 = r612335 - r612329;
        double r612337 = r612333 / r612336;
        double r612338 = r612329 + r612337;
        double r612339 = 1.0;
        double r612340 = r612329 + r612339;
        double r612341 = r612338 / r612340;
        return r612341;
}
↓
double f(double x, double y, double z, double t) {
        double r612342 = z;
        double r612343 = -6.517647809609127e+137;
        bool r612344 = r612342 <= r612343;
        double r612345 = x;
        double r612346 = y;
        double r612347 = t;
        double r612348 = r612346 / r612347;
        double r612349 = r612345 + r612348;
        double r612350 = 1.0;
        double r612351 = r612345 + r612350;
        double r612352 = r612349 / r612351;
        double r612353 = 1.0004945769368506e+65;
        bool r612354 = r612342 <= r612353;
        double r612355 = r612346 * r612342;
        double r612356 = r612355 - r612345;
        double r612357 = 1.0;
        double r612358 = r612347 * r612342;
        double r612359 = r612358 - r612345;
        double r612360 = r612357 / r612359;
        double r612361 = r612356 * r612360;
        double r612362 = r612345 + r612361;
        double r612363 = r612362 / r612351;
        double r612364 = r612346 / r612359;
        double r612365 = fma(r612364, r612342, r612345);
        double r612366 = r612345 / r612359;
        double r612367 = r612365 - r612366;
        double r612368 = r612357 / r612351;
        double r612369 = r612367 * r612368;
        double r612370 = r612354 ? r612363 : r612369;
        double r612371 = r612344 ? r612352 : r612370;
        return r612371;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.1
Target0.3
Herbie3.2
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.517647809609127e+137

    1. Initial program 21.1

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

    2. Taylor expanded around inf 7.5

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

    if -6.517647809609127e+137 < z < 1.0004945769368506e+65

    1. Initial program 1.2

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

    3. Applied div-inv1.3

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

    if 1.0004945769368506e+65 < z

    1. Initial program 18.0

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

    3. Applied div-sub18.0

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

    4. Applied associate-+r-18.0

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

    5. Simplified6.9

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

    7. Applied div-inv7.0

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.5176478096091269 \cdot 10^{137}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 1.00049457693685056 \cdot 10^{65}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))