Average Error: 6.1 → 0.5
Time: 19.0s
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 r14493537 = x;
        double r14493538 = y;
        double r14493539 = z;
        double r14493540 = t;
        double r14493541 = r14493539 - r14493540;
        double r14493542 = r14493538 * r14493541;
        double r14493543 = a;
        double r14493544 = r14493542 / r14493543;
        double r14493545 = r14493537 - r14493544;
        return r14493545;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14493546 = z;
        double r14493547 = t;
        double r14493548 = r14493546 - r14493547;
        double r14493549 = y;
        double r14493550 = r14493548 * r14493549;
        double r14493551 = -1.4996251390377065e+235;
        bool r14493552 = r14493550 <= r14493551;
        double r14493553 = x;
        double r14493554 = a;
        double r14493555 = r14493554 / r14493548;
        double r14493556 = r14493549 / r14493555;
        double r14493557 = r14493553 - r14493556;
        double r14493558 = 3.7199348004809193e+172;
        bool r14493559 = r14493550 <= r14493558;
        double r14493560 = 1.0;
        double r14493561 = -1.0;
        double r14493562 = r14493561 / r14493554;
        double r14493563 = r14493550 * r14493562;
        double r14493564 = fma(r14493560, r14493553, r14493563);
        double r14493565 = r14493560 / r14493554;
        double r14493566 = r14493550 * r14493565;
        double r14493567 = fma(r14493562, r14493550, r14493566);
        double r14493568 = r14493564 + r14493567;
        double r14493569 = r14493547 / r14493554;
        double r14493570 = r14493546 / r14493554;
        double r14493571 = r14493569 - r14493570;
        double r14493572 = r14493549 * r14493571;
        double r14493573 = r14493572 + r14493553;
        double r14493574 = r14493559 ? r14493568 : r14493573;
        double r14493575 = r14493552 ? r14493557 : r14493574;
        return r14493575;
}

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)))