Average Error: 7.5 → 3.7
Time: 20.7s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.440078247488523162933428867079497205001 \cdot 10^{157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \sqrt[3]{\frac{x}{t \cdot z - x} \cdot \left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1}\\ \mathbf{elif}\;z \le 1.514816533856499451078635282534599335268 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t \cdot z - x} \cdot y, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.440078247488523162933428867079497205001 \cdot 10^{157}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \sqrt[3]{\frac{x}{t \cdot z - x} \cdot \left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1}\\

\mathbf{elif}\;z \le 1.514816533856499451078635282534599335268 \cdot 10^{-166}:\\
\;\;\;\;\frac{\frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)} + x}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r26218279 = x;
        double r26218280 = y;
        double r26218281 = z;
        double r26218282 = r26218280 * r26218281;
        double r26218283 = r26218282 - r26218279;
        double r26218284 = t;
        double r26218285 = r26218284 * r26218281;
        double r26218286 = r26218285 - r26218279;
        double r26218287 = r26218283 / r26218286;
        double r26218288 = r26218279 + r26218287;
        double r26218289 = 1.0;
        double r26218290 = r26218279 + r26218289;
        double r26218291 = r26218288 / r26218290;
        return r26218291;
}
↓
double f(double x, double y, double z, double t) {
        double r26218292 = z;
        double r26218293 = -5.440078247488523e+157;
        bool r26218294 = r26218292 <= r26218293;
        double r26218295 = y;
        double r26218296 = t;
        double r26218297 = r26218296 * r26218292;
        double r26218298 = x;
        double r26218299 = r26218297 - r26218298;
        double r26218300 = r26218295 / r26218299;
        double r26218301 = fma(r26218300, r26218292, r26218298);
        double r26218302 = r26218298 / r26218299;
        double r26218303 = r26218302 * r26218302;
        double r26218304 = r26218302 * r26218303;
        double r26218305 = cbrt(r26218304);
        double r26218306 = r26218301 - r26218305;
        double r26218307 = 1.0;
        double r26218308 = r26218298 + r26218307;
        double r26218309 = r26218306 / r26218308;
        double r26218310 = 1.5148165338564995e-166;
        bool r26218311 = r26218292 <= r26218310;
        double r26218312 = r26218295 * r26218292;
        double r26218313 = r26218312 - r26218298;
        double r26218314 = -r26218298;
        double r26218315 = fma(r26218296, r26218292, r26218314);
        double r26218316 = r26218313 / r26218315;
        double r26218317 = r26218316 + r26218298;
        double r26218318 = r26218317 / r26218308;
        double r26218319 = 1.0;
        double r26218320 = r26218319 / r26218299;
        double r26218321 = r26218320 * r26218295;
        double r26218322 = fma(r26218321, r26218292, r26218298);
        double r26218323 = r26218322 - r26218302;
        double r26218324 = r26218323 / r26218308;
        double r26218325 = r26218311 ? r26218318 : r26218324;
        double r26218326 = r26218294 ? r26218309 : r26218325;
        return r26218326;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.5
Target0.4
Herbie3.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.440078247488523e+157

    1. Initial program 23.5

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

    3. Applied div-sub23.5

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

    4. Applied associate-+r-23.5

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

    5. Simplified9.1

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

    7. Applied add-cbrt-cube9.1

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

    if -5.440078247488523e+157 < z < 1.5148165338564995e-166

    1. Initial program 1.8

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

    3. Applied fma-neg1.8

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

    if 1.5148165338564995e-166 < z

    1. Initial program 9.7

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

    3. Applied div-sub9.7

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

    4. Applied associate-+r-9.7

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

    5. Simplified4.4

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

    7. Applied div-inv4.5

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.440078247488523162933428867079497205001 \cdot 10^{157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \sqrt[3]{\frac{x}{t \cdot z - x} \cdot \left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1}\\ \mathbf{elif}\;z \le 1.514816533856499451078635282534599335268 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t \cdot z - x} \cdot y, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))