FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2

Average Error: 37.5 → 26.3

Time: 10.6s

Precision: 64

Math

\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.568144212234696205195188572784059781203 \cdot 10^{245}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le -4.327299985686750281759353025814658479884 \cdot 10^{237}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \le -2.103726633190398316891354491617718523432 \cdot 10^{78}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le 9.751071527140869009283550824686405505817 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\ \mathbf{elif}\;z \le 2.515595516499310151012610474474805715008 \cdot 10^{-188}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \le 6.991472626016894069691041216962870797951 \cdot 10^{102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r536465 = x;
        double r536466 = r536465 * r536465;
        double r536467 = y;
        double r536468 = r536467 * r536467;
        double r536469 = r536466 + r536468;
        double r536470 = z;
        double r536471 = r536470 * r536470;
        double r536472 = r536469 + r536471;
        double r536473 = sqrt(r536472);
        return r536473;
}

↓

double f(double x, double y, double z) {
        double r536474 = z;
        double r536475 = -8.568144212234696e+245;
        bool r536476 = r536474 <= r536475;
        double r536477 = -r536474;
        double r536478 = -4.32729998568675e+237;
        bool r536479 = r536474 <= r536478;
        double r536480 = y;
        double r536481 = -2.1037266331903983e+78;
        bool r536482 = r536474 <= r536481;
        double r536483 = 9.751071527140869e-230;
        bool r536484 = r536474 <= r536483;
        double r536485 = x;
        double r536486 = r536480 * r536480;
        double r536487 = fma(r536485, r536485, r536486);
        double r536488 = fma(r536474, r536474, r536487);
        double r536489 = sqrt(r536488);
        double r536490 = 2.51559551649931e-188;
        bool r536491 = r536474 <= r536490;
        double r536492 = 6.991472626016894e+102;
        bool r536493 = r536474 <= r536492;
        double r536494 = r536493 ? r536489 : r536474;
        double r536495 = r536491 ? r536480 : r536494;
        double r536496 = r536484 ? r536489 : r536495;
        double r536497 = r536482 ? r536477 : r536496;
        double r536498 = r536479 ? r536480 : r536497;
        double r536499 = r536476 ? r536477 : r536498;
        return r536499;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	37.5
Target	25.3
Herbie	26.3

\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -8.568144212234696e+245 or -4.32729998568675e+237 < z < -2.1037266331903983e+78

   1. Initial program 51.1

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

   2. Simplified 51.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}}\]

   3. Taylor expanded around -inf 20.5

      \[\leadsto \color{blue}{-1 \cdot z}\]

   4. Simplified 20.5

      \[\leadsto \color{blue}{-z}\]

   if -8.568144212234696e+245 < z < -4.32729998568675e+237 or 9.751071527140869e-230 < z < 2.51559551649931e-188

   1. Initial program 36.1

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

   2. Simplified 36.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}}\]

   3. Taylor expanded around 0 47.3

      \[\leadsto \color{blue}{y}\]

   if -2.1037266331903983e+78 < z < 9.751071527140869e-230 or 2.51559551649931e-188 < z < 6.991472626016894e+102

   1. Initial program 29.0

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

   2. Simplified 29.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}}\]

   if 6.991472626016894e+102 < z

   1. Initial program 54.7

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

   2. Simplified 54.7

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}}\]

   3. Taylor expanded around inf 18.1

      \[\leadsto \color{blue}{z}\]
3. Recombined 4 regimes into one program.

4. Final simplification 26.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.568144212234696205195188572784059781203 \cdot 10^{245}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le -4.327299985686750281759353025814658479884 \cdot 10^{237}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \le -2.103726633190398316891354491617718523432 \cdot 10^{78}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le 9.751071527140869009283550824686405505817 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\ \mathbf{elif}\;z \le 2.515595516499310151012610474474805715008 \cdot 10^{-188}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \le 6.991472626016894069691041216962870797951 \cdot 10^{102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))