Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 6.3 → 5.5

Time: 15.5s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.523760054213747597180523810708067618358 \cdot 10^{81}:\\ \;\;\;\;\left(\left(\left(\left(x - \frac{1}{2}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \left(\left(\frac{7320129949063637}{9223372036854775808} \cdot \frac{{z}^{2}}{x} - \frac{3202559735019045}{1152921504606846976} \cdot \frac{z}{x}\right) + \frac{\frac{6004799503160637}{72057594037927936}}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r382245 = x;
        double r382246 = 0.5;
        double r382247 = r382245 - r382246;
        double r382248 = log(r382245);
        double r382249 = r382247 * r382248;
        double r382250 = r382249 - r382245;
        double r382251 = 0.91893853320467;
        double r382252 = r382250 + r382251;
        double r382253 = y;
        double r382254 = 0.0007936500793651;
        double r382255 = r382253 + r382254;
        double r382256 = z;
        double r382257 = r382255 * r382256;
        double r382258 = 0.0027777777777778;
        double r382259 = r382257 - r382258;
        double r382260 = r382259 * r382256;
        double r382261 = 0.083333333333333;
        double r382262 = r382260 + r382261;
        double r382263 = r382262 / r382245;
        double r382264 = r382252 + r382263;
        return r382264;
}

↓

double f(double x, double y, double z) {
        double r382265 = x;
        double r382266 = 1.5237600542137476e+81;
        bool r382267 = r382265 <= r382266;
        double r382268 = 1.0;
        double r382269 = 2.0;
        double r382270 = r382268 / r382269;
        double r382271 = r382265 - r382270;
        double r382272 = 2.0;
        double r382273 = cbrt(r382265);
        double r382274 = log(r382273);
        double r382275 = r382272 * r382274;
        double r382276 = r382271 * r382275;
        double r382277 = r382271 * r382274;
        double r382278 = r382276 + r382277;
        double r382279 = r382278 - r382265;
        double r382280 = 2069265617858471.0;
        double r382281 = 2251799813685248.0;
        double r382282 = r382280 / r382281;
        double r382283 = r382279 + r382282;
        double r382284 = y;
        double r382285 = 7320129949063637.0;
        double r382286 = 9.223372036854776e+18;
        double r382287 = r382285 / r382286;
        double r382288 = r382284 + r382287;
        double r382289 = z;
        double r382290 = r382288 * r382289;
        double r382291 = 3202559735019045.0;
        double r382292 = 1.152921504606847e+18;
        double r382293 = r382291 / r382292;
        double r382294 = r382290 - r382293;
        double r382295 = r382294 * r382289;
        double r382296 = 6004799503160637.0;
        double r382297 = 7.205759403792794e+16;
        double r382298 = r382296 / r382297;
        double r382299 = r382295 + r382298;
        double r382300 = r382299 / r382265;
        double r382301 = r382283 + r382300;
        double r382302 = r382273 * r382273;
        double r382303 = log(r382302);
        double r382304 = r382271 * r382303;
        double r382305 = 0.3333333333333333;
        double r382306 = pow(r382265, r382305);
        double r382307 = log(r382306);
        double r382308 = r382307 * r382271;
        double r382309 = r382308 - r382265;
        double r382310 = r382309 + r382282;
        double r382311 = r382304 + r382310;
        double r382312 = pow(r382289, r382272);
        double r382313 = r382312 / r382265;
        double r382314 = r382287 * r382313;
        double r382315 = r382289 / r382265;
        double r382316 = r382293 * r382315;
        double r382317 = r382314 - r382316;
        double r382318 = r382298 / r382265;
        double r382319 = r382317 + r382318;
        double r382320 = r382311 + r382319;
        double r382321 = r382267 ? r382301 : r382320;
        return r382321;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	1.2
Herbie	5.5

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 1.5237600542137476e+81

   1. Initial program 0.9

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   2. Simplified 0.9

      \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 0.9

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   5. Applied log-prod 0.9

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   6. Applied distribute-lft-in 0.9

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   7. Simplified 0.9

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   if 1.5237600542137476e+81 < x

   1. Initial program 12.6

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   2. Simplified 12.6

      \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 12.6

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   5. Applied log-prod 12.7

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   6. Applied distribute-lft-in 12.7

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   7. Applied associate--l+ 12.6

      \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right)\right)} + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   8. Applied associate-+l+ 12.6

      \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right)} + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   9. Simplified 12.6

      \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \color{blue}{\left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]
   10. Using strategy rm

   11. Applied pow1/3 12.6

       \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\log \color{blue}{\left({x}^{\frac{1}{3}}\right)} \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

   12. Taylor expanded around 0 10.9

       \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.08333333333333299564049667651488562114537 \cdot \frac{1}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\]

   13. Simplified 10.9

       \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \color{blue}{\left(\left(\frac{7320129949063637}{9223372036854775808} \cdot \frac{{z}^{2}}{x} - \frac{3202559735019045}{1152921504606846976} \cdot \frac{z}{x}\right) + \frac{\frac{6004799503160637}{72057594037927936}}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.523760054213747597180523810708067618358 \cdot 10^{81}:\\ \;\;\;\;\left(\left(\left(\left(x - \frac{1}{2}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \frac{1}{2}\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \left(\left(\frac{7320129949063637}{9223372036854775808} \cdot \frac{{z}^{2}}{x} - \frac{3202559735019045}{1152921504606846976} \cdot \frac{z}{x}\right) + \frac{\frac{6004799503160637}{72057594037927936}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467001 x)) (/ 0.0833333333333329956 x)) (* (/ z x) (- (* z (+ y 7.93650079365100015e-4)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467001) (/ (+ (* (- (* (+ y 7.93650079365100015e-4) z) 0.0027777777777778) z) 0.0833333333333329956) x)))