Average Error: 7.2 → 3.6
Time: 4.6s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.71987773590663113 \cdot 10^{-63} \lor \neg \left(x \le 4.88595928443932435 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t \cdot z - x} \cdot \sqrt[3]{t \cdot z - x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.71987773590663113 \cdot 10^{-63} \lor \neg \left(x \le 4.88595928443932435 \cdot 10^{-20}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t \cdot z - x} \cdot \sqrt[3]{t \cdot z - x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t \cdot z - x}}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r613529 = x;
        double r613530 = y;
        double r613531 = z;
        double r613532 = r613530 * r613531;
        double r613533 = r613532 - r613529;
        double r613534 = t;
        double r613535 = r613534 * r613531;
        double r613536 = r613535 - r613529;
        double r613537 = r613533 / r613536;
        double r613538 = r613529 + r613537;
        double r613539 = 1.0;
        double r613540 = r613529 + r613539;
        double r613541 = r613538 / r613540;
        return r613541;
}
↓
double f(double x, double y, double z, double t) {
        double r613542 = x;
        double r613543 = -1.719877735906631e-63;
        bool r613544 = r613542 <= r613543;
        double r613545 = 4.8859592844393243e-20;
        bool r613546 = r613542 <= r613545;
        double r613547 = !r613546;
        bool r613548 = r613544 || r613547;
        double r613549 = y;
        double r613550 = t;
        double r613551 = z;
        double r613552 = r613550 * r613551;
        double r613553 = r613552 - r613542;
        double r613554 = r613549 / r613553;
        double r613555 = fma(r613554, r613551, r613542);
        double r613556 = 1.0;
        double r613557 = r613542 + r613556;
        double r613558 = 1.0;
        double r613559 = r613557 * r613558;
        double r613560 = r613555 / r613559;
        double r613561 = cbrt(r613542);
        double r613562 = r613561 * r613561;
        double r613563 = cbrt(r613553);
        double r613564 = r613563 * r613563;
        double r613565 = r613562 / r613564;
        double r613566 = r613561 / r613563;
        double r613567 = r613565 * r613566;
        double r613568 = r613567 / r613557;
        double r613569 = r613560 - r613568;
        double r613570 = r613549 * r613551;
        double r613571 = r613570 - r613542;
        double r613572 = r613558 / r613553;
        double r613573 = r613571 * r613572;
        double r613574 = r613542 + r613573;
        double r613575 = r613574 / r613557;
        double r613576 = r613548 ? r613569 : r613575;
        return r613576;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.2
Target0.4
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 x < -1.719877735906631e-63 or 4.8859592844393243e-20 < x

    1. Initial program 7.3

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

    3. Applied div-sub7.3

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

    4. Applied associate-+r-7.3

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

    5. Applied div-sub7.3

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

    6. Simplified1.1

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

    8. Applied add-cube-cbrt1.1

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{\color{blue}{\left(\sqrt[3]{t \cdot z - x} \cdot \sqrt[3]{t \cdot z - x}\right) \cdot \sqrt[3]{t \cdot z - x}}}}{x + 1}\]

    9. Applied add-cube-cbrt1.1

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{t \cdot z - x} \cdot \sqrt[3]{t \cdot z - x}\right) \cdot \sqrt[3]{t \cdot z - x}}}{x + 1}\]

    10. Applied times-frac1.1

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t \cdot z - x} \cdot \sqrt[3]{t \cdot z - x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t \cdot z - x}}}}{x + 1}\]

    if -1.719877735906631e-63 < x < 4.8859592844393243e-20

    1. Initial program 7.0

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

    3. Applied div-inv7.1

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

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.71987773590663113 \cdot 10^{-63} \lor \neg \left(x \le 4.88595928443932435 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t \cdot z - x} \cdot \sqrt[3]{t \cdot z - x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))