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

Average Error: 29.8 → 1.1

Time: 8.9s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.7165119779502947 \cdot 10^{24} \lor \neg \left(z \le 3.87772220842748429 \cdot 10^{27}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot y + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r363315 = x;
        double r363316 = y;
        double r363317 = z;
        double r363318 = 3.13060547623;
        double r363319 = r363317 * r363318;
        double r363320 = 11.1667541262;
        double r363321 = r363319 + r363320;
        double r363322 = r363321 * r363317;
        double r363323 = t;
        double r363324 = r363322 + r363323;
        double r363325 = r363324 * r363317;
        double r363326 = a;
        double r363327 = r363325 + r363326;
        double r363328 = r363327 * r363317;
        double r363329 = b;
        double r363330 = r363328 + r363329;
        double r363331 = r363316 * r363330;
        double r363332 = 15.234687407;
        double r363333 = r363317 + r363332;
        double r363334 = r363333 * r363317;
        double r363335 = 31.4690115749;
        double r363336 = r363334 + r363335;
        double r363337 = r363336 * r363317;
        double r363338 = 11.9400905721;
        double r363339 = r363337 + r363338;
        double r363340 = r363339 * r363317;
        double r363341 = 0.607771387771;
        double r363342 = r363340 + r363341;
        double r363343 = r363331 / r363342;
        double r363344 = r363315 + r363343;
        return r363344;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r363345 = z;
        double r363346 = -3.7165119779502947e+24;
        bool r363347 = r363345 <= r363346;
        double r363348 = 3.8777222084274843e+27;
        bool r363349 = r363345 <= r363348;
        double r363350 = !r363349;
        bool r363351 = r363347 || r363350;
        double r363352 = y;
        double r363353 = 3.13060547623;
        double r363354 = t;
        double r363355 = 2.0;
        double r363356 = pow(r363345, r363355);
        double r363357 = r363354 / r363356;
        double r363358 = r363353 + r363357;
        double r363359 = x;
        double r363360 = fma(r363352, r363358, r363359);
        double r363361 = 11.1667541262;
        double r363362 = fma(r363345, r363353, r363361);
        double r363363 = fma(r363362, r363345, r363354);
        double r363364 = a;
        double r363365 = fma(r363363, r363345, r363364);
        double r363366 = b;
        double r363367 = fma(r363365, r363345, r363366);
        double r363368 = 15.234687407;
        double r363369 = r363345 + r363368;
        double r363370 = 31.4690115749;
        double r363371 = fma(r363369, r363345, r363370);
        double r363372 = 11.9400905721;
        double r363373 = fma(r363371, r363345, r363372);
        double r363374 = 0.607771387771;
        double r363375 = fma(r363373, r363345, r363374);
        double r363376 = r363367 / r363375;
        double r363377 = r363376 * r363352;
        double r363378 = r363377 + r363359;
        double r363379 = r363351 ? r363360 : r363378;
        return r363379;
}

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	29.8
Target	1.1
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -3.7165119779502947e+24 or 3.8777222084274843e+27 < z

   1. Initial program 58.5

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

   2. Simplified 56.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

   3. Taylor expanded around inf 9.0

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547622999996 \cdot y\right)}\]

   4. Simplified 1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]

   if -3.7165119779502947e+24 < z < 3.8777222084274843e+27

   1. Initial program 0.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]
   3. Using strategy rm

   4. Applied clear-num 0.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right) + x}\]

   7. Simplified 0.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}} + x\]
   8. Using strategy rm

   9. Applied associate-/r/ 0.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot y} + x\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.7165119779502947 \cdot 10^{24} \lor \neg \left(z \le 3.87772220842748429 \cdot 10^{27}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)} \cdot y + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :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))))