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

Average Error: 6.5 → 4.4

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 85213128.80680434405803680419921875:\\ \;\;\;\;\left(\left(\sqrt{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt{\left(x - 0.5\right) \cdot \log x} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\log \left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right) - x\right)\right)\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r332632 = x;
        double r332633 = 0.5;
        double r332634 = r332632 - r332633;
        double r332635 = log(r332632);
        double r332636 = r332634 * r332635;
        double r332637 = r332636 - r332632;
        double r332638 = 0.91893853320467;
        double r332639 = r332637 + r332638;
        double r332640 = y;
        double r332641 = 0.0007936500793651;
        double r332642 = r332640 + r332641;
        double r332643 = z;
        double r332644 = r332642 * r332643;
        double r332645 = 0.0027777777777778;
        double r332646 = r332644 - r332645;
        double r332647 = r332646 * r332643;
        double r332648 = 0.083333333333333;
        double r332649 = r332647 + r332648;
        double r332650 = r332649 / r332632;
        double r332651 = r332639 + r332650;
        return r332651;
}

↓

double f(double x, double y, double z) {
        double r332652 = x;
        double r332653 = 85213128.80680434;
        bool r332654 = r332652 <= r332653;
        double r332655 = 0.5;
        double r332656 = r332652 - r332655;
        double r332657 = log(r332652);
        double r332658 = r332656 * r332657;
        double r332659 = sqrt(r332658);
        double r332660 = r332659 * r332659;
        double r332661 = r332660 - r332652;
        double r332662 = 0.91893853320467;
        double r332663 = r332661 + r332662;
        double r332664 = y;
        double r332665 = 0.0007936500793651;
        double r332666 = r332664 + r332665;
        double r332667 = z;
        double r332668 = r332666 * r332667;
        double r332669 = 0.0027777777777778;
        double r332670 = r332668 - r332669;
        double r332671 = r332670 * r332667;
        double r332672 = 0.083333333333333;
        double r332673 = r332671 + r332672;
        double r332674 = r332673 / r332652;
        double r332675 = r332663 + r332674;
        double r332676 = sqrt(r332652);
        double r332677 = log(r332676);
        double r332678 = r332656 * r332677;
        double r332679 = cbrt(r332652);
        double r332680 = r332679 * r332679;
        double r332681 = sqrt(r332680);
        double r332682 = log(r332681);
        double r332683 = r332682 * r332656;
        double r332684 = sqrt(r332679);
        double r332685 = log(r332684);
        double r332686 = r332656 * r332685;
        double r332687 = r332686 - r332652;
        double r332688 = r332683 + r332687;
        double r332689 = r332678 + r332688;
        double r332690 = r332689 + r332662;
        double r332691 = 2.0;
        double r332692 = pow(r332667, r332691);
        double r332693 = r332692 / r332652;
        double r332694 = r332693 * r332666;
        double r332695 = r332667 / r332652;
        double r332696 = r332669 * r332695;
        double r332697 = r332694 - r332696;
        double r332698 = r332690 + r332697;
        double r332699 = r332654 ? r332675 : r332698;
        return r332699;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	1.4
Herbie	4.4

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

Derivation

1. Split input into 2 regimes

2. if x < 85213128.80680434

   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. Using strategy rm

   3. Applied add-sqr-sqrt 0.2

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

   if 85213128.80680434 < x

   1. Initial program 11.2

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

   3. Applied add-sqr-sqrt 11.2

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

   4. Applied log-prod 11.2

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

   5. Applied distribute-lft-in 11.2

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

   6. Applied associate--l+ 11.2

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

   8. Applied add-cube-cbrt 11.2

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

   9. Applied sqrt-prod 11.2

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

   10. Applied log-prod 11.2

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

   11. Applied distribute-rgt-in 11.2

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

   12. Applied associate--l+ 11.2

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

   13. Simplified 11.2

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

   14. Taylor expanded around inf 11.2

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

   15. Simplified 7.5

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

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 85213128.80680434405803680419921875:\\ \;\;\;\;\left(\left(\sqrt{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt{\left(x - 0.5\right) \cdot \log x} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\log \left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{x}}\right) - x\right)\right)\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))