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

Average Error: 5.5 → 4.8

Time: 9.3s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.27804641027178125 \cdot 10^{148}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot 2\right) + \left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot 2 + \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r469492 = x;
        double r469493 = 0.5;
        double r469494 = r469492 - r469493;
        double r469495 = log(r469492);
        double r469496 = r469494 * r469495;
        double r469497 = r469496 - r469492;
        double r469498 = 0.91893853320467;
        double r469499 = r469497 + r469498;
        double r469500 = y;
        double r469501 = 0.0007936500793651;
        double r469502 = r469500 + r469501;
        double r469503 = z;
        double r469504 = r469502 * r469503;
        double r469505 = 0.0027777777777778;
        double r469506 = r469504 - r469505;
        double r469507 = r469506 * r469503;
        double r469508 = 0.083333333333333;
        double r469509 = r469507 + r469508;
        double r469510 = r469509 / r469492;
        double r469511 = r469499 + r469510;
        return r469511;
}

↓

double f(double x, double y, double z) {
        double r469512 = x;
        double r469513 = 1.2780464102717813e+148;
        bool r469514 = r469512 <= r469513;
        double r469515 = 0.5;
        double r469516 = r469512 - r469515;
        double r469517 = cbrt(r469512);
        double r469518 = r469517 * r469517;
        double r469519 = cbrt(r469518);
        double r469520 = log(r469519);
        double r469521 = 2.0;
        double r469522 = r469520 * r469521;
        double r469523 = r469516 * r469522;
        double r469524 = cbrt(r469517);
        double r469525 = log(r469524);
        double r469526 = r469525 * r469521;
        double r469527 = log(r469517);
        double r469528 = r469526 + r469527;
        double r469529 = r469516 * r469528;
        double r469530 = r469523 + r469529;
        double r469531 = r469530 - r469512;
        double r469532 = 0.91893853320467;
        double r469533 = r469531 + r469532;
        double r469534 = y;
        double r469535 = 0.0007936500793651;
        double r469536 = r469534 + r469535;
        double r469537 = z;
        double r469538 = r469536 * r469537;
        double r469539 = 0.0027777777777778;
        double r469540 = r469538 - r469539;
        double r469541 = r469540 * r469537;
        double r469542 = 0.083333333333333;
        double r469543 = r469541 + r469542;
        double r469544 = r469543 / r469512;
        double r469545 = r469533 + r469544;
        double r469546 = r469521 * r469527;
        double r469547 = r469516 * r469546;
        double r469548 = r469516 * r469527;
        double r469549 = r469547 + r469548;
        double r469550 = r469549 - r469512;
        double r469551 = r469550 + r469532;
        double r469552 = pow(r469537, r469521);
        double r469553 = r469552 / r469512;
        double r469554 = r469535 * r469553;
        double r469555 = 1.0;
        double r469556 = r469555 / r469512;
        double r469557 = r469542 * r469556;
        double r469558 = r469554 + r469557;
        double r469559 = r469537 / r469512;
        double r469560 = r469539 * r469559;
        double r469561 = r469558 - r469560;
        double r469562 = r469551 + r469561;
        double r469563 = r469514 ? r469545 : r469562;
        return r469563;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.5
Target	1.2
Herbie	4.8

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 1.2780464102717813e+148

   1. Initial program 2.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   4. Applied log-prod 2.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   5. Applied distribute-lft-in 2.1

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   6. Simplified 2.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 2.1

      \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   9. Applied cbrt-prod 2.1

      \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   10. Applied log-prod 2.1

       \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right)}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   11. Applied distribute-rgt-in 2.1

       \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot 2 + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot 2\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   12. Applied distribute-lft-in 2.1

       \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot 2\right) + \left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot 2\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   13. Applied associate-+l+ 2.1

       \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot 2\right) + \left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot 2\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   14. Simplified 2.1

       \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot 2\right) + \color{blue}{\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot 2 + \log \left(\sqrt[3]{x}\right)\right)}\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   if 1.2780464102717813e+148 < x

   1. Initial program 12.8

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 12.8

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   4. Applied log-prod 12.9

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   5. Applied distribute-lft-in 12.9

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   6. Simplified 12.9

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   7. Taylor expanded around 0 10.6

      \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.27804641027178125 \cdot 10^{148}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot 2\right) + \left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot 2 + \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))