Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.8 → 11.1

Time: 53.4s

Precision: binary64

Cost: 9032

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.95739283321287 \cdot 10^{+61} \lor \neg \left(y \leq 1.969901205390635 \cdot 10^{+54}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + \left(a \cdot \left(y \cdot y\right) + \left({y}^{3} + y \cdot b\right)\right)\right)}\\ \end{array}\]

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 (or (<= y -5.95739283321287e+61) (not (<= y 1.969901205390635e+54)))
   (- (+ x (/ z y)) (/ (* x a) y))
   (/
    (+ t (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
    (+ i (* y (+ c (+ (* a (* y y)) (+ (pow y 3.0) (* y b)))))))))

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 <= -5.95739283321287e+61) || !(y <= 1.969901205390635e+54)) {
		tmp = (x + (z / y)) - ((x * a) / y);
	} else {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + ((a * (y * y)) + (pow(y, 3.0) + (y * b))))));
	}
	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

Alternatives

Alternative 1
Error	11.0
Cost	2440

\[\begin{array}{l} \mathbf{if}\;y \leq -1.0651750119814148 \cdot 10^{+62} \lor \neg \left(y \leq 5.458935799521085 \cdot 10^{+54}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \end{array}\]

Alternative 2
Error	13.7
Cost	2884

\[\begin{array}{l} \mathbf{if}\;y \leq -5.037219563747754 \cdot 10^{+61}:\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{elif}\;y \leq -2.0122531788412233 \cdot 10^{-32}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 6.947262880171667 \cdot 10^{+39}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{elif}\;y \leq 1.9494790515558104 \cdot 10^{+61}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \end{array}\]

Alternative 3
Error	14.1
Cost	2056

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8576710324559692 \cdot 10^{+45} \lor \neg \left(y \leq 2.6225868897624954 \cdot 10^{+46}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \end{array}\]

Alternative 4
Error	16.5
Cost	2435

\[\begin{array}{l} \mathbf{if}\;y \leq -4.052967791373937 \cdot 10^{+53}:\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{elif}\;y \leq 3.379665162233978 \cdot 10^{-116}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{elif}\;y \leq 4.187628943681715 \cdot 10^{+46}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \end{array}\]

Alternative 5
Error	17.3
Cost	1800

\[\begin{array}{l} \mathbf{if}\;y \leq -7.430207020246087 \cdot 10^{+44} \lor \neg \left(y \leq 3.819383754524252 \cdot 10^{+46}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot c}\\ \end{array}\]

Alternative 6
Error	25.1
Cost	1672

\[\begin{array}{l} \mathbf{if}\;y \leq -3537.2386425231316 \lor \neg \left(y \leq 2.2102597879494187 \cdot 10^{+21}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)\right) \cdot \frac{1}{i}\\ \end{array}\]

Alternative 7
Error	28.4
Cost	1544

\[\begin{array}{l} \mathbf{if}\;y \leq -1.0346588861157264 \cdot 10^{-13} \lor \neg \left(y \leq 3.232918153529015 \cdot 10^{-37}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(230661.510616 \cdot \frac{y}{i} + \frac{t}{i}\right) - \frac{t \cdot \left(y \cdot c\right)}{i \cdot i}\\ \end{array}\]

Alternative 8
Error	29.0
Cost	1032

\[\begin{array}{l} \mathbf{if}\;y \leq -4.039223868146957 \cdot 10^{+42} \lor \neg \left(y \leq 1.2708586045076855 \cdot 10^{+61}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i}\\ \end{array}\]

Alternative 9
Error	32.8
Cost	834

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6208311032025923 \cdot 10^{+46}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.40289655192886 \cdot 10^{+48}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 10
Error	45.6
Cost	706

\[\begin{array}{l} \mathbf{if}\;y \leq -1.695635190650733 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.0571860638329912 \cdot 10^{+55}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 11
Error	59.3
Cost	64

\[0\]

Alternative 12
Error	61.5
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if y < -5.95739283321286985e61 or 1.96990120539063488e54 < y

   1. Initial program 62.5

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

   2. Taylor expanded around inf 19.9

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

   3. Simplified 19.9

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

   4. Simplified 19.9

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

   if -5.95739283321286985e61 < y < 1.96990120539063488e54

   1. Initial program 4.7

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

   2. Taylor expanded around 0 4.7

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

   3. Simplified 4.7

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

   4. Simplified 4.7

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.95739283321287 \cdot 10^{+61} \lor \neg \left(y \leq 1.969901205390635 \cdot 10^{+54}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + \left(a \cdot \left(y \cdot y\right) + \left({y}^{3} + y \cdot b\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))