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

Average Error: 5.8 → 0.3

Time: 29.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 21506604914.314178466796875:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right) + 0.08333333333333299564049667651488562114537}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt[3]{\left(x - 0.5\right) \cdot \log x}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{z}{x} \cdot 0.002777777777777800001512975569539776188321\right) + \left(\left(\left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r17653530 = x;
        double r17653531 = 0.5;
        double r17653532 = r17653530 - r17653531;
        double r17653533 = log(r17653530);
        double r17653534 = r17653532 * r17653533;
        double r17653535 = r17653534 - r17653530;
        double r17653536 = 0.91893853320467;
        double r17653537 = r17653535 + r17653536;
        double r17653538 = y;
        double r17653539 = 0.0007936500793651;
        double r17653540 = r17653538 + r17653539;
        double r17653541 = z;
        double r17653542 = r17653540 * r17653541;
        double r17653543 = 0.0027777777777778;
        double r17653544 = r17653542 - r17653543;
        double r17653545 = r17653544 * r17653541;
        double r17653546 = 0.083333333333333;
        double r17653547 = r17653545 + r17653546;
        double r17653548 = r17653547 / r17653530;
        double r17653549 = r17653537 + r17653548;
        return r17653549;
}

↓

double f(double x, double y, double z) {
        double r17653550 = x;
        double r17653551 = 21506604914.31418;
        bool r17653552 = r17653550 <= r17653551;
        double r17653553 = z;
        double r17653554 = y;
        double r17653555 = 0.0007936500793651;
        double r17653556 = r17653554 + r17653555;
        double r17653557 = r17653553 * r17653556;
        double r17653558 = 0.0027777777777778;
        double r17653559 = r17653557 - r17653558;
        double r17653560 = r17653553 * r17653559;
        double r17653561 = 0.083333333333333;
        double r17653562 = r17653560 + r17653561;
        double r17653563 = r17653562 / r17653550;
        double r17653564 = 0.91893853320467;
        double r17653565 = 0.5;
        double r17653566 = r17653550 - r17653565;
        double r17653567 = log(r17653550);
        double r17653568 = r17653566 * r17653567;
        double r17653569 = cbrt(r17653568);
        double r17653570 = r17653569 * r17653569;
        double r17653571 = r17653569 * r17653570;
        double r17653572 = r17653571 - r17653550;
        double r17653573 = r17653564 + r17653572;
        double r17653574 = r17653563 + r17653573;
        double r17653575 = r17653550 / r17653553;
        double r17653576 = r17653553 / r17653575;
        double r17653577 = r17653576 * r17653556;
        double r17653578 = r17653553 / r17653550;
        double r17653579 = r17653578 * r17653558;
        double r17653580 = r17653577 - r17653579;
        double r17653581 = 1.0;
        double r17653582 = r17653581 / r17653550;
        double r17653583 = -0.3333333333333333;
        double r17653584 = pow(r17653582, r17653583);
        double r17653585 = log(r17653584);
        double r17653586 = r17653585 * r17653566;
        double r17653587 = r17653586 - r17653550;
        double r17653588 = r17653587 + r17653564;
        double r17653589 = cbrt(r17653550);
        double r17653590 = r17653589 * r17653589;
        double r17653591 = log(r17653590);
        double r17653592 = r17653591 * r17653566;
        double r17653593 = r17653588 + r17653592;
        double r17653594 = r17653580 + r17653593;
        double r17653595 = r17653552 ? r17653574 : r17653594;
        return r17653595;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 21506604914.31418

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.2

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

   if 21506604914.31418 < x

   1. Initial program 10.2

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

   3. Applied add-cube-cbrt 10.2

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

   4. Applied log-prod 10.2

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

   5. Applied distribute-lft-in 10.2

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

   6. Applied associate--l+ 10.2

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

   7. Applied associate-+l+ 10.2

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

   8. Taylor expanded around inf 10.1

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)} - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   9. Taylor expanded around inf 10.2

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\]

   10. Simplified 0.4

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z}{x} \cdot 0.002777777777777800001512975569539776188321\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 21506604914.314178466796875:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right) + 0.08333333333333299564049667651488562114537}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt[3]{\left(x - 0.5\right) \cdot \log x}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{z}{x} \cdot 0.002777777777777800001512975569539776188321\right) + \left(\left(\left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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)))