Average Error: 5.8 → 0.4
Time: 3.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.22194845951310805539173405538279367801 \cdot 10^{299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 8.372301841351568798694844482890536564686 \cdot 10^{181}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.22194845951310805539173405538279367801 \cdot 10^{299}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 8.372301841351568798694844482890536564686 \cdot 10^{181}:\\
\;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r387160 = x;
        double r387161 = y;
        double r387162 = z;
        double r387163 = t;
        double r387164 = r387162 - r387163;
        double r387165 = r387161 * r387164;
        double r387166 = a;
        double r387167 = r387165 / r387166;
        double r387168 = r387160 - r387167;
        return r387168;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r387169 = y;
        double r387170 = z;
        double r387171 = t;
        double r387172 = r387170 - r387171;
        double r387173 = r387169 * r387172;
        double r387174 = -3.221948459513108e+299;
        bool r387175 = r387173 <= r387174;
        double r387176 = a;
        double r387177 = r387169 / r387176;
        double r387178 = r387171 - r387170;
        double r387179 = x;
        double r387180 = fma(r387177, r387178, r387179);
        double r387181 = 8.372301841351569e+181;
        bool r387182 = r387173 <= r387181;
        double r387183 = 1.0;
        double r387184 = r387183 / r387176;
        double r387185 = r387173 * r387184;
        double r387186 = r387179 - r387185;
        double r387187 = r387176 / r387172;
        double r387188 = r387169 / r387187;
        double r387189 = r387179 - r387188;
        double r387190 = r387182 ? r387186 : r387189;
        double r387191 = r387175 ? r387180 : r387190;
        return r387191;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.8
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)) < -3.221948459513108e+299

    1. Initial program 59.4

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

    2. Simplified0.2

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

    if -3.221948459513108e+299 < (* y (- z t)) < 8.372301841351569e+181

    1. Initial program 0.3

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied div-inv0.4

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

    if 8.372301841351569e+181 < (* y (- z t))

    1. Initial program 23.4

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied associate-/l*0.9

      \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.22194845951310805539173405538279367801 \cdot 10^{299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 8.372301841351568798694844482890536564686 \cdot 10^{181}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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 -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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