Average Error: 7.3 → 2.7
Time: 4.2s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.31147958709434157 \cdot 10^{283}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{\frac{y \cdot z - x}{z}} - \frac{x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.31147958709434157 \cdot 10^{283}:\\
\;\;\;\;\frac{x + \frac{1}{\frac{t}{\frac{y \cdot z - x}{z}} - \frac{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 r623518 = x;
        double r623519 = y;
        double r623520 = z;
        double r623521 = r623519 * r623520;
        double r623522 = r623521 - r623518;
        double r623523 = t;
        double r623524 = r623523 * r623520;
        double r623525 = r623524 - r623518;
        double r623526 = r623522 / r623525;
        double r623527 = r623518 + r623526;
        double r623528 = 1.0;
        double r623529 = r623518 + r623528;
        double r623530 = r623527 / r623529;
        return r623530;
}
↓
double f(double x, double y, double z, double t) {
        double r623531 = x;
        double r623532 = y;
        double r623533 = z;
        double r623534 = r623532 * r623533;
        double r623535 = r623534 - r623531;
        double r623536 = t;
        double r623537 = r623536 * r623533;
        double r623538 = r623537 - r623531;
        double r623539 = r623535 / r623538;
        double r623540 = r623531 + r623539;
        double r623541 = 1.0;
        double r623542 = r623531 + r623541;
        double r623543 = r623540 / r623542;
        double r623544 = 1.3114795870943416e+283;
        bool r623545 = r623543 <= r623544;
        double r623546 = 1.0;
        double r623547 = r623535 / r623533;
        double r623548 = r623536 / r623547;
        double r623549 = r623531 / r623535;
        double r623550 = r623548 - r623549;
        double r623551 = r623546 / r623550;
        double r623552 = r623531 + r623551;
        double r623553 = r623552 / r623542;
        double r623554 = r623532 / r623536;
        double r623555 = r623531 + r623554;
        double r623556 = r623555 / r623542;
        double r623557 = r623545 ? r623553 : r623556;
        return r623557;
}

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

Original7.3
Target0.4
Herbie2.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 1.3114795870943416e+283

    1. Initial program 2.7

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

    3. Applied clear-num2.7

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

    5. Applied div-sub2.7

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

    7. Applied associate-/l*2.1

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

    if 1.3114795870943416e+283 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program 61.5

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.31147958709434157 \cdot 10^{283}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{\frac{y \cdot z - x}{z}} - \frac{x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))