Average Error: 29.5 → 0.9
Time: 27.8s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.559639319434067 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, \mathsf{fma}\left(457.9610022158428, \frac{y}{z \cdot z}, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880642\right)\\ \mathbf{elif}\;z \leq 4.5472185148746684 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\ \end{array} \]
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 -3.559639319434067 \cdot 10^{+26}:\\
\;\;\;\;\mathsf{fma}\left(3.13060547623, y, \mathsf{fma}\left(457.9610022158428, \frac{y}{z \cdot z}, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880642\right)\\

\mathbf{elif}\;z \leq 4.5472185148746684 \cdot 10^{+23}:\\
\;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\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 -3.559639319434067e+26)
   (+
    (fma 3.13060547623 y (fma 457.9610022158428 (/ y (* z z)) x))
    (* (/ y z) (- (/ t z) 36.52704169880642)))
   (if (<= z 4.5472185148746684e+23)
     (fma
      y
      (*
       (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
       (/
        1.0
        (fma
         z
         (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
         0.607771387771)))
      x)
     (fma
      y
      (-
       (+
        3.13060547623
        (+ (/ t (* z z)) (+ (/ 457.9610022158428 (* z z)) (/ a (pow z 3.0)))))
       (fma
        15.234687407
        (/ t (pow z 3.0))
        (+ (/ 36.52704169880642 z) (/ 5864.8025282699045 (pow z 3.0)))))
      x))))
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 <= -3.559639319434067e+26) {
		tmp = fma(3.13060547623, y, fma(457.9610022158428, (y / (z * z)), x)) + ((y / z) * ((t / z) - 36.52704169880642));
	} else if (z <= 4.5472185148746684e+23) {
		tmp = fma(y, (fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) * (1.0 / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771))), x);
	} else {
		tmp = fma(y, ((3.13060547623 + ((t / (z * z)) + ((457.9610022158428 / (z * z)) + (a / pow(z, 3.0))))) - fma(15.234687407, (t / pow(z, 3.0)), ((36.52704169880642 / z) + (5864.8025282699045 / pow(z, 3.0))))), x);
	}
	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

Target

Original29.5
Target1.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.5596393194340673e26

    1. Initial program 59.0

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

    2. Simplified55.9

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

    3. Taylor expanded in z around inf 9.4

      \[\leadsto \color{blue}{\left(\frac{y \cdot t}{{z}^{2}} + \left(3.13060547623 \cdot y + \left(457.9610022158428 \cdot \frac{y}{{z}^{2}} + x\right)\right)\right) - 36.52704169880642 \cdot \frac{y}{z}} \]

    4. Simplified1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, \mathsf{fma}\left(457.9610022158428, \frac{y}{z \cdot z}, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880642\right)} \]

    if -3.5596393194340673e26 < z < 4.5472185148746684e23

    1. Initial program 0.7

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

    2. Simplified0.4

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

    3. Applied div-inv_binary640.3

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

    if 4.5472185148746684e23 < z

    1. Initial program 58.0

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

    2. Simplified55.3

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

    3. Taylor expanded in z around inf 0.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) - \left(15.234687407 \cdot \frac{t}{{z}^{3}} + \left(5864.8025282699045 \cdot \frac{1}{{z}^{3}} + 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]

    4. Simplified0.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right)}, x\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.559639319434067 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, \mathsf{fma}\left(457.9610022158428, \frac{y}{z \cdot z}, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880642\right)\\ \mathbf{elif}\;z \leq 4.5472185148746684 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))