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

Average Error: 20.1 → 0.1

Time: 5.9s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -199552179246.06909 \lor \neg \left(z \le 405500.713644677133\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}\right) \cdot y\right) \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r393303 = x;
        double r393304 = y;
        double r393305 = z;
        double r393306 = 0.0692910599291889;
        double r393307 = r393305 * r393306;
        double r393308 = 0.4917317610505968;
        double r393309 = r393307 + r393308;
        double r393310 = r393309 * r393305;
        double r393311 = 0.279195317918525;
        double r393312 = r393310 + r393311;
        double r393313 = r393304 * r393312;
        double r393314 = 6.012459259764103;
        double r393315 = r393305 + r393314;
        double r393316 = r393315 * r393305;
        double r393317 = 3.350343815022304;
        double r393318 = r393316 + r393317;
        double r393319 = r393313 / r393318;
        double r393320 = r393303 + r393319;
        return r393320;
}

↓

double f(double x, double y, double z) {
        double r393321 = z;
        double r393322 = -199552179246.0691;
        bool r393323 = r393321 <= r393322;
        double r393324 = 405500.71364467713;
        bool r393325 = r393321 <= r393324;
        double r393326 = !r393325;
        bool r393327 = r393323 || r393326;
        double r393328 = x;
        double r393329 = 0.07512208616047561;
        double r393330 = y;
        double r393331 = r393330 / r393321;
        double r393332 = r393329 * r393331;
        double r393333 = 0.0692910599291889;
        double r393334 = r393333 * r393330;
        double r393335 = r393332 + r393334;
        double r393336 = 0.40462203869992125;
        double r393337 = 2.0;
        double r393338 = pow(r393321, r393337);
        double r393339 = r393330 / r393338;
        double r393340 = r393336 * r393339;
        double r393341 = r393335 - r393340;
        double r393342 = r393328 + r393341;
        double r393343 = r393321 * r393333;
        double r393344 = 0.4917317610505968;
        double r393345 = r393343 + r393344;
        double r393346 = r393345 * r393321;
        double r393347 = 0.279195317918525;
        double r393348 = r393346 + r393347;
        double r393349 = cbrt(r393348);
        double r393350 = r393349 * r393349;
        double r393351 = r393350 * r393330;
        double r393352 = 6.012459259764103;
        double r393353 = r393321 + r393352;
        double r393354 = r393353 * r393321;
        double r393355 = 3.350343815022304;
        double r393356 = r393354 + r393355;
        double r393357 = r393349 / r393356;
        double r393358 = r393351 * r393357;
        double r393359 = r393328 + r393358;
        double r393360 = r393327 ? r393342 : r393359;
        return r393360;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.1
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -199552179246.0691 or 405500.71364467713 < z

   1. Initial program 41.0

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)}\]

   if -199552179246.0691 < z < 405500.71364467713

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.2

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

   4. Applied times-frac 0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

   5. Simplified 0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 0.1

      \[\leadsto x + y \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

   8. Applied add-cube-cbrt 0.2

      \[\leadsto x + y \cdot \frac{\color{blue}{\left(\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}\right) \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}\]

   9. Applied times-frac 0.2

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{1} \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\right)}\]

   10. Applied associate-*r* 0.2

       \[\leadsto x + \color{blue}{\left(y \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{1}\right) \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

   11. Simplified 0.2

       \[\leadsto x + \color{blue}{\left(\left(\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}\right) \cdot y\right)} \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -199552179246.06909 \lor \neg \left(z \le 405500.713644677133\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977} \cdot \sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}\right) \cdot y\right) \cdot \frac{\sqrt[3]{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))