Average Error: 7.1 → 4.2
Time: 4.0s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.329769589365412156618691488056306963715 \cdot 10^{-21}:\\ \;\;\;\;1 \cdot \frac{\mathsf{fma}\left(y \cdot \frac{1}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;x \le -1.087436073798729316230006245761988192998 \cdot 10^{-304}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.329769589365412156618691488056306963715 \cdot 10^{-21}:\\
\;\;\;\;1 \cdot \frac{\mathsf{fma}\left(y \cdot \frac{1}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\

\mathbf{elif}\;x \le -1.087436073798729316230006245761988192998 \cdot 10^{-304}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r599311 = x;
        double r599312 = y;
        double r599313 = z;
        double r599314 = r599312 * r599313;
        double r599315 = r599314 - r599311;
        double r599316 = t;
        double r599317 = r599316 * r599313;
        double r599318 = r599317 - r599311;
        double r599319 = r599315 / r599318;
        double r599320 = r599311 + r599319;
        double r599321 = 1.0;
        double r599322 = r599311 + r599321;
        double r599323 = r599320 / r599322;
        return r599323;
}
↓
double f(double x, double y, double z, double t) {
        double r599324 = x;
        double r599325 = -4.329769589365412e-21;
        bool r599326 = r599324 <= r599325;
        double r599327 = 1.0;
        double r599328 = y;
        double r599329 = t;
        double r599330 = z;
        double r599331 = r599329 * r599330;
        double r599332 = r599331 - r599324;
        double r599333 = r599327 / r599332;
        double r599334 = r599328 * r599333;
        double r599335 = fma(r599334, r599330, r599324);
        double r599336 = 1.0;
        double r599337 = r599324 + r599336;
        double r599338 = r599337 * r599327;
        double r599339 = r599335 / r599338;
        double r599340 = r599327 * r599339;
        double r599341 = r599324 / r599332;
        double r599342 = r599341 / r599337;
        double r599343 = r599340 - r599342;
        double r599344 = -1.0874360737987293e-304;
        bool r599345 = r599324 <= r599344;
        double r599346 = r599328 * r599330;
        double r599347 = r599346 - r599324;
        double r599348 = r599347 * r599333;
        double r599349 = r599324 + r599348;
        double r599350 = r599349 / r599337;
        double r599351 = r599328 / r599332;
        double r599352 = fma(r599351, r599330, r599324);
        double r599353 = r599337 / r599352;
        double r599354 = r599327 / r599353;
        double r599355 = r599327 * r599354;
        double r599356 = r599355 - r599342;
        double r599357 = r599345 ? r599350 : r599356;
        double r599358 = r599326 ? r599343 : r599357;
        return r599358;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.1
Target0.4
Herbie4.2
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.329769589365412e-21

    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. Simplified1.2

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

    8. Applied *-un-lft-identity1.2

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

    10. Applied div-inv1.2

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

    if -4.329769589365412e-21 < x < -1.0874360737987293e-304

    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-inv7.4

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

    if -1.0874360737987293e-304 < x

    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-sub7.0

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

    4. Applied associate-+r-7.0

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

    5. Applied div-sub7.0

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

    6. Simplified4.5

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

    8. Applied *-un-lft-identity4.5

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

    10. Applied clear-num4.6

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

    11. Simplified4.6

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.329769589365412156618691488056306963715 \cdot 10^{-21}:\\ \;\;\;\;1 \cdot \frac{\mathsf{fma}\left(y \cdot \frac{1}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;x \le -1.087436073798729316230006245761988192998 \cdot 10^{-304}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))