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

Average Error: 6.0 → 4.3

Time: 29.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 3.342421783116303450309756095958266932238 \cdot 10^{49}:\\ \;\;\;\;\left(\left(\log x \cdot x - 1.5 \cdot \left(\log x \cdot \frac{1}{3}\right)\right) + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r317423 = x;
        double r317424 = 0.5;
        double r317425 = r317423 - r317424;
        double r317426 = log(r317423);
        double r317427 = r317425 * r317426;
        double r317428 = r317427 - r317423;
        double r317429 = 0.91893853320467;
        double r317430 = r317428 + r317429;
        double r317431 = y;
        double r317432 = 0.0007936500793651;
        double r317433 = r317431 + r317432;
        double r317434 = z;
        double r317435 = r317433 * r317434;
        double r317436 = 0.0027777777777778;
        double r317437 = r317435 - r317436;
        double r317438 = r317437 * r317434;
        double r317439 = 0.083333333333333;
        double r317440 = r317438 + r317439;
        double r317441 = r317440 / r317423;
        double r317442 = r317430 + r317441;
        return r317442;
}

↓

double f(double x, double y, double z) {
        double r317443 = x;
        double r317444 = 3.3424217831163035e+49;
        bool r317445 = r317443 <= r317444;
        double r317446 = log(r317443);
        double r317447 = r317446 * r317443;
        double r317448 = 1.5;
        double r317449 = 0.3333333333333333;
        double r317450 = r317446 * r317449;
        double r317451 = r317448 * r317450;
        double r317452 = r317447 - r317451;
        double r317453 = 0.91893853320467;
        double r317454 = r317453 - r317443;
        double r317455 = r317452 + r317454;
        double r317456 = y;
        double r317457 = 0.0007936500793651;
        double r317458 = r317456 + r317457;
        double r317459 = z;
        double r317460 = r317458 * r317459;
        double r317461 = 0.0027777777777778;
        double r317462 = r317460 - r317461;
        double r317463 = r317462 * r317459;
        double r317464 = 0.083333333333333;
        double r317465 = r317463 + r317464;
        double r317466 = r317465 / r317443;
        double r317467 = r317455 + r317466;
        double r317468 = 0.5;
        double r317469 = r317443 - r317468;
        double r317470 = r317469 * r317446;
        double r317471 = r317470 - r317443;
        double r317472 = r317471 + r317453;
        double r317473 = 2.0;
        double r317474 = pow(r317459, r317473);
        double r317475 = r317474 / r317443;
        double r317476 = r317475 * r317458;
        double r317477 = r317459 / r317443;
        double r317478 = r317461 * r317477;
        double r317479 = r317476 - r317478;
        double r317480 = r317472 + r317479;
        double r317481 = r317445 ? r317467 : r317480;
        return r317481;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.4
Herbie	4.3

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

Derivation

1. Split input into 2 regimes

2. if x < 3.3424217831163035e+49

   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 sub-neg 0.4

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

   4. Applied associate-+l+ 0.4

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

   5. Simplified 0.4

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

   7. Applied add-cube-cbrt 0.4

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

   8. Applied log-prod 0.4

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

   9. Applied distribute-lft-in 0.4

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

   10. Simplified 0.4

       \[\leadsto \left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   11. Taylor expanded around inf 0.4

       \[\leadsto \left(\color{blue}{\left(3 \cdot \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right) - 1.5 \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right)} + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   12. Simplified 0.4

       \[\leadsto \left(\color{blue}{\left(x \cdot \left(-\left(-\log x\right)\right) - \left(\left(-\log x\right) \cdot \frac{-1}{3}\right) \cdot 1.5\right)} + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   if 3.3424217831163035e+49 < x

   1. Initial program 11.5

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

   2. Taylor expanded around inf 11.6

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

   3. Simplified 8.1

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.342421783116303450309756095958266932238 \cdot 10^{49}:\\ \;\;\;\;\left(\left(\log x \cdot x - 1.5 \cdot \left(\log x \cdot \frac{1}{3}\right)\right) + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))