Average Error: 7.6 → 4.3
Time: 5.2s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.07004524106268236 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(\frac{y}{t \cdot z - x} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} + x}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;x \le 3.9751503704831254 \cdot 10^{-120}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.07004524106268236 \cdot 10^{-305}:\\
\;\;\;\;\frac{\left(\frac{y}{t \cdot z - x} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} + x}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\

\mathbf{elif}\;x \le 3.9751503704831254 \cdot 10^{-120}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r785436 = x;
        double r785437 = y;
        double r785438 = z;
        double r785439 = r785437 * r785438;
        double r785440 = r785439 - r785436;
        double r785441 = t;
        double r785442 = r785441 * r785438;
        double r785443 = r785442 - r785436;
        double r785444 = r785440 / r785443;
        double r785445 = r785436 + r785444;
        double r785446 = 1.0;
        double r785447 = r785436 + r785446;
        double r785448 = r785445 / r785447;
        return r785448;
}
↓
double f(double x, double y, double z, double t) {
        double r785449 = x;
        double r785450 = -2.0700452410626824e-305;
        bool r785451 = r785449 <= r785450;
        double r785452 = y;
        double r785453 = t;
        double r785454 = z;
        double r785455 = r785453 * r785454;
        double r785456 = r785455 - r785449;
        double r785457 = r785452 / r785456;
        double r785458 = cbrt(r785454);
        double r785459 = r785458 * r785458;
        double r785460 = r785457 * r785459;
        double r785461 = r785460 * r785458;
        double r785462 = r785461 + r785449;
        double r785463 = 1.0;
        double r785464 = r785449 + r785463;
        double r785465 = 1.0;
        double r785466 = r785464 * r785465;
        double r785467 = r785462 / r785466;
        double r785468 = r785449 / r785456;
        double r785469 = r785468 / r785464;
        double r785470 = r785467 - r785469;
        double r785471 = 3.9751503704831254e-120;
        bool r785472 = r785449 <= r785471;
        double r785473 = r785452 * r785454;
        double r785474 = r785473 - r785449;
        double r785475 = r785474 / r785456;
        double r785476 = r785449 + r785475;
        double r785477 = r785449 * r785449;
        double r785478 = r785463 * r785463;
        double r785479 = r785477 - r785478;
        double r785480 = r785476 / r785479;
        double r785481 = r785449 - r785463;
        double r785482 = r785480 * r785481;
        double r785483 = fma(r785457, r785454, r785449);
        double r785484 = r785483 / r785466;
        double r785485 = 3.0;
        double r785486 = pow(r785468, r785485);
        double r785487 = cbrt(r785486);
        double r785488 = r785487 / r785464;
        double r785489 = r785484 - r785488;
        double r785490 = r785472 ? r785482 : r785489;
        double r785491 = r785451 ? r785470 : r785490;
        return r785491;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.6
Target0.4
Herbie4.3
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.0700452410626824e-305

    1. Initial program 7.8

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

    3. Applied div-sub7.8

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

    4. Applied associate-+r-7.8

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

    5. Applied div-sub7.8

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

    6. Simplified4.8

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

    8. Applied fma-udef4.8

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

    10. Applied add-cube-cbrt5.0

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

    11. Applied associate-*r*5.0

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

    if -2.0700452410626824e-305 < x < 3.9751503704831254e-120

    1. Initial program 7.7

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

    3. Applied flip-+7.7

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

    4. Applied associate-/r/7.7

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

    if 3.9751503704831254e-120 < x

    1. Initial program 7.2

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

    3. Applied div-sub7.2

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

    4. Applied associate-+r-7.2

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

    5. Applied div-sub7.2

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

    6. Simplified1.9

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

    8. Applied add-cbrt-cube2.3

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

    9. Applied add-cbrt-cube34.0

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

    10. Applied cbrt-undiv34.0

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

    11. Simplified1.9

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.07004524106268236 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(\frac{y}{t \cdot z - x} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} + x}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;x \le 3.9751503704831254 \cdot 10^{-120}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))