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

Average Error: 6.0 → 0.3

Time: 47.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 20160595213046037610496:\\ \;\;\;\;\frac{0.08333333333333299564049667651488562114537 + \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log x\right) \cdot \left(\sqrt{x} + \sqrt{0.5}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{0.002777777777777800001512975569539776188321}{\frac{x}{z}}\right) + \left(0.9189385332046700050057097541866824030876 + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r21379557 = x;
        double r21379558 = 0.5;
        double r21379559 = r21379557 - r21379558;
        double r21379560 = log(r21379557);
        double r21379561 = r21379559 * r21379560;
        double r21379562 = r21379561 - r21379557;
        double r21379563 = 0.91893853320467;
        double r21379564 = r21379562 + r21379563;
        double r21379565 = y;
        double r21379566 = 0.0007936500793651;
        double r21379567 = r21379565 + r21379566;
        double r21379568 = z;
        double r21379569 = r21379567 * r21379568;
        double r21379570 = 0.0027777777777778;
        double r21379571 = r21379569 - r21379570;
        double r21379572 = r21379571 * r21379568;
        double r21379573 = 0.083333333333333;
        double r21379574 = r21379572 + r21379573;
        double r21379575 = r21379574 / r21379557;
        double r21379576 = r21379564 + r21379575;
        return r21379576;
}

↓

double f(double x, double y, double z) {
        double r21379577 = x;
        double r21379578 = 2.0160595213046038e+22;
        bool r21379579 = r21379577 <= r21379578;
        double r21379580 = 0.083333333333333;
        double r21379581 = y;
        double r21379582 = 0.0007936500793651;
        double r21379583 = r21379581 + r21379582;
        double r21379584 = z;
        double r21379585 = r21379583 * r21379584;
        double r21379586 = 0.0027777777777778;
        double r21379587 = r21379585 - r21379586;
        double r21379588 = r21379587 * r21379584;
        double r21379589 = r21379580 + r21379588;
        double r21379590 = r21379589 / r21379577;
        double r21379591 = 0.91893853320467;
        double r21379592 = sqrt(r21379577);
        double r21379593 = 0.5;
        double r21379594 = sqrt(r21379593);
        double r21379595 = r21379592 - r21379594;
        double r21379596 = log(r21379577);
        double r21379597 = r21379595 * r21379596;
        double r21379598 = r21379592 + r21379594;
        double r21379599 = r21379597 * r21379598;
        double r21379600 = r21379599 - r21379577;
        double r21379601 = r21379591 + r21379600;
        double r21379602 = r21379590 + r21379601;
        double r21379603 = r21379577 / r21379584;
        double r21379604 = r21379584 / r21379603;
        double r21379605 = r21379604 * r21379583;
        double r21379606 = r21379586 / r21379603;
        double r21379607 = r21379605 - r21379606;
        double r21379608 = cbrt(r21379577);
        double r21379609 = r21379608 * r21379608;
        double r21379610 = log(r21379609);
        double r21379611 = r21379577 - r21379593;
        double r21379612 = r21379610 * r21379611;
        double r21379613 = 0.3333333333333333;
        double r21379614 = pow(r21379577, r21379613);
        double r21379615 = log(r21379614);
        double r21379616 = r21379611 * r21379615;
        double r21379617 = r21379616 - r21379577;
        double r21379618 = r21379612 + r21379617;
        double r21379619 = r21379591 + r21379618;
        double r21379620 = r21379607 + r21379619;
        double r21379621 = r21379579 ? r21379602 : r21379620;
        return r21379621;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 2.0160595213046038e+22

   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

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

   4. Applied add-sqr-sqrt 0.2

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

   5. Applied difference-of-squares 0.2

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

   6. Applied associate-*l* 0.2

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

   if 2.0160595213046038e+22 < x

   1. Initial program 10.8

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

   3. Applied add-cube-cbrt 10.8

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

   4. Applied log-prod 10.9

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

   5. Applied distribute-lft-in 10.9

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

   6. Applied associate--l+ 10.9

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

   8. Applied pow1/3 10.8

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

   9. Taylor expanded around inf 10.9

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

   10. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 20160595213046037610496:\\ \;\;\;\;\frac{0.08333333333333299564049667651488562114537 + \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log x\right) \cdot \left(\sqrt{x} + \sqrt{0.5}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{0.002777777777777800001512975569539776188321}{\frac{x}{z}}\right) + \left(0.9189385332046700050057097541866824030876 + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))