Average Error: 6.7 → 0.4
Time: 13.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.035064490176370167637368676280506465886 \cdot 10^{200}:\\ \;\;\;\;\frac{t - z}{\frac{a}{y}} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.143948490603520686127081681315653192418 \cdot 10^{204}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.035064490176370167637368676280506465886 \cdot 10^{200}:\\
\;\;\;\;\frac{t - z}{\frac{a}{y}} + x\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.143948490603520686127081681315653192418 \cdot 10^{204}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r254319 = x;
        double r254320 = y;
        double r254321 = z;
        double r254322 = t;
        double r254323 = r254321 - r254322;
        double r254324 = r254320 * r254323;
        double r254325 = a;
        double r254326 = r254324 / r254325;
        double r254327 = r254319 - r254326;
        return r254327;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r254328 = y;
        double r254329 = z;
        double r254330 = t;
        double r254331 = r254329 - r254330;
        double r254332 = r254328 * r254331;
        double r254333 = -2.03506449017637e+200;
        bool r254334 = r254332 <= r254333;
        double r254335 = r254330 - r254329;
        double r254336 = a;
        double r254337 = r254336 / r254328;
        double r254338 = r254335 / r254337;
        double r254339 = x;
        double r254340 = r254338 + r254339;
        double r254341 = 2.1439484906035207e+204;
        bool r254342 = r254332 <= r254341;
        double r254343 = r254332 / r254336;
        double r254344 = r254339 - r254343;
        double r254345 = r254328 / r254336;
        double r254346 = fma(r254345, r254335, r254339);
        double r254347 = r254342 ? r254344 : r254346;
        double r254348 = r254334 ? r254340 : r254347;
        return r254348;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.7
Target0.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -2.03506449017637e+200

    1. Initial program 30.1

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

    2. Simplified0.5

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

    4. Applied fma-udef0.5

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

    5. Simplified0.5

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

    if -2.03506449017637e+200 < (* y (- z t)) < 2.1439484906035207e+204

    1. Initial program 0.4

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

    if 2.1439484906035207e+204 < (* y (- z t))

    1. Initial program 31.6

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

    2. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.035064490176370167637368676280506465886 \cdot 10^{200}:\\ \;\;\;\;\frac{t - z}{\frac{a}{y}} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.143948490603520686127081681315653192418 \cdot 10^{204}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))