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

Average Error: 5.9 → 4.0

Time: 20.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 291216481186222482904729899761664:\\ \;\;\;\;\frac{{\left(\left(x - 0.5\right) \cdot \log x - x\right)}^{3} + {0.9189385332046700050057097541866824030876}^{3}}{0.9189385332046700050057097541866824030876 \cdot \left(0.9189385332046700050057097541866824030876 - \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(x - 0.5\right) \cdot \log x - x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r379512 = x;
        double r379513 = 0.5;
        double r379514 = r379512 - r379513;
        double r379515 = log(r379512);
        double r379516 = r379514 * r379515;
        double r379517 = r379516 - r379512;
        double r379518 = 0.91893853320467;
        double r379519 = r379517 + r379518;
        double r379520 = y;
        double r379521 = 0.0007936500793651;
        double r379522 = r379520 + r379521;
        double r379523 = z;
        double r379524 = r379522 * r379523;
        double r379525 = 0.0027777777777778;
        double r379526 = r379524 - r379525;
        double r379527 = r379526 * r379523;
        double r379528 = 0.083333333333333;
        double r379529 = r379527 + r379528;
        double r379530 = r379529 / r379512;
        double r379531 = r379519 + r379530;
        return r379531;
}

↓

double f(double x, double y, double z) {
        double r379532 = x;
        double r379533 = 2.912164811862225e+32;
        bool r379534 = r379532 <= r379533;
        double r379535 = 0.5;
        double r379536 = r379532 - r379535;
        double r379537 = log(r379532);
        double r379538 = r379536 * r379537;
        double r379539 = r379538 - r379532;
        double r379540 = 3.0;
        double r379541 = pow(r379539, r379540);
        double r379542 = 0.91893853320467;
        double r379543 = pow(r379542, r379540);
        double r379544 = r379541 + r379543;
        double r379545 = r379542 - r379539;
        double r379546 = r379542 * r379545;
        double r379547 = r379539 * r379539;
        double r379548 = r379546 + r379547;
        double r379549 = r379544 / r379548;
        double r379550 = y;
        double r379551 = 0.0007936500793651;
        double r379552 = r379550 + r379551;
        double r379553 = z;
        double r379554 = r379552 * r379553;
        double r379555 = 0.0027777777777778;
        double r379556 = r379554 - r379555;
        double r379557 = r379556 * r379553;
        double r379558 = 0.083333333333333;
        double r379559 = r379557 + r379558;
        double r379560 = r379559 / r379532;
        double r379561 = r379549 + r379560;
        double r379562 = cbrt(r379532);
        double r379563 = r379562 * r379562;
        double r379564 = log(r379563);
        double r379565 = r379536 * r379564;
        double r379566 = log(r379562);
        double r379567 = r379536 * r379566;
        double r379568 = r379567 - r379532;
        double r379569 = r379568 + r379542;
        double r379570 = r379565 + r379569;
        double r379571 = 2.0;
        double r379572 = pow(r379553, r379571);
        double r379573 = r379572 / r379532;
        double r379574 = r379573 * r379552;
        double r379575 = r379553 / r379532;
        double r379576 = r379555 * r379575;
        double r379577 = r379574 - r379576;
        double r379578 = r379570 + r379577;
        double r379579 = r379534 ? r379561 : r379578;
        return r379579;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.1
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if x < 2.912164811862225e+32

   1. Initial program 0.4

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

   3. Applied flip3-+ 0.4

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

   4. Simplified 0.4

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

   if 2.912164811862225e+32 < 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 10.7

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

   5. Applied distribute-lft-in 10.7

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

   6. Applied associate--l+ 10.7

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

   7. Applied associate-+l+ 10.7

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

   8. Taylor expanded around inf 10.8

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

   9. Simplified 7.1

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 291216481186222482904729899761664:\\ \;\;\;\;\frac{{\left(\left(x - 0.5\right) \cdot \log x - x\right)}^{3} + {0.9189385332046700050057097541866824030876}^{3}}{0.9189385332046700050057097541866824030876 \cdot \left(0.9189385332046700050057097541866824030876 - \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) + \left(\left(x - 0.5\right) \cdot \log x - x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))