Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.3 → 9.8

Time: 1.1min

Precision: binary64

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -6.951769050563065 \cdot 10^{+74}:\\ \;\;\;\;\left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \mathbf{elif}\;y \leq -15305887873.150782:\\ \;\;\;\;230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(x \cdot \frac{{y}^{4}}{\left(i + y \cdot \left(c + y \cdot b\right)\right) + {y}^{3} \cdot \left(y + a\right)} + \frac{z}{\frac{c}{{y}^{2}} + \left(a + \left(\frac{i}{{y}^{3}} + \frac{b}{y}\right)\right)}\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.3871402978406024 \cdot 10^{-18}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(x \cdot {y}^{3} + \left(y \cdot 27464.7644705 + z \cdot {y}^{2}\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 3.162362327673477 \cdot 10^{+68}:\\ \;\;\;\;230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(x \cdot \frac{{y}^{4}}{\left(i + y \cdot \left(c + y \cdot b\right)\right) + {y}^{3} \cdot \left(y + a\right)} + \frac{z}{\frac{c}{{y}^{2}} + \left(a + \left(\frac{i}{{y}^{3}} + \frac{b}{y}\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \end{array}\]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -6.951769050563065e+74)
   (- (+ (/ z y) x) (/ (* x a) y))
   (if (<= y -15305887873.150782)
     (+
      (*
       230661.510616
       (/
        y
        (+ (pow y 4.0) (+ (* a (pow y 3.0)) (+ (* (* y y) b) (+ (* y c) i))))))
      (+
       (/
        t
        (+ (pow y 4.0) (+ (* a (pow y 3.0)) (+ (* (* y y) b) (+ (* y c) i)))))
       (+
        (*
         27464.7644705
         (/
          (* y y)
          (+
           (pow y 4.0)
           (+ (* a (pow y 3.0)) (+ (* (* y y) b) (+ (* y c) i))))))
        (+
         (*
          x
          (/
           (pow y 4.0)
           (+ (+ i (* y (+ c (* y b)))) (* (pow y 3.0) (+ y a)))))
         (/ z (+ (/ c (pow y 2.0)) (+ a (+ (/ i (pow y 3.0)) (/ b y)))))))))
     (if (<= y 3.3871402978406024e-18)
       (/
        (+
         t
         (*
          y
          (+
           230661.510616
           (+ (* x (pow y 3.0)) (+ (* y 27464.7644705) (* z (pow y 2.0)))))))
        (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))
       (if (<= y 3.162362327673477e+68)
         (+
          (*
           230661.510616
           (/
            y
            (+
             (pow y 4.0)
             (+ (* a (pow y 3.0)) (+ (* (* y y) b) (+ (* y c) i))))))
          (+
           (/
            t
            (+
             (pow y 4.0)
             (+ (* a (pow y 3.0)) (+ (* (* y y) b) (+ (* y c) i)))))
           (+
            (*
             27464.7644705
             (/
              (* y y)
              (+
               (pow y 4.0)
               (+ (* a (pow y 3.0)) (+ (* (* y y) b) (+ (* y c) i))))))
            (+
             (*
              x
              (/
               (pow y 4.0)
               (+ (+ i (* y (+ c (* y b)))) (* (pow y 3.0) (+ y a)))))
             (/
              z
              (+ (/ c (pow y 2.0)) (+ a (+ (/ i (pow y 3.0)) (/ b y)))))))))
         (- (+ (/ z y) x) (/ (* x a) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -6.951769050563065e+74) {
		tmp = ((z / y) + x) - ((x * a) / y);
	} else if (y <= -15305887873.150782) {
		tmp = (230661.510616 * (y / (pow(y, 4.0) + ((a * pow(y, 3.0)) + (((y * y) * b) + ((y * c) + i)))))) + ((t / (pow(y, 4.0) + ((a * pow(y, 3.0)) + (((y * y) * b) + ((y * c) + i))))) + ((27464.7644705 * ((y * y) / (pow(y, 4.0) + ((a * pow(y, 3.0)) + (((y * y) * b) + ((y * c) + i)))))) + ((x * (pow(y, 4.0) / ((i + (y * (c + (y * b)))) + (pow(y, 3.0) * (y + a))))) + (z / ((c / pow(y, 2.0)) + (a + ((i / pow(y, 3.0)) + (b / y))))))));
	} else if (y <= 3.3871402978406024e-18) {
		tmp = (t + (y * (230661.510616 + ((x * pow(y, 3.0)) + ((y * 27464.7644705) + (z * pow(y, 2.0))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else if (y <= 3.162362327673477e+68) {
		tmp = (230661.510616 * (y / (pow(y, 4.0) + ((a * pow(y, 3.0)) + (((y * y) * b) + ((y * c) + i)))))) + ((t / (pow(y, 4.0) + ((a * pow(y, 3.0)) + (((y * y) * b) + ((y * c) + i))))) + ((27464.7644705 * ((y * y) / (pow(y, 4.0) + ((a * pow(y, 3.0)) + (((y * y) * b) + ((y * c) + i)))))) + ((x * (pow(y, 4.0) / ((i + (y * (c + (y * b)))) + (pow(y, 3.0) * (y + a))))) + (z / ((c / pow(y, 2.0)) + (a + ((i / pow(y, 3.0)) + (b / y))))))));
	} else {
		tmp = ((z / y) + x) - ((x * a) / y);
	}
	return tmp;
}

