Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Average Error: 29.3 → 1.3

Time: 16.9s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1889722895859.1785:\\ \;\;\;\;x + y \cdot \left(\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}\right)\\ \mathbf{elif}\;z \le 9.640368061021876 \cdot 10^{+38}:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(a + \left(z \cdot \left(3.13060547623 \cdot z + 11.1667541262\right) + t\right) \cdot z\right) + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y \cdot 3.13060547623 - 36.527041698806414 \cdot \frac{y}{z}\right) + \frac{t}{z} \cdot \frac{y}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r9504204 = x;
        double r9504205 = y;
        double r9504206 = z;
        double r9504207 = 3.13060547623;
        double r9504208 = r9504206 * r9504207;
        double r9504209 = 11.1667541262;
        double r9504210 = r9504208 + r9504209;
        double r9504211 = r9504210 * r9504206;
        double r9504212 = t;
        double r9504213 = r9504211 + r9504212;
        double r9504214 = r9504213 * r9504206;
        double r9504215 = a;
        double r9504216 = r9504214 + r9504215;
        double r9504217 = r9504216 * r9504206;
        double r9504218 = b;
        double r9504219 = r9504217 + r9504218;
        double r9504220 = r9504205 * r9504219;
        double r9504221 = 15.234687407;
        double r9504222 = r9504206 + r9504221;
        double r9504223 = r9504222 * r9504206;
        double r9504224 = 31.4690115749;
        double r9504225 = r9504223 + r9504224;
        double r9504226 = r9504225 * r9504206;
        double r9504227 = 11.9400905721;
        double r9504228 = r9504226 + r9504227;
        double r9504229 = r9504228 * r9504206;
        double r9504230 = 0.607771387771;
        double r9504231 = r9504229 + r9504230;
        double r9504232 = r9504220 / r9504231;
        double r9504233 = r9504204 + r9504232;
        return r9504233;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r9504234 = z;
        double r9504235 = -1889722895859.1785;
        bool r9504236 = r9504234 <= r9504235;
        double r9504237 = x;
        double r9504238 = y;
        double r9504239 = 3.13060547623;
        double r9504240 = t;
        double r9504241 = r9504234 * r9504234;
        double r9504242 = r9504240 / r9504241;
        double r9504243 = r9504239 + r9504242;
        double r9504244 = 36.527041698806414;
        double r9504245 = r9504244 / r9504234;
        double r9504246 = r9504243 - r9504245;
        double r9504247 = r9504238 * r9504246;
        double r9504248 = r9504237 + r9504247;
        double r9504249 = 9.640368061021876e+38;
        bool r9504250 = r9504234 <= r9504249;
        double r9504251 = a;
        double r9504252 = r9504239 * r9504234;
        double r9504253 = 11.1667541262;
        double r9504254 = r9504252 + r9504253;
        double r9504255 = r9504234 * r9504254;
        double r9504256 = r9504255 + r9504240;
        double r9504257 = r9504256 * r9504234;
        double r9504258 = r9504251 + r9504257;
        double r9504259 = r9504234 * r9504258;
        double r9504260 = b;
        double r9504261 = r9504259 + r9504260;
        double r9504262 = 15.234687407;
        double r9504263 = r9504234 + r9504262;
        double r9504264 = r9504263 * r9504234;
        double r9504265 = 31.4690115749;
        double r9504266 = r9504264 + r9504265;
        double r9504267 = r9504266 * r9504234;
        double r9504268 = 11.9400905721;
        double r9504269 = r9504267 + r9504268;
        double r9504270 = r9504269 * r9504234;
        double r9504271 = 0.607771387771;
        double r9504272 = r9504270 + r9504271;
        double r9504273 = r9504261 / r9504272;
        double r9504274 = r9504238 * r9504273;
        double r9504275 = r9504237 + r9504274;
        double r9504276 = r9504238 * r9504239;
        double r9504277 = r9504238 / r9504234;
        double r9504278 = r9504244 * r9504277;
        double r9504279 = r9504276 - r9504278;
        double r9504280 = r9504240 / r9504234;
        double r9504281 = r9504280 * r9504277;
        double r9504282 = r9504279 + r9504281;
        double r9504283 = r9504237 + r9504282;
        double r9504284 = r9504250 ? r9504275 : r9504283;
        double r9504285 = r9504236 ? r9504248 : r9504284;
        return r9504285;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.3
Target	1.1
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -1889722895859.1785

   1. Initial program 55.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 55.1

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771\right)}}\]

   4. Applied times-frac 52.5

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}}\]

   5. Simplified 52.5

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

   6. Taylor expanded around inf 2.4

      \[\leadsto x + y \cdot \color{blue}{\left(\left(\frac{t}{{z}^{2}} + 3.13060547623\right) - 36.527041698806414 \cdot \frac{1}{z}\right)}\]

   7. Simplified 2.4

      \[\leadsto x + y \cdot \color{blue}{\left(\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}\right)}\]

   if -1889722895859.1785 < z < 9.640368061021876e+38

   1. Initial program 1.2

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.2

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771\right)}}\]

   4. Applied times-frac 0.6

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}}\]

   5. Simplified 0.6

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

   if 9.640368061021876e+38 < z

   1. Initial program 59.3

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 59.3

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771\right)}}\]

   4. Applied times-frac 56.9

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}}\]

   5. Simplified 56.9

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

   6. Taylor expanded around inf 7.8

      \[\leadsto x + \color{blue}{\left(\left(\frac{t \cdot y}{{z}^{2}} + 3.13060547623 \cdot y\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

   7. Simplified 1.6

      \[\leadsto x + \color{blue}{\left(\frac{t}{z} \cdot \frac{y}{z} + \left(y \cdot 3.13060547623 - \frac{y}{z} \cdot 36.527041698806414\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1889722895859.1785:\\ \;\;\;\;x + y \cdot \left(\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}\right)\\ \mathbf{elif}\;z \le 9.640368061021876 \cdot 10^{+38}:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(a + \left(z \cdot \left(3.13060547623 \cdot z + 11.1667541262\right) + t\right) \cdot z\right) + b}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y \cdot 3.13060547623 - 36.527041698806414 \cdot \frac{y}{z}\right) + \frac{t}{z} \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))