Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 27.2 → 0.6

Time: 15.8s

Precision: 64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.0944108871381443 \cdot 10^{65} \lor \neg \left(x \le 7.2564691634128278 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot \sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003}\right) \cdot \left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot x\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r1626 = x;
        double r1627 = 2.0;
        double r1628 = r1626 - r1627;
        double r1629 = 4.16438922228;
        double r1630 = r1626 * r1629;
        double r1631 = 78.6994924154;
        double r1632 = r1630 + r1631;
        double r1633 = r1632 * r1626;
        double r1634 = 137.519416416;
        double r1635 = r1633 + r1634;
        double r1636 = r1635 * r1626;
        double r1637 = y;
        double r1638 = r1636 + r1637;
        double r1639 = r1638 * r1626;
        double r1640 = z;
        double r1641 = r1639 + r1640;
        double r1642 = r1628 * r1641;
        double r1643 = 43.3400022514;
        double r1644 = r1626 + r1643;
        double r1645 = r1644 * r1626;
        double r1646 = 263.505074721;
        double r1647 = r1645 + r1646;
        double r1648 = r1647 * r1626;
        double r1649 = 313.399215894;
        double r1650 = r1648 + r1649;
        double r1651 = r1650 * r1626;
        double r1652 = 47.066876606;
        double r1653 = r1651 + r1652;
        double r1654 = r1642 / r1653;
        return r1654;
}

↓

double f(double x, double y, double z) {
        double r1655 = x;
        double r1656 = -9.094410887138144e+65;
        bool r1657 = r1655 <= r1656;
        double r1658 = 7.256469163412828e+43;
        bool r1659 = r1655 <= r1658;
        double r1660 = !r1659;
        bool r1661 = r1657 || r1660;
        double r1662 = y;
        double r1663 = 2.0;
        double r1664 = pow(r1655, r1663);
        double r1665 = r1662 / r1664;
        double r1666 = 4.16438922228;
        double r1667 = r1666 * r1655;
        double r1668 = r1665 + r1667;
        double r1669 = 110.1139242984811;
        double r1670 = r1668 - r1669;
        double r1671 = 2.0;
        double r1672 = r1655 - r1671;
        double r1673 = r1655 * r1666;
        double r1674 = 78.6994924154;
        double r1675 = r1673 + r1674;
        double r1676 = r1675 * r1655;
        double r1677 = 137.519416416;
        double r1678 = r1676 + r1677;
        double r1679 = r1678 * r1655;
        double r1680 = r1679 + r1662;
        double r1681 = r1680 * r1655;
        double r1682 = z;
        double r1683 = r1681 + r1682;
        double r1684 = 43.3400022514;
        double r1685 = r1655 + r1684;
        double r1686 = r1685 * r1655;
        double r1687 = 263.505074721;
        double r1688 = r1686 + r1687;
        double r1689 = cbrt(r1688);
        double r1690 = r1689 * r1689;
        double r1691 = r1689 * r1655;
        double r1692 = r1690 * r1691;
        double r1693 = 313.399215894;
        double r1694 = r1692 + r1693;
        double r1695 = r1694 * r1655;
        double r1696 = 47.066876606;
        double r1697 = r1695 + r1696;
        double r1698 = r1683 / r1697;
        double r1699 = r1672 * r1698;
        double r1700 = r1661 ? r1670 : r1699;
        return r1700;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.2
Target	0.4
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -9.094410887138144e+65 or 7.256469163412828e+43 < x

   1. Initial program 62.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

   if -9.094410887138144e+65 < x < 7.256469163412828e+43

   1. Initial program 1.7

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.7

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 0.6

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 0.6

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.7

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\left(\left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot \sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003}\right) \cdot \sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003}\right)} \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   8. Applied associate-*l* 0.7

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot \sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003}\right) \cdot \left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot x\right)} + 313.399215894\right) \cdot x + 47.066876606000001}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.0944108871381443 \cdot 10^{65} \lor \neg \left(x \le 7.2564691634128278 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot \sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003}\right) \cdot \left(\sqrt[3]{\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003} \cdot x\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))