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

Bits error versus c

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if y < -6.951769050563065e74 or 3.16236232767347694e68 < y

   1. Initial program 63.3

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

   2. Taylor expanded around inf 18.8

      \[\leadsto \color{blue}{\left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y}}\]

   if -6.951769050563065e74 < y < -15305887873.1507816 or 3.3871402978406024e-18 < y < 3.16236232767347694e68

   1. Initial program 29.9

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

   2. Taylor expanded around 0 29.9

      \[\leadsto \color{blue}{230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left({y}^{2} \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left({y}^{2} \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{{y}^{2}}{{y}^{4} + \left(a \cdot {y}^{3} + \left({y}^{2} \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{{y}^{4} + \left(a \cdot {y}^{3} + \left({y}^{2} \cdot b + \left(y \cdot c + i\right)\right)\right)} + \frac{z \cdot {y}^{3}}{{y}^{4} + \left(a \cdot {y}^{3} + \left({y}^{2} \cdot b + \left(y \cdot c + i\right)\right)\right)}\right)\right)\right)}\]

   3. Simplified 29.9

      \[\leadsto \color{blue}{230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \frac{z \cdot {y}^{3}}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)}\right)\right)\right)}\]
   4. Using strategy rm

   5. Applied associate-/l*_binary64_705 25.7

      \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \color{blue}{\frac{z}{\frac{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)}{{y}^{3}}}}\right)\right)\right)\]

   6. Simplified 25.7

      \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \frac{z}{\color{blue}{\frac{{y}^{3} \cdot \left(y + a\right) + \left(i + y \cdot \left(y \cdot b + c\right)\right)}{{y}^{3}}}}\right)\right)\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity_binary64_760 25.7

      \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{\color{blue}{1 \cdot \left({y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)\right)}} + \frac{z}{\frac{{y}^{3} \cdot \left(y + a\right) + \left(i + y \cdot \left(y \cdot b + c\right)\right)}{{y}^{3}}}\right)\right)\right)\]

   9. Applied times-frac_binary64_766 17.8

      \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\color{blue}{\frac{x}{1} \cdot \frac{{y}^{4}}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)}} + \frac{z}{\frac{{y}^{3} \cdot \left(y + a\right) + \left(i + y \cdot \left(y \cdot b + c\right)\right)}{{y}^{3}}}\right)\right)\right)\]

   10. Simplified 17.8

       \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\color{blue}{x} \cdot \frac{{y}^{4}}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \frac{z}{\frac{{y}^{3} \cdot \left(y + a\right) + \left(i + y \cdot \left(y \cdot b + c\right)\right)}{{y}^{3}}}\right)\right)\right)\]

   11. Simplified 17.8

       \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(x \cdot \color{blue}{\frac{{y}^{4}}{\left(i + y \cdot \left(c + y \cdot b\right)\right) + {y}^{3} \cdot \left(y + a\right)}} + \frac{z}{\frac{{y}^{3} \cdot \left(y + a\right) + \left(i + y \cdot \left(y \cdot b + c\right)\right)}{{y}^{3}}}\right)\right)\right)\]

   12. Taylor expanded around 0 16.6

       \[\leadsto 230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(x \cdot \frac{{y}^{4}}{\left(i + y \cdot \left(c + y \cdot b\right)\right) + {y}^{3} \cdot \left(y + a\right)} + \frac{z}{\color{blue}{\frac{c}{{y}^{2}} + \left(a + \left(\frac{i}{{y}^{3}} + \frac{b}{y}\right)\right)}}\right)\right)\right)\]

   if -15305887873.1507816 < y < 3.3871402978406024e-18

   1. Initial program 0.3

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

   2. Taylor expanded around 0 0.3

      \[\leadsto \frac{\left(\color{blue}{\left(x \cdot {y}^{3} + \left(27464.7644705 \cdot y + z \cdot {y}^{2}\right)\right)} + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.951769050563065 \cdot 10^{+74}:\\ \;\;\;\;\left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \mathbf{elif}\;y \leq -15305887873.150782:\\ \;\;\;\;230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(x \cdot \frac{{y}^{4}}{\left(i + y \cdot \left(c + y \cdot b\right)\right) + {y}^{3} \cdot \left(y + a\right)} + \frac{z}{\frac{c}{{y}^{2}} + \left(a + \left(\frac{i}{{y}^{3}} + \frac{b}{y}\right)\right)}\right)\right)\right)\\ \mathbf{elif}\;y \leq 3.3871402978406024 \cdot 10^{-18}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(x \cdot {y}^{3} + \left(y \cdot 27464.7644705 + z \cdot {y}^{2}\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 3.162362327673477 \cdot 10^{+68}:\\ \;\;\;\;230661.510616 \cdot \frac{y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(\frac{t}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(27464.7644705 \cdot \frac{y \cdot y}{{y}^{4} + \left(a \cdot {y}^{3} + \left(\left(y \cdot y\right) \cdot b + \left(y \cdot c + i\right)\right)\right)} + \left(x \cdot \frac{{y}^{4}}{\left(i + y \cdot \left(c + y \cdot b\right)\right) + {y}^{3} \cdot \left(y + a\right)} + \frac{z}{\frac{c}{{y}^{2}} + \left(a + \left(\frac{i}{{y}^{3}} + \frac{b}{y}\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))