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

Average Error: 5.8 → 4.1

Time: 27.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 2.42723405450260913 \cdot 10^{26}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right) - x\right) + 0.91893853320467001} \cdot \sqrt{\left(2 \cdot \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right) - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right)\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r1199554 = x;
        double r1199555 = 0.5;
        double r1199556 = r1199554 - r1199555;
        double r1199557 = log(r1199554);
        double r1199558 = r1199556 * r1199557;
        double r1199559 = r1199558 - r1199554;
        double r1199560 = 0.91893853320467;
        double r1199561 = r1199559 + r1199560;
        double r1199562 = y;
        double r1199563 = 0.0007936500793651;
        double r1199564 = r1199562 + r1199563;
        double r1199565 = z;
        double r1199566 = r1199564 * r1199565;
        double r1199567 = 0.0027777777777778;
        double r1199568 = r1199566 - r1199567;
        double r1199569 = r1199568 * r1199565;
        double r1199570 = 0.083333333333333;
        double r1199571 = r1199569 + r1199570;
        double r1199572 = r1199571 / r1199554;
        double r1199573 = r1199561 + r1199572;
        return r1199573;
}

↓

double f(double x, double y, double z) {
        double r1199574 = x;
        double r1199575 = 2.427234054502609e+26;
        bool r1199576 = r1199574 <= r1199575;
        double r1199577 = 2.0;
        double r1199578 = 0.5;
        double r1199579 = r1199574 - r1199578;
        double r1199580 = sqrt(r1199574);
        double r1199581 = log(r1199580);
        double r1199582 = r1199579 * r1199581;
        double r1199583 = r1199577 * r1199582;
        double r1199584 = r1199583 - r1199574;
        double r1199585 = 0.91893853320467;
        double r1199586 = r1199584 + r1199585;
        double r1199587 = sqrt(r1199586);
        double r1199588 = r1199587 * r1199587;
        double r1199589 = y;
        double r1199590 = 0.0007936500793651;
        double r1199591 = r1199589 + r1199590;
        double r1199592 = z;
        double r1199593 = r1199591 * r1199592;
        double r1199594 = 0.0027777777777778;
        double r1199595 = r1199593 - r1199594;
        double r1199596 = r1199595 * r1199592;
        double r1199597 = 0.083333333333333;
        double r1199598 = r1199596 + r1199597;
        double r1199599 = r1199598 / r1199574;
        double r1199600 = r1199588 + r1199599;
        double r1199601 = cbrt(r1199574);
        double r1199602 = r1199601 * r1199601;
        double r1199603 = log(r1199602);
        double r1199604 = r1199603 * r1199579;
        double r1199605 = log(r1199601);
        double r1199606 = r1199579 * r1199605;
        double r1199607 = r1199606 - r1199574;
        double r1199608 = r1199604 + r1199607;
        double r1199609 = r1199608 + r1199585;
        double r1199610 = pow(r1199592, r1199577);
        double r1199611 = r1199610 / r1199574;
        double r1199612 = r1199611 * r1199591;
        double r1199613 = r1199592 / r1199574;
        double r1199614 = r1199594 * r1199613;
        double r1199615 = r1199612 - r1199614;
        double r1199616 = r1199609 + r1199615;
        double r1199617 = r1199576 ? r1199600 : r1199616;
        return r1199617;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.1
Herbie	4.1

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

Derivation

1. Split input into 2 regimes

2. if x < 2.427234054502609e+26

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt 0.2

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

   4. Applied log-prod 0.2

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

   5. Applied distribute-lft-in 0.2

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

   6. Applied associate--l+ 0.2

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

   7. Applied associate-+l+ 0.2

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

   9. Applied add-sqr-sqrt 0.3

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

   10. Simplified 0.3

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

   11. Simplified 0.3

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

   if 2.427234054502609e+26 < x

   1. Initial program 10.4

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

   3. Applied add-cube-cbrt 10.4

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

   4. Applied log-prod 10.5

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

   5. Applied distribute-rgt-in 10.5

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

   6. Applied associate--l+ 10.4

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

   7. Simplified 10.4

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

   8. Taylor expanded around inf 10.5

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

   9. Simplified 7.1

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.42723405450260913 \cdot 10^{26}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right) - x\right) + 0.91893853320467001} \cdot \sqrt{\left(2 \cdot \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right) - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right)\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(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)))