Average Error: 29.9 → 1.4
Time: 8.6s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.8010994585646794 \cdot 10^{+25}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y + y \cdot \frac{\frac{t}{z} - 36.527041698806414}{z}\right)\\ \mathbf{elif}\;z \leq 8583854952815885:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y + \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\ \end{array}\]
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 -1.8010994585646794 \cdot 10^{+25}:\\
\;\;\;\;x + \left(3.13060547623 \cdot y + y \cdot \frac{\frac{t}{z} - 36.527041698806414}{z}\right)\\

\mathbf{elif}\;z \leq 8583854952815885:\\
\;\;\;\;x + \frac{y \cdot \left(z \cdot \left(z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\

\mathbf{else}:\\
\;\;\;\;x + \left(3.13060547623 \cdot y + \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\

\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 (<= z -1.8010994585646794e+25)
   (+ x (+ (* 3.13060547623 y) (* y (/ (- (/ t z) 36.527041698806414) z))))
   (if (<= z 8583854952815885.0)
     (+
      x
      (/
       (*
        y
        (+
         (* z (+ (* z (+ t (* z (+ (* z 3.13060547623) 11.1667541262)))) a))
         b))
       (+
        (*
         z
         (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
        0.607771387771)))
     (+
      x
      (+ (* 3.13060547623 y) (* (- (/ t z) 36.527041698806414) (/ y z)))))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x + (((double) (y * ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (z * 3.13060547623)) + 11.1667541262)) * z)) + t)) * z)) + a)) * z)) + b)))) / ((double) (((double) (((double) (((double) (((double) (((double) (((double) (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 <= -1.8010994585646794e+25)) {
		tmp = ((double) (x + ((double) (((double) (3.13060547623 * y)) + ((double) (y * (((double) ((t / z) - 36.527041698806414)) / z)))))));
	} else {
		double tmp_1;
		if ((z <= 8583854952815885.0)) {
			tmp_1 = ((double) (x + (((double) (y * ((double) (((double) (z * ((double) (((double) (z * ((double) (t + ((double) (z * ((double) (((double) (z * 3.13060547623)) + 11.1667541262)))))))) + a)))) + b)))) / ((double) (((double) (z * ((double) (((double) (z * ((double) (((double) (z * ((double) (z + 15.234687407)))) + 31.4690115749)))) + 11.9400905721)))) + 0.607771387771)))));
		} else {
			tmp_1 = ((double) (x + ((double) (((double) (3.13060547623 * y)) + ((double) (((double) ((t / z) - 36.527041698806414)) * (y / z)))))));
		}
		tmp = tmp_1;
	}
	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

Original29.9
Target1.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.8010994585646794e25

    1. Initial program Error: 58.1 bits

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

    2. SimplifiedError: 54.6 bits

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

    3. Taylor expanded around inf Error: 9.8 bits

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

    4. SimplifiedError: 2.7 bits

      \[\leadsto x + \color{blue}{\left(3.13060547623 \cdot y + \frac{y}{z} \cdot \left(\frac{t}{z} - 36.527041698806414\right)\right)}\]
    5. Using strategy rm

    6. Applied div-invError: 2.7 bits

      \[\leadsto x + \left(3.13060547623 \cdot y + \color{blue}{\left(y \cdot \frac{1}{z}\right)} \cdot \left(\frac{t}{z} - 36.527041698806414\right)\right)\]

    7. Applied associate-*l*Error: 2.6 bits

      \[\leadsto x + \left(3.13060547623 \cdot y + \color{blue}{y \cdot \left(\frac{1}{z} \cdot \left(\frac{t}{z} - 36.527041698806414\right)\right)}\right)\]

    8. SimplifiedError: 2.5 bits

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

    if -1.8010994585646794e25 < z < 8583854952815885

    1. Initial program Error: 0.5 bits

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

    if 8583854952815885 < z

    1. Initial program Error: 57.3 bits

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

    2. SimplifiedError: 54.4 bits

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

    3. Taylor expanded around inf Error: 9.2 bits

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

    4. SimplifiedError: 2.1 bits

      \[\leadsto x + \color{blue}{\left(3.13060547623 \cdot y + \frac{y}{z} \cdot \left(\frac{t}{z} - 36.527041698806414\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 1.4 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8010994585646794 \cdot 10^{+25}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y + y \cdot \frac{\frac{t}{z} - 36.527041698806414}{z}\right)\\ \mathbf{elif}\;z \leq 8583854952815885:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 \cdot y + \left(\frac{t}{z} - 36.527041698806414\right) \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))