Average Error: 29.7 → 0.6
Time: 21.7s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.0257316015877942 \cdot 10^{+32} \lor \neg \left(z \leq 9.227591230584217 \cdot 10^{+41}\right):\\ \;\;\;\;x + y \cdot \left(\left(\frac{457.96100221584277}{z \cdot z} + \left(\frac{t}{z \cdot z} + \left(3.13060547623 + \frac{a}{{z}^{3}}\right)\right)\right) - \left(\frac{5864.802528269903}{{z}^{3}} + \left(\frac{36.527041698806414}{z} + 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(a + z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right)\right) + b}}\\ \end{array}\]
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 \leq -3.0257316015877942 \cdot 10^{+32} \lor \neg \left(z \leq 9.227591230584217 \cdot 10^{+41}\right):\\
\;\;\;\;x + y \cdot \left(\left(\frac{457.96100221584277}{z \cdot z} + \left(\frac{t}{z \cdot z} + \left(3.13060547623 + \frac{a}{{z}^{3}}\right)\right)\right) - \left(\frac{5864.802528269903}{{z}^{3}} + \left(\frac{36.527041698806414}{z} + 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(a + z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right)\right) + b}}\\

\end{array}
(FPCore (x y z t a b)
 :precision binary64
 (+
  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))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.0257316015877942e+32) (not (<= z 9.227591230584217e+41)))
   (+
    x
    (*
     y
     (-
      (+
       (/ 457.96100221584277 (* z z))
       (+ (/ t (* z z)) (+ 3.13060547623 (/ a (pow z 3.0)))))
      (+
       (/ 5864.802528269903 (pow z 3.0))
       (+ (/ 36.527041698806414 z) (* 15.234687407 (/ t (pow z 3.0))))))))
   (+
    x
    (/
     y
     (/
      (+
       (* z (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
       0.607771387771)
      (+
       (* z (+ a (* z (+ t (* z (+ (* z 3.13060547623) 11.1667541262))))))
       b))))))
double code(double x, double y, double z, double t, double a, double b) {
	return 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));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.0257316015877942e+32) || !(z <= 9.227591230584217e+41)) {
		tmp = x + (y * (((457.96100221584277 / (z * z)) + ((t / (z * z)) + (3.13060547623 + (a / pow(z, 3.0))))) - ((5864.802528269903 / pow(z, 3.0)) + ((36.527041698806414 / z) + (15.234687407 * (t / pow(z, 3.0)))))));
	} else {
		tmp = x + (y / (((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771) / ((z * (a + (z * (t + (z * ((z * 3.13060547623) + 11.1667541262)))))) + 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target1.0
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.0257316015877942e32 or 9.22759123058421736e41 < z

    1. Initial program 59.6

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

    3. Applied *-un-lft-identity_binary64_1167259.6

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

    4. Applied times-frac_binary64_1167857.0

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

    5. Simplified57.0

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

    6. Simplified57.0

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

    7. Taylor expanded around inf 0.6

      \[\leadsto x + y \cdot \color{blue}{\left(\left(457.96100221584277 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + \left(\frac{a}{{z}^{3}} + 3.13060547623\right)\right)\right) - \left(5864.802528269903 \cdot \frac{1}{{z}^{3}} + \left(15.234687407 \cdot \frac{t}{{z}^{3}} + 36.527041698806414 \cdot \frac{1}{z}\right)\right)\right)}\]

    8. Simplified0.6

      \[\leadsto x + y \cdot \color{blue}{\left(\left(\frac{457.96100221584277}{z \cdot z} + \left(\frac{t}{z \cdot z} + \left(3.13060547623 + \frac{a}{{z}^{3}}\right)\right)\right) - \left(\frac{5864.802528269903}{{z}^{3}} + \left(\frac{36.527041698806414}{z} + 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right)}\]

    if -3.0257316015877942e32 < z < 9.22759123058421736e41

    1. Initial program 1.4

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

    3. Applied associate-/l*_binary64_116170.6

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.0257316015877942 \cdot 10^{+32} \lor \neg \left(z \leq 9.227591230584217 \cdot 10^{+41}\right):\\ \;\;\;\;x + y \cdot \left(\left(\frac{457.96100221584277}{z \cdot z} + \left(\frac{t}{z \cdot z} + \left(3.13060547623 + \frac{a}{{z}^{3}}\right)\right)\right) - \left(\frac{5864.802528269903}{{z}^{3}} + \left(\frac{36.527041698806414}{z} + 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(a + z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right)\right) + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021069 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (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 1.0)))))

  (+ 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))))