Average Error: 6.1 → 0.5
Time: 18.3s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.4996251390377065 \cdot 10^{+235}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 3.7199348004809193 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \left(\left(z - t\right) \cdot y\right) \cdot \frac{-1}{a}\right) + \mathsf{fma}\left(\frac{-1}{a}, \left(z - t\right) \cdot y, \left(\left(z - t\right) \cdot y\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{t}{a} - \frac{z}{a}\right) + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -1.4996251390377065 \cdot 10^{+235}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le 3.7199348004809193 \cdot 10^{+172}:\\
\;\;\;\;\mathsf{fma}\left(1, x, \left(\left(z - t\right) \cdot y\right) \cdot \frac{-1}{a}\right) + \mathsf{fma}\left(\frac{-1}{a}, \left(z - t\right) \cdot y, \left(\left(z - t\right) \cdot y\right) \cdot \frac{1}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;y \cdot \left(\frac{t}{a} - \frac{z}{a}\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13093209 = x;
        double r13093210 = y;
        double r13093211 = z;
        double r13093212 = t;
        double r13093213 = r13093211 - r13093212;
        double r13093214 = r13093210 * r13093213;
        double r13093215 = a;
        double r13093216 = r13093214 / r13093215;
        double r13093217 = r13093209 - r13093216;
        return r13093217;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13093218 = z;
        double r13093219 = t;
        double r13093220 = r13093218 - r13093219;
        double r13093221 = y;
        double r13093222 = r13093220 * r13093221;
        double r13093223 = -1.4996251390377065e+235;
        bool r13093224 = r13093222 <= r13093223;
        double r13093225 = x;
        double r13093226 = a;
        double r13093227 = r13093226 / r13093220;
        double r13093228 = r13093221 / r13093227;
        double r13093229 = r13093225 - r13093228;
        double r13093230 = 3.7199348004809193e+172;
        bool r13093231 = r13093222 <= r13093230;
        double r13093232 = 1.0;
        double r13093233 = -1.0;
        double r13093234 = r13093233 / r13093226;
        double r13093235 = r13093222 * r13093234;
        double r13093236 = fma(r13093232, r13093225, r13093235);
        double r13093237 = r13093232 / r13093226;
        double r13093238 = r13093222 * r13093237;
        double r13093239 = fma(r13093234, r13093222, r13093238);
        double r13093240 = r13093236 + r13093239;
        double r13093241 = r13093219 / r13093226;
        double r13093242 = r13093218 / r13093226;
        double r13093243 = r13093241 - r13093242;
        double r13093244 = r13093221 * r13093243;
        double r13093245 = r13093244 + r13093225;
        double r13093246 = r13093231 ? r13093240 : r13093245;
        double r13093247 = r13093224 ? r13093229 : r13093246;
        return r13093247;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.4996251390377065e+235

    1. Initial program 36.3

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

    3. Applied associate-/l*0.2

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

    if -1.4996251390377065e+235 < (* y (- z t)) < 3.7199348004809193e+172

    1. Initial program 0.4

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

    3. Applied div-inv0.4

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

    4. Applied *-un-lft-identity0.4

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

    5. Applied prod-diff0.4

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

    if 3.7199348004809193e+172 < (* y (- z t))

    1. Initial program 23.2

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

    2. Taylor expanded around 0 23.2

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

    3. Simplified1.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.4996251390377065 \cdot 10^{+235}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 3.7199348004809193 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \left(\left(z - t\right) \cdot y\right) \cdot \frac{-1}{a}\right) + \mathsf{fma}\left(\frac{-1}{a}, \left(z - t\right) \cdot y, \left(\left(z - t\right) \cdot y\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{t}{a} - \frac{z}{a}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :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)))