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

Average Error: 6.3 → 0.6

Time: 13.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.386011167286209797538090785742809716412 \cdot 10^{-28} \lor \neg \left(a \le 4635323820368954973487104\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r208030 = x;
        double r208031 = y;
        double r208032 = z;
        double r208033 = t;
        double r208034 = r208032 - r208033;
        double r208035 = r208031 * r208034;
        double r208036 = a;
        double r208037 = r208035 / r208036;
        double r208038 = r208030 + r208037;
        return r208038;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r208039 = a;
        double r208040 = -5.38601116728621e-28;
        bool r208041 = r208039 <= r208040;
        double r208042 = 4.635323820368955e+24;
        bool r208043 = r208039 <= r208042;
        double r208044 = !r208043;
        bool r208045 = r208041 || r208044;
        double r208046 = x;
        double r208047 = y;
        double r208048 = z;
        double r208049 = t;
        double r208050 = r208048 - r208049;
        double r208051 = r208050 / r208039;
        double r208052 = r208047 * r208051;
        double r208053 = r208046 + r208052;
        double r208054 = r208047 * r208048;
        double r208055 = -r208049;
        double r208056 = r208047 * r208055;
        double r208057 = r208054 + r208056;
        double r208058 = r208057 / r208039;
        double r208059 = r208046 + r208058;
        double r208060 = r208045 ? r208053 : r208059;
        return r208060;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.3
Target	0.6
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -5.38601116728621e-28 or 4.635323820368955e+24 < a

   1. Initial program 9.8

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

   3. Applied *-un-lft-identity 9.8

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

   4. Applied times-frac 0.6

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

   5. Simplified 0.6

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

   if -5.38601116728621e-28 < a < 4.635323820368955e+24

   1. Initial program 0.6

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

   3. Applied sub-neg 0.6

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

   4. Applied distribute-lft-in 0.6

      \[\leadsto x + \frac{\color{blue}{y \cdot z + y \cdot \left(-t\right)}}{a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.386011167286209797538090785742809716412 \cdot 10^{-28} \lor \neg \left(a \le 4635323820368954973487104\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))