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

Average Error: 6.1 → 4.5

Time: 26.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 177333351812856.5:\\ \;\;\;\;\sqrt{\log \left(\sqrt{x}\right) \cdot \left(\left(x - 0.5\right) + \left(x - 0.5\right)\right) - \left(x - 0.9189385332046700050057097541866824030876\right)} \cdot \sqrt{\log \left(\sqrt{x}\right) \cdot \left(\left(x - 0.5\right) + \left(x - 0.5\right)\right) - \left(x - 0.9189385332046700050057097541866824030876\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r344562 = x;
        double r344563 = 0.5;
        double r344564 = r344562 - r344563;
        double r344565 = log(r344562);
        double r344566 = r344564 * r344565;
        double r344567 = r344566 - r344562;
        double r344568 = 0.91893853320467;
        double r344569 = r344567 + r344568;
        double r344570 = y;
        double r344571 = 0.0007936500793651;
        double r344572 = r344570 + r344571;
        double r344573 = z;
        double r344574 = r344572 * r344573;
        double r344575 = 0.0027777777777778;
        double r344576 = r344574 - r344575;
        double r344577 = r344576 * r344573;
        double r344578 = 0.083333333333333;
        double r344579 = r344577 + r344578;
        double r344580 = r344579 / r344562;
        double r344581 = r344569 + r344580;
        return r344581;
}

↓

double f(double x, double y, double z) {
        double r344582 = x;
        double r344583 = 177333351812856.5;
        bool r344584 = r344582 <= r344583;
        double r344585 = sqrt(r344582);
        double r344586 = log(r344585);
        double r344587 = 0.5;
        double r344588 = r344582 - r344587;
        double r344589 = r344588 + r344588;
        double r344590 = r344586 * r344589;
        double r344591 = 0.91893853320467;
        double r344592 = r344582 - r344591;
        double r344593 = r344590 - r344592;
        double r344594 = sqrt(r344593);
        double r344595 = r344594 * r344594;
        double r344596 = y;
        double r344597 = 0.0007936500793651;
        double r344598 = r344596 + r344597;
        double r344599 = z;
        double r344600 = r344598 * r344599;
        double r344601 = 0.0027777777777778;
        double r344602 = r344600 - r344601;
        double r344603 = r344602 * r344599;
        double r344604 = 0.083333333333333;
        double r344605 = r344603 + r344604;
        double r344606 = r344605 / r344582;
        double r344607 = r344595 + r344606;
        double r344608 = cbrt(r344582);
        double r344609 = r344608 * r344608;
        double r344610 = log(r344609);
        double r344611 = r344610 * r344588;
        double r344612 = log(r344608);
        double r344613 = r344612 * r344588;
        double r344614 = r344613 - r344582;
        double r344615 = r344614 + r344591;
        double r344616 = r344611 + r344615;
        double r344617 = 2.0;
        double r344618 = pow(r344599, r344617);
        double r344619 = r344618 / r344582;
        double r344620 = r344619 * r344598;
        double r344621 = r344599 / r344582;
        double r344622 = r344601 * r344621;
        double r344623 = r344620 - r344622;
        double r344624 = r344616 + r344623;
        double r344625 = r344584 ? r344607 : r344624;
        return r344625;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	1.2
Herbie	4.5

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

Derivation

1. Split input into 2 regimes

2. if x < 177333351812856.5

   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-sqr-sqrt 0.2

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

   4. Applied log-prod 0.2

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

   5. Applied distribute-lft-in 0.2

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

   6. Applied associate--l+ 0.2

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

   7. Applied associate-+l+ 0.2

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

   8. Simplified 0.2

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

   10. Applied add-sqr-sqrt 0.3

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

   11. Simplified 0.3

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

   12. Simplified 0.3

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

   if 177333351812856.5 < x

   1. Initial program 10.7

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

   3. Applied add-cube-cbrt 10.7

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

   4. Applied log-prod 10.8

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

   5. Applied distribute-rgt-in 10.8

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

   6. Applied associate--l+ 10.7

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

   7. Applied associate-+l+ 10.7

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

   8. Taylor expanded around inf 10.8

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

   9. Simplified 7.6

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

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 177333351812856.5:\\ \;\;\;\;\sqrt{\log \left(\sqrt{x}\right) \cdot \left(\left(x - 0.5\right) + \left(x - 0.5\right)\right) - \left(x - 0.9189385332046700050057097541866824030876\right)} \cdot \sqrt{\log \left(\sqrt{x}\right) \cdot \left(\left(x - 0.5\right) + \left(x - 0.5\right)\right) - \left(x - 0.9189385332046700050057097541866824030876\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)) (- 0.91893853320467 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)))