Average Error: 7.1 → 3.6
Time: 4.1s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.550785285630013736322572265977521807997 \cdot 10^{177}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 9.56138289533363135301744860674886759669 \cdot 10^{-188}:\\ \;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{x \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.550785285630013736322572265977521807997 \cdot 10^{177}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\mathbf{elif}\;z \le 9.56138289533363135301744860674886759669 \cdot 10^{-188}:\\
\;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r985534 = x;
        double r985535 = y;
        double r985536 = z;
        double r985537 = r985535 * r985536;
        double r985538 = r985537 - r985534;
        double r985539 = t;
        double r985540 = r985539 * r985536;
        double r985541 = r985540 - r985534;
        double r985542 = r985538 / r985541;
        double r985543 = r985534 + r985542;
        double r985544 = 1.0;
        double r985545 = r985534 + r985544;
        double r985546 = r985543 / r985545;
        return r985546;
}
↓
double f(double x, double y, double z, double t) {
        double r985547 = z;
        double r985548 = -3.5507852856300137e+177;
        bool r985549 = r985547 <= r985548;
        double r985550 = x;
        double r985551 = y;
        double r985552 = t;
        double r985553 = r985551 / r985552;
        double r985554 = r985550 + r985553;
        double r985555 = 1.0;
        double r985556 = r985550 + r985555;
        double r985557 = r985554 / r985556;
        double r985558 = 9.561382895333631e-188;
        bool r985559 = r985547 <= r985558;
        double r985560 = r985551 * r985547;
        double r985561 = r985560 - r985550;
        double r985562 = r985552 * r985547;
        double r985563 = r985562 - r985550;
        double r985564 = r985561 / r985563;
        double r985565 = r985550 + r985564;
        double r985566 = 1.0;
        double r985567 = r985566 / r985556;
        double r985568 = r985565 * r985567;
        double r985569 = r985551 / r985563;
        double r985570 = fma(r985569, r985547, r985550);
        double r985571 = r985556 * r985566;
        double r985572 = r985570 / r985571;
        double r985573 = r985566 / r985563;
        double r985574 = r985550 * r985573;
        double r985575 = r985574 / r985556;
        double r985576 = r985572 - r985575;
        double r985577 = r985559 ? r985568 : r985576;
        double r985578 = r985549 ? r985557 : r985577;
        return r985578;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.1
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 3 regimes

  2. if z < -3.5507852856300137e+177

    1. Initial program 22.1

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

    2. Taylor expanded around inf 6.0

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

    if -3.5507852856300137e+177 < z < 9.561382895333631e-188

    1. Initial program 2.4

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

    3. Applied div-inv2.5

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

    if 9.561382895333631e-188 < z

    1. Initial program 9.2

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

    3. Applied div-sub9.2

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

    4. Applied associate-+r-9.2

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

    5. Applied div-sub9.2

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

    6. Simplified4.3

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

    8. Applied div-inv4.3

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

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.550785285630013736322572265977521807997 \cdot 10^{177}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 9.56138289533363135301744860674886759669 \cdot 10^{-188}:\\ \;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{x \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))