Average Error: 29.3 → 0.7
Time: 58.0s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -4204316794228.3296 \lor \neg \left(z \leq 5.255145995936815 \cdot 10^{+26}\right):\\ \;\;\;\;x + \left(\left(457.96100221584277 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + \left(\frac{a}{{z}^{3}} + 3.13060547623\right)\right)\right) + \left(5864.802528269903 \cdot \frac{-1}{{z}^{3}} + \left(36.527041698806414 \cdot \frac{-1}{z} - 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \cdot \left(z \cdot \left(a + z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right)\right) + b\right)\\ \end{array}\]
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 -4204316794228.3296 \lor \neg \left(z \leq 5.255145995936815 \cdot 10^{+26}\right):\\
\;\;\;\;x + \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(36.527041698806414 \cdot \frac{-1}{z} - 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \cdot \left(z \cdot \left(a + z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right)\right) + b\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 (or (<= z -4204316794228.3296) (not (<= z 5.255145995936815e+26)))
   (+
    x
    (*
     (+
      (+
       (* 457.96100221584277 (/ 1.0 (pow z 2.0)))
       (+ (/ t (pow z 2.0)) (+ (/ a (pow z 3.0)) 3.13060547623)))
      (+
       (* 5864.802528269903 (/ -1.0 (pow z 3.0)))
       (-
        (* 36.527041698806414 (/ -1.0 z))
        (* 15.234687407 (/ t (pow z 3.0))))))
     y))
   (+
    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 <= -4204316794228.3296) || !(z <= 5.255145995936815e+26)) {
		tmp = x + ((((457.96100221584277 * (1.0 / pow(z, 2.0))) + ((t / pow(z, 2.0)) + ((a / pow(z, 3.0)) + 3.13060547623))) + ((5864.802528269903 * (-1.0 / pow(z, 3.0))) + ((36.527041698806414 * (-1.0 / z)) - (15.234687407 * (t / pow(z, 3.0)))))) * y);
	} 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

Original29.3
Target1.0
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4204316794228.32959 or 5.2551459959368147e26 < z

    1. Initial program 57.6

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

    2. Simplified54.3

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

    3. Taylor expanded around inf 1.1

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

    if -4204316794228.32959 < z < 5.2551459959368147e26

    1. Initial program 0.6

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

    2. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4204316794228.3296 \lor \neg \left(z \leq 5.255145995936815 \cdot 10^{+26}\right):\\ \;\;\;\;x + \left(\left(457.96100221584277 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + \left(\frac{a}{{z}^{3}} + 3.13060547623\right)\right)\right) + \left(5864.802528269903 \cdot \frac{-1}{{z}^{3}} + \left(36.527041698806414 \cdot \frac{-1}{z} - 15.234687407 \cdot \frac{t}{{z}^{3}}\right)\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \cdot \left(z \cdot \left(a + z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right)\right) + b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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))))