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

Average Error: 5.8 → 5.9

Time: 22.4s

Precision: 64

Math

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

↓

\[\left(x - \frac{1}{2}\right) \cdot \left(\log \left(\sqrt[3]{x} \cdot {x}^{\left(\frac{1}{3}\right)}\right) + \log \left({x}^{\left(\frac{1}{3}\right)}\right)\right) - \left(\left(x - \frac{2069265617858471}{2251799813685248}\right) - \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\right)\]

C

double f(double x, double y, double z) {
        double r524396 = x;
        double r524397 = 0.5;
        double r524398 = r524396 - r524397;
        double r524399 = log(r524396);
        double r524400 = r524398 * r524399;
        double r524401 = r524400 - r524396;
        double r524402 = 0.91893853320467;
        double r524403 = r524401 + r524402;
        double r524404 = y;
        double r524405 = 0.0007936500793651;
        double r524406 = r524404 + r524405;
        double r524407 = z;
        double r524408 = r524406 * r524407;
        double r524409 = 0.0027777777777778;
        double r524410 = r524408 - r524409;
        double r524411 = r524410 * r524407;
        double r524412 = 0.083333333333333;
        double r524413 = r524411 + r524412;
        double r524414 = r524413 / r524396;
        double r524415 = r524403 + r524414;
        return r524415;
}

↓

double f(double x, double y, double z) {
        double r524416 = x;
        double r524417 = 1.0;
        double r524418 = 2.0;
        double r524419 = r524417 / r524418;
        double r524420 = r524416 - r524419;
        double r524421 = cbrt(r524416);
        double r524422 = 1.0;
        double r524423 = 3.0;
        double r524424 = r524422 / r524423;
        double r524425 = pow(r524416, r524424);
        double r524426 = r524421 * r524425;
        double r524427 = log(r524426);
        double r524428 = log(r524425);
        double r524429 = r524427 + r524428;
        double r524430 = r524420 * r524429;
        double r524431 = 2069265617858471.0;
        double r524432 = 2251799813685248.0;
        double r524433 = r524431 / r524432;
        double r524434 = r524416 - r524433;
        double r524435 = y;
        double r524436 = 7320129949063637.0;
        double r524437 = 9.223372036854776e+18;
        double r524438 = r524436 / r524437;
        double r524439 = r524435 + r524438;
        double r524440 = z;
        double r524441 = r524439 * r524440;
        double r524442 = 3202559735019045.0;
        double r524443 = 1.152921504606847e+18;
        double r524444 = r524442 / r524443;
        double r524445 = r524441 - r524444;
        double r524446 = r524445 * r524440;
        double r524447 = 6004799503160637.0;
        double r524448 = 7.205759403792794e+16;
        double r524449 = r524447 / r524448;
        double r524450 = r524446 + r524449;
        double r524451 = r524450 / r524416;
        double r524452 = r524434 - r524451;
        double r524453 = r524430 - r524452;
        return r524453;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.4
Herbie	5.9

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

Derivation

1. Initial program 5.8

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

2. Simplified 5.8

   \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 5.8

   \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

5. Applied log-prod 5.8

   \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

6. Applied distribute-rgt-in 5.8

   \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right)\right)} - x\right) + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

7. Applied associate--l+ 5.8

   \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) - x\right)\right)} + \frac{2069265617858471}{2251799813685248}\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

8. Applied associate-+l+ 5.8

   \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right)} + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]
9. Using strategy rm

10. Applied pow1/3 5.8

    \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right) \cdot \left(x - \frac{1}{2}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

11. Taylor expanded around 0 5.8

    \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot {x}^{\frac{1}{3}}\right) \cdot \left(x - \frac{1}{2}\right) + \left(\left(\log \color{blue}{\left({x}^{\frac{1}{3}}\right)} \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

12. Simplified 5.8

    \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot {x}^{\frac{1}{3}}\right) \cdot \left(x - \frac{1}{2}\right) + \left(\left(\log \color{blue}{\left({x}^{\left(\frac{1}{3}\right)}\right)} \cdot \left(x - \frac{1}{2}\right) - x\right) + \frac{2069265617858471}{2251799813685248}\right)\right) + \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\]

13. Final simplification 5.9

    \[\leadsto \left(x - \frac{1}{2}\right) \cdot \left(\log \left(\sqrt[3]{x} \cdot {x}^{\left(\frac{1}{3}\right)}\right) + \log \left({x}^{\left(\frac{1}{3}\right)}\right)\right) - \left(\left(x - \frac{2069265617858471}{2251799813685248}\right) - \frac{\left(\left(y + \frac{7320129949063637}{9223372036854775808}\right) \cdot z - \frac{3202559735019045}{1152921504606846976}\right) \cdot z + \frac{6004799503160637}{72057594037927936}}{x}\right)\]

Reproduce

herbie shell --seed 2019303 
(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)))