Average Error: 7.0 → 2.6
Time: 18.8s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.129992465558795124685168747083463536285 \cdot 10^{-8} \lor \neg \left(z \le 1.105778529535857557068218568937650049807 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{x + 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{t \cdot z - x}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.129992465558795124685168747083463536285 \cdot 10^{-8} \lor \neg \left(z \le 1.105778529535857557068218568937650049807 \cdot 10^{-83}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{x + 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r479265 = x;
        double r479266 = y;
        double r479267 = z;
        double r479268 = r479266 * r479267;
        double r479269 = r479268 - r479265;
        double r479270 = t;
        double r479271 = r479270 * r479267;
        double r479272 = r479271 - r479265;
        double r479273 = r479269 / r479272;
        double r479274 = r479265 + r479273;
        double r479275 = 1.0;
        double r479276 = r479265 + r479275;
        double r479277 = r479274 / r479276;
        return r479277;
}
↓
double f(double x, double y, double z, double t) {
        double r479278 = z;
        double r479279 = -2.129992465558795e-08;
        bool r479280 = r479278 <= r479279;
        double r479281 = 1.1057785295358576e-83;
        bool r479282 = r479278 <= r479281;
        double r479283 = !r479282;
        bool r479284 = r479280 || r479283;
        double r479285 = y;
        double r479286 = t;
        double r479287 = r479286 * r479278;
        double r479288 = x;
        double r479289 = r479287 - r479288;
        double r479290 = r479285 / r479289;
        double r479291 = fma(r479290, r479278, r479288);
        double r479292 = 1.0;
        double r479293 = r479288 + r479292;
        double r479294 = r479291 / r479293;
        double r479295 = r479288 / r479289;
        double r479296 = 3.0;
        double r479297 = pow(r479295, r479296);
        double r479298 = cbrt(r479297);
        double r479299 = r479298 / r479293;
        double r479300 = r479294 - r479299;
        double r479301 = 1.0;
        double r479302 = -r479288;
        double r479303 = fma(r479285, r479278, r479302);
        double r479304 = r479303 / r479289;
        double r479305 = r479288 + r479304;
        double r479306 = r479293 / r479305;
        double r479307 = r479301 / r479306;
        double r479308 = r479284 ? r479300 : r479307;
        return r479308;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.0
Target0.3
Herbie2.6
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.129992465558795e-08 or 1.1057785295358576e-83 < z

    1. Initial program 12.6

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

    3. Applied div-sub12.6

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

    4. Applied associate-+r-12.6

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

    5. Applied div-sub12.6

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

    6. Simplified4.4

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

    8. Applied add-cbrt-cube6.4

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

    9. Applied add-cbrt-cube30.6

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

    10. Applied cbrt-undiv30.6

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

    11. Simplified4.6

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

    if -2.129992465558795e-08 < z < 1.1057785295358576e-83

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    5. Applied clear-num0.2

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

    6. Simplified0.1

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.129992465558795124685168747083463536285 \cdot 10^{-8} \lor \neg \left(z \le 1.105778529535857557068218568937650049807 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{x + 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{t \cdot z - x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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)))