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

Average Error: 5.8 → 2.1

Time: 28.4s

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}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) = -\infty:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{elif}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le 2.413037256038914791422685382082884363974 \cdot 10^{296}:\\ \;\;\;\;\frac{1}{\frac{x}{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r19764469 = x;
        double r19764470 = 0.5;
        double r19764471 = r19764469 - r19764470;
        double r19764472 = log(r19764469);
        double r19764473 = r19764471 * r19764472;
        double r19764474 = r19764473 - r19764469;
        double r19764475 = 0.91893853320467;
        double r19764476 = r19764474 + r19764475;
        double r19764477 = y;
        double r19764478 = 0.0007936500793651;
        double r19764479 = r19764477 + r19764478;
        double r19764480 = z;
        double r19764481 = r19764479 * r19764480;
        double r19764482 = 0.0027777777777778;
        double r19764483 = r19764481 - r19764482;
        double r19764484 = r19764483 * r19764480;
        double r19764485 = 0.083333333333333;
        double r19764486 = r19764484 + r19764485;
        double r19764487 = r19764486 / r19764469;
        double r19764488 = r19764476 + r19764487;
        return r19764488;
}

↓

double f(double x, double y, double z) {
        double r19764489 = z;
        double r19764490 = 0.0007936500793651;
        double r19764491 = y;
        double r19764492 = r19764490 + r19764491;
        double r19764493 = r19764492 * r19764489;
        double r19764494 = 0.0027777777777778;
        double r19764495 = r19764493 - r19764494;
        double r19764496 = r19764489 * r19764495;
        double r19764497 = -inf.0;
        bool r19764498 = r19764496 <= r19764497;
        double r19764499 = 1.0;
        double r19764500 = x;
        double r19764501 = r19764500 / r19764489;
        double r19764502 = r19764501 / r19764489;
        double r19764503 = r19764502 / r19764491;
        double r19764504 = r19764489 * r19764491;
        double r19764505 = r19764504 * r19764504;
        double r19764506 = r19764500 / r19764505;
        double r19764507 = 6.298804484762296e-07;
        double r19764508 = r19764507 / r19764491;
        double r19764509 = r19764506 * r19764508;
        double r19764510 = r19764503 + r19764509;
        double r19764511 = r19764506 * r19764490;
        double r19764512 = r19764510 - r19764511;
        double r19764513 = r19764499 / r19764512;
        double r19764514 = cbrt(r19764500);
        double r19764515 = r19764514 * r19764514;
        double r19764516 = log(r19764515);
        double r19764517 = 0.5;
        double r19764518 = r19764500 - r19764517;
        double r19764519 = r19764516 * r19764518;
        double r19764520 = 0.91893853320467;
        double r19764521 = 0.3333333333333333;
        double r19764522 = pow(r19764500, r19764521);
        double r19764523 = log(r19764522);
        double r19764524 = r19764518 * r19764523;
        double r19764525 = r19764524 - r19764500;
        double r19764526 = r19764520 + r19764525;
        double r19764527 = r19764519 + r19764526;
        double r19764528 = r19764513 + r19764527;
        double r19764529 = 2.4130372560389148e+296;
        bool r19764530 = r19764496 <= r19764529;
        double r19764531 = 0.083333333333333;
        double r19764532 = r19764531 + r19764496;
        double r19764533 = r19764500 / r19764532;
        double r19764534 = r19764499 / r19764533;
        double r19764535 = r19764534 + r19764527;
        double r19764536 = r19764530 ? r19764535 : r19764528;
        double r19764537 = r19764498 ? r19764528 : r19764536;
        return r19764537;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.2
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -inf.0 or 2.4130372560389148e+296 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

   1. Initial program 60.5

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

   3. Applied add-cube-cbrt 60.5

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

   4. Applied log-prod 60.5

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

   5. Applied distribute-rgt-in 60.5

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

   6. Applied associate--l+ 60.5

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

   7. Applied associate-+l+ 60.5

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

   9. Applied pow1/3 60.5

      \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \color{blue}{\left({x}^{\frac{1}{3}}\right)} \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
   10. Using strategy rm

   11. Applied clear-num 60.5

       \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}}}\]

   12. Taylor expanded around inf 62.1

       \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{1}{\color{blue}{\left(6.29880448476229567390436391016717010416 \cdot 10^{-7} \cdot \frac{x}{{z}^{2} \cdot {y}^{3}} + \frac{x}{{z}^{2} \cdot y}\right) - 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{x}{{z}^{2} \cdot {y}^{2}}}}\]

   13. Simplified 19.7

       \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{1}{\color{blue}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(y \cdot z\right) \cdot \left(y \cdot z\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{x}{\left(y \cdot z\right) \cdot \left(y \cdot z\right)}}}\]

   if -inf.0 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 2.4130372560389148e+296

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt 0.2

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

   4. Applied log-prod 0.3

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

   5. Applied distribute-rgt-in 0.3

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

   6. Applied associate--l+ 0.3

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

   7. Applied associate-+l+ 0.3

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

   9. Applied pow1/3 0.2

      \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \color{blue}{\left({x}^{\frac{1}{3}}\right)} \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
   10. Using strategy rm

   11. Applied clear-num 0.3

       \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) = -\infty:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{elif}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le 2.413037256038914791422685382082884363974 \cdot 10^{296}:\\ \;\;\;\;\frac{1}{\frac{x}{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))