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

Average Error: 5.9 → 5.9

Time: 21.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r350506 = x;
        double r350507 = 0.5;
        double r350508 = r350506 - r350507;
        double r350509 = log(r350506);
        double r350510 = r350508 * r350509;
        double r350511 = r350510 - r350506;
        double r350512 = 0.91893853320467;
        double r350513 = r350511 + r350512;
        double r350514 = y;
        double r350515 = 0.0007936500793651;
        double r350516 = r350514 + r350515;
        double r350517 = z;
        double r350518 = r350516 * r350517;
        double r350519 = 0.0027777777777778;
        double r350520 = r350518 - r350519;
        double r350521 = r350520 * r350517;
        double r350522 = 0.083333333333333;
        double r350523 = r350521 + r350522;
        double r350524 = r350523 / r350506;
        double r350525 = r350513 + r350524;
        return r350525;
}

↓

double f(double x, double y, double z) {
        double r350526 = x;
        double r350527 = 0.5;
        double r350528 = r350526 - r350527;
        double r350529 = sqrt(r350526);
        double r350530 = log(r350529);
        double r350531 = 2.0;
        double r350532 = sqrt(r350529);
        double r350533 = log(r350532);
        double r350534 = r350531 * r350533;
        double r350535 = r350530 + r350534;
        double r350536 = r350528 * r350535;
        double r350537 = 0.91893853320467;
        double r350538 = y;
        double r350539 = 0.0007936500793651;
        double r350540 = r350538 + r350539;
        double r350541 = z;
        double r350542 = r350540 * r350541;
        double r350543 = 0.0027777777777778;
        double r350544 = r350542 - r350543;
        double r350545 = r350544 * r350541;
        double r350546 = 0.083333333333333;
        double r350547 = r350545 + r350546;
        double r350548 = r350547 / r350526;
        double r350549 = r350537 + r350548;
        double r350550 = r350526 - r350549;
        double r350551 = r350536 - r350550;
        return r350551;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.1
Herbie	5.9

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

Derivation

1. Initial program 5.9

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

3. Applied add-sqr-sqrt 5.9

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

4. Applied log-prod 5.9

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

5. Applied distribute-lft-in 5.9

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

6. Applied associate--l+ 6.0

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

7. Applied associate-+l+ 6.0

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

8. Simplified 6.0

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

10. Applied add-sqr-sqrt 6.0

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

11. Applied sqrt-prod 6.0

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

12. Applied log-prod 6.0

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

13. Applied distribute-lft-in 6.0

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

14. Applied associate-+l+ 6.0

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

15. Simplified 6.0

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

16. Final simplification 5.9

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

Reproduce

herbie shell --seed 2019294 
(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)))