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

Average Error: 6.1 → 0.6

Time: 22.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1000364.972141867387108504772186279296875:\\ \;\;\;\;\sqrt{\log x \cdot \left(x - 0.5\right) - x} \cdot \sqrt{\log x \cdot \left(x - 0.5\right) - x} + \left(\frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} + 0.9189385332046700050057097541866824030876\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(\sqrt{\sqrt[3]{x}}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \left(\sqrt{\frac{2}{3}} \cdot \log x\right) \cdot \sqrt{\frac{2}{3}}\right) \cdot \left(x - 0.5\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{0.002777777777777800001512975569539776188321}{\frac{x}{z}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r358293 = x;
        double r358294 = 0.5;
        double r358295 = r358293 - r358294;
        double r358296 = log(r358293);
        double r358297 = r358295 * r358296;
        double r358298 = r358297 - r358293;
        double r358299 = 0.91893853320467;
        double r358300 = r358298 + r358299;
        double r358301 = y;
        double r358302 = 0.0007936500793651;
        double r358303 = r358301 + r358302;
        double r358304 = z;
        double r358305 = r358303 * r358304;
        double r358306 = 0.0027777777777778;
        double r358307 = r358305 - r358306;
        double r358308 = r358307 * r358304;
        double r358309 = 0.083333333333333;
        double r358310 = r358308 + r358309;
        double r358311 = r358310 / r358293;
        double r358312 = r358300 + r358311;
        return r358312;
}

↓

double f(double x, double y, double z) {
        double r358313 = x;
        double r358314 = 1000364.9721418674;
        bool r358315 = r358313 <= r358314;
        double r358316 = log(r358313);
        double r358317 = 0.5;
        double r358318 = r358313 - r358317;
        double r358319 = r358316 * r358318;
        double r358320 = r358319 - r358313;
        double r358321 = sqrt(r358320);
        double r358322 = r358321 * r358321;
        double r358323 = y;
        double r358324 = 0.0007936500793651;
        double r358325 = r358323 + r358324;
        double r358326 = z;
        double r358327 = r358325 * r358326;
        double r358328 = 0.0027777777777778;
        double r358329 = r358327 - r358328;
        double r358330 = r358329 * r358326;
        double r358331 = 0.083333333333333;
        double r358332 = r358330 + r358331;
        double r358333 = r358332 / r358313;
        double r358334 = 0.91893853320467;
        double r358335 = r358333 + r358334;
        double r358336 = r358322 + r358335;
        double r358337 = cbrt(r358313);
        double r358338 = sqrt(r358337);
        double r358339 = log(r358338);
        double r358340 = r358339 * r358318;
        double r358341 = 0.6666666666666666;
        double r358342 = sqrt(r358341);
        double r358343 = r358342 * r358316;
        double r358344 = r358343 * r358342;
        double r358345 = r358339 + r358344;
        double r358346 = r358345 * r358318;
        double r358347 = r358340 + r358346;
        double r358348 = r358347 - r358313;
        double r358349 = r358313 / r358326;
        double r358350 = r358326 / r358349;
        double r358351 = r358350 * r358325;
        double r358352 = r358328 / r358349;
        double r358353 = r358351 - r358352;
        double r358354 = r358334 + r358353;
        double r358355 = r358348 + r358354;
        double r358356 = r358315 ? r358336 : r358355;
        return r358356;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	1.4
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < 1000364.9721418674

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.4

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

   5. Simplified 0.4

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

   6. Simplified 0.4

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

   if 1000364.9721418674 < x

   1. Initial program 10.5

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

   2. Simplified 10.5

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log x - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 10.5

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   5. Applied log-prod 10.6

      \[\leadsto \left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   6. Applied distribute-lft-in 10.6

      \[\leadsto \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   7. Simplified 10.6

      \[\leadsto \left(\left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 10.6

      \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \log \color{blue}{\left(\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right)}\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   10. Applied log-prod 10.6

       \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \log \left(\sqrt{\sqrt[3]{x}}\right)\right)}\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   11. Applied distribute-lft-in 10.6

       \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right)}\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   12. Applied associate-+r+ 10.6

       \[\leadsto \left(\color{blue}{\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right)} - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   13. Simplified 10.6

       \[\leadsto \left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \frac{2}{3} \cdot \log x\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]
   14. Using strategy rm

   15. Applied add-sqr-sqrt 10.6

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \color{blue}{\left(\sqrt{\frac{2}{3}} \cdot \sqrt{\frac{2}{3}}\right)} \cdot \log x\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   16. Applied associate-*l* 10.6

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \color{blue}{\sqrt{\frac{2}{3}} \cdot \left(\sqrt{\frac{2}{3}} \cdot \log x\right)}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   17. Simplified 10.6

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \sqrt{\frac{2}{3}} \cdot \color{blue}{\left(\log x \cdot \sqrt{\frac{2}{3}}\right)}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\right)\]

   18. Taylor expanded around inf 10.8

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \sqrt{\frac{2}{3}} \cdot \left(\log x \cdot \sqrt{\frac{2}{3}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\right)\]

   19. Simplified 0.7

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \sqrt{\frac{2}{3}} \cdot \left(\log x \cdot \sqrt{\frac{2}{3}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{0.002777777777777800001512975569539776188321}{\frac{x}{z}}\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1000364.972141867387108504772186279296875:\\ \;\;\;\;\sqrt{\log x \cdot \left(x - 0.5\right) - x} \cdot \sqrt{\log x \cdot \left(x - 0.5\right) - x} + \left(\frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} + 0.9189385332046700050057097541866824030876\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(\sqrt{\sqrt[3]{x}}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt{\sqrt[3]{x}}\right) + \left(\sqrt{\frac{2}{3}} \cdot \log x\right) \cdot \sqrt{\frac{2}{3}}\right) \cdot \left(x - 0.5\right)\right) - x\right) + \left(0.9189385332046700050057097541866824030876 + \left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{0.002777777777777800001512975569539776188321}{\frac{x}{z}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))