Average Error: 6.0 → 0.7
Time: 9.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.7422788828719246 \cdot 10^{161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.424193312752178 \cdot 10^{192}:\\ \;\;\;\;x + \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.7422788828719246 \cdot 10^{161}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.424193312752178 \cdot 10^{192}:\\
\;\;\;\;x + \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r382209 = x;
        double r382210 = y;
        double r382211 = z;
        double r382212 = t;
        double r382213 = r382211 - r382212;
        double r382214 = r382210 * r382213;
        double r382215 = a;
        double r382216 = r382214 / r382215;
        double r382217 = r382209 + r382216;
        return r382217;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r382218 = y;
        double r382219 = z;
        double r382220 = t;
        double r382221 = r382219 - r382220;
        double r382222 = r382218 * r382221;
        double r382223 = -3.7422788828719246e+161;
        bool r382224 = r382222 <= r382223;
        double r382225 = a;
        double r382226 = r382221 / r382225;
        double r382227 = x;
        double r382228 = fma(r382226, r382218, r382227);
        double r382229 = 2.424193312752178e+192;
        bool r382230 = r382222 <= r382229;
        double r382231 = 1.0;
        double r382232 = r382225 / r382222;
        double r382233 = r382231 / r382232;
        double r382234 = r382227 + r382233;
        double r382235 = r382218 / r382225;
        double r382236 = fma(r382235, r382221, r382227);
        double r382237 = r382230 ? r382234 : r382236;
        double r382238 = r382224 ? r382228 : r382237;
        return r382238;
}

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 (* y (- z t)) < -3.7422788828719246e+161

    1. Initial program 21.2

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

    2. Taylor expanded around 0 21.2

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

    3. Simplified1.3

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

    4. Taylor expanded around 0 21.2

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

    5. Simplified1.7

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

    if -3.7422788828719246e+161 < (* y (- z t)) < 2.424193312752178e+192

    1. Initial program 0.4

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

    3. Applied clear-num0.4

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

    if 2.424193312752178e+192 < (* y (- z t))

    1. Initial program 26.2

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

    2. Simplified1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.7422788828719246 \cdot 10^{161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.424193312752178 \cdot 10^{192}:\\ \;\;\;\;x + \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]

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, E"
  :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)))