Average Error: 6.0 → 0.7
Time: 14.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.0552175198629431 \cdot 10^{51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{t - z}}, y, x\right)\\ \mathbf{elif}\;a \le 6.63097420197497467 \cdot 10^{28}:\\ \;\;\;\;\left(x + \frac{t \cdot y}{a}\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.0552175198629431 \cdot 10^{51}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{t - z}}, y, x\right)\\

\mathbf{elif}\;a \le 6.63097420197497467 \cdot 10^{28}:\\
\;\;\;\;\left(x + \frac{t \cdot y}{a}\right) - \frac{z \cdot y}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r320187 = x;
        double r320188 = y;
        double r320189 = z;
        double r320190 = t;
        double r320191 = r320189 - r320190;
        double r320192 = r320188 * r320191;
        double r320193 = a;
        double r320194 = r320192 / r320193;
        double r320195 = r320187 - r320194;
        return r320195;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r320196 = a;
        double r320197 = -2.055217519862943e+51;
        bool r320198 = r320196 <= r320197;
        double r320199 = 1.0;
        double r320200 = t;
        double r320201 = z;
        double r320202 = r320200 - r320201;
        double r320203 = r320196 / r320202;
        double r320204 = r320199 / r320203;
        double r320205 = y;
        double r320206 = x;
        double r320207 = fma(r320204, r320205, r320206);
        double r320208 = 6.630974201974975e+28;
        bool r320209 = r320196 <= r320208;
        double r320210 = r320200 * r320205;
        double r320211 = r320210 / r320196;
        double r320212 = r320206 + r320211;
        double r320213 = r320201 * r320205;
        double r320214 = r320213 / r320196;
        double r320215 = r320212 - r320214;
        double r320216 = r320202 / r320196;
        double r320217 = fma(r320216, r320205, r320206);
        double r320218 = r320209 ? r320215 : r320217;
        double r320219 = r320198 ? r320207 : r320218;
        return r320219;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -2.055217519862943e+51

    1. Initial program 9.9

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

    2. Simplified0.5

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

    4. Applied clear-num0.6

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

    if -2.055217519862943e+51 < a < 6.630974201974975e+28

    1. Initial program 1.0

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

    2. Simplified12.8

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

    4. Applied clear-num12.9

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

    6. Applied div-inv12.9

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

    7. Applied associate-/r*12.9

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

    8. Taylor expanded around inf 1.0

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

    if 6.630974201974975e+28 < a

    1. Initial program 10.0

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

    2. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.0552175198629431 \cdot 10^{51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{t - z}}, y, x\right)\\ \mathbf{elif}\;a \le 6.63097420197497467 \cdot 10^{28}:\\ \;\;\;\;\left(x + \frac{t \cdot y}{a}\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))