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

Average Error: 28.6 → 1.2

Time: 27.8s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.3430881614108444 \cdot 10^{+53}:\\ \;\;\;\;x + \mathsf{fma}\left(3.13060547623, y, \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 3.456748167446755 \cdot 10^{+62}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), 11.9400905721\right), z, 0.607771387771\right)} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), b\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(3.13060547623, y, \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r15173785 = x;
        double r15173786 = y;
        double r15173787 = z;
        double r15173788 = 3.13060547623;
        double r15173789 = r15173787 * r15173788;
        double r15173790 = 11.1667541262;
        double r15173791 = r15173789 + r15173790;
        double r15173792 = r15173791 * r15173787;
        double r15173793 = t;
        double r15173794 = r15173792 + r15173793;
        double r15173795 = r15173794 * r15173787;
        double r15173796 = a;
        double r15173797 = r15173795 + r15173796;
        double r15173798 = r15173797 * r15173787;
        double r15173799 = b;
        double r15173800 = r15173798 + r15173799;
        double r15173801 = r15173786 * r15173800;
        double r15173802 = 15.234687407;
        double r15173803 = r15173787 + r15173802;
        double r15173804 = r15173803 * r15173787;
        double r15173805 = 31.4690115749;
        double r15173806 = r15173804 + r15173805;
        double r15173807 = r15173806 * r15173787;
        double r15173808 = 11.9400905721;
        double r15173809 = r15173807 + r15173808;
        double r15173810 = r15173809 * r15173787;
        double r15173811 = 0.607771387771;
        double r15173812 = r15173810 + r15173811;
        double r15173813 = r15173801 / r15173812;
        double r15173814 = r15173785 + r15173813;
        return r15173814;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r15173815 = z;
        double r15173816 = -1.3430881614108444e+53;
        bool r15173817 = r15173815 <= r15173816;
        double r15173818 = x;
        double r15173819 = 3.13060547623;
        double r15173820 = y;
        double r15173821 = t;
        double r15173822 = r15173821 / r15173815;
        double r15173823 = 36.527041698806414;
        double r15173824 = r15173822 - r15173823;
        double r15173825 = r15173820 / r15173815;
        double r15173826 = r15173824 * r15173825;
        double r15173827 = fma(r15173819, r15173820, r15173826);
        double r15173828 = r15173818 + r15173827;
        double r15173829 = 3.456748167446755e+62;
        bool r15173830 = r15173815 <= r15173829;
        double r15173831 = 15.234687407;
        double r15173832 = r15173831 + r15173815;
        double r15173833 = 31.4690115749;
        double r15173834 = fma(r15173832, r15173815, r15173833);
        double r15173835 = 11.9400905721;
        double r15173836 = fma(r15173815, r15173834, r15173835);
        double r15173837 = 0.607771387771;
        double r15173838 = fma(r15173836, r15173815, r15173837);
        double r15173839 = r15173820 / r15173838;
        double r15173840 = 11.1667541262;
        double r15173841 = fma(r15173819, r15173815, r15173840);
        double r15173842 = fma(r15173841, r15173815, r15173821);
        double r15173843 = a;
        double r15173844 = fma(r15173842, r15173815, r15173843);
        double r15173845 = b;
        double r15173846 = fma(r15173815, r15173844, r15173845);
        double r15173847 = r15173839 * r15173846;
        double r15173848 = r15173847 + r15173818;
        double r15173849 = r15173830 ? r15173848 : r15173828;
        double r15173850 = r15173817 ? r15173828 : r15173849;
        return r15173850;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	28.6
Target	1.0
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.3430881614108444e+53 or 3.456748167446755e+62 < z

   1. Initial program 60.5

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

   2. Simplified 59.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]
   3. Using strategy rm

   4. Applied div-inv 59.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 59.1

      \[\leadsto \color{blue}{y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right) + x}\]

   7. Simplified 59.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), b\right)}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z + 15.234687407, z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{y}}} + x\]

   8. Taylor expanded around inf 7.2

      \[\leadsto \color{blue}{\left(\left(\frac{t \cdot y}{{z}^{2}} + 3.13060547623 \cdot y\right) - 36.527041698806414 \cdot \frac{y}{z}\right)} + x\]

   9. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, \frac{y}{z} \cdot \left(\frac{t}{z} - 36.527041698806414\right)\right)} + x\]

   if -1.3430881614108444e+53 < z < 3.456748167446755e+62

   1. Initial program 3.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

   2. Simplified 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]
   3. Using strategy rm

   4. Applied div-inv 1.5

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 1.5

      \[\leadsto \color{blue}{y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right) + x}\]

   7. Simplified 1.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), b\right)}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z + 15.234687407, z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{y}}} + x\]
   8. Using strategy rm

   9. Applied div-inv 1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), b\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z + 15.234687407, z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{y}}} + x\]

   10. Simplified 1.6

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), b\right) \cdot \color{blue}{\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z + 15.234687407, z, 31.4690115749\right), 11.9400905721\right), z, 0.607771387771\right)}} + x\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.3430881614108444 \cdot 10^{+53}:\\ \;\;\;\;x + \mathsf{fma}\left(3.13060547623, y, \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 3.456748167446755 \cdot 10^{+62}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), 11.9400905721\right), z, 0.607771387771\right)} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), b\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(3.13060547623, y, \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))