Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Average Error: 6.7 → 1.1

Time: 18.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -2.72790853210520445783593734651044130646 \cdot 10^{251}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le 6.41978194333992067450854588504563580042 \cdot 10^{300}:\\ \;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r16518210 = x;
        double r16518211 = y;
        double r16518212 = z;
        double r16518213 = r16518212 - r16518210;
        double r16518214 = r16518211 * r16518213;
        double r16518215 = t;
        double r16518216 = r16518214 / r16518215;
        double r16518217 = r16518210 + r16518216;
        return r16518217;
}

↓

double f(double x, double y, double z, double t) {
        double r16518218 = x;
        double r16518219 = z;
        double r16518220 = r16518219 - r16518218;
        double r16518221 = y;
        double r16518222 = r16518220 * r16518221;
        double r16518223 = t;
        double r16518224 = r16518222 / r16518223;
        double r16518225 = r16518218 + r16518224;
        double r16518226 = -2.7279085321052045e+251;
        bool r16518227 = r16518225 <= r16518226;
        double r16518228 = r16518221 / r16518223;
        double r16518229 = fma(r16518228, r16518220, r16518218);
        double r16518230 = 6.419781943339921e+300;
        bool r16518231 = r16518225 <= r16518230;
        double r16518232 = r16518220 / r16518223;
        double r16518233 = fma(r16518232, r16518221, r16518218);
        double r16518234 = r16518231 ? r16518225 : r16518233;
        double r16518235 = r16518227 ? r16518229 : r16518234;
        return r16518235;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.7
Target	1.9
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* y (- z x)) t)) < -2.7279085321052045e+251

   1. Initial program 31.1

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

   2. Taylor expanded around 0 31.1

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

   3. Simplified 2.6

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

   if -2.7279085321052045e+251 < (+ x (/ (* y (- z x)) t)) < 6.419781943339921e+300

   1. Initial program 0.8

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

   if 6.419781943339921e+300 < (+ x (/ (* y (- z x)) t))

   1. Initial program 56.1

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

   2. Simplified 3.4

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -2.72790853210520445783593734651044130646 \cdot 10^{251}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le 6.41978194333992067450854588504563580042 \cdot 10^{300}:\\ \;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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