Average Error: 29.7 → 0.8
Time: 17.7s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
↓
\[\begin{array}{l} t_1 := 3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\\ \mathbf{if}\;z \leq -1381235740.2720332:\\ \;\;\;\;\mathsf{fma}\left(y, t_1 - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\ \mathbf{elif}\;z \leq 111344112.01846091:\\ \;\;\;\;\begin{array}{l} t_2 := 0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)\right)\\ \left(\mathsf{fma}\left(3.13060547623, \frac{y \cdot {z}^{4}}{t_2}, x\right) + \frac{y}{t_2} \cdot \mathsf{fma}\left(a, z, b\right)\right) + \mathsf{fma}\left(11.1667541262, \frac{y \cdot {z}^{3}}{t_2}, \frac{y \cdot \left(t \cdot \left(z \cdot z\right)\right)}{t_2}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t_1 - \frac{36.52704169880642}{z}, x\right)\\ \end{array} \]
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}
t_1 := 3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\\
\mathbf{if}\;z \leq -1381235740.2720332:\\
\;\;\;\;\mathsf{fma}\left(y, t_1 - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\

\mathbf{elif}\;z \leq 111344112.01846091:\\
\;\;\;\;\begin{array}{l}
t_2 := 0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)\right)\\
\left(\mathsf{fma}\left(3.13060547623, \frac{y \cdot {z}^{4}}{t_2}, x\right) + \frac{y}{t_2} \cdot \mathsf{fma}\left(a, z, b\right)\right) + \mathsf{fma}\left(11.1667541262, \frac{y \cdot {z}^{3}}{t_2}, \frac{y \cdot \left(t \cdot \left(z \cdot z\right)\right)}{t_2}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, t_1 - \frac{36.52704169880642}{z}, 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
 (let* ((t_1
         (+
          3.13060547623
          (+
           (/ t (* z z))
           (+ (/ 457.9610022158428 (* z z)) (/ a (pow z 3.0)))))))
   (if (<= z -1381235740.2720332)
     (fma
      y
      (-
       t_1
       (fma
        15.234687407
        (/ t (pow z 3.0))
        (+ (/ 36.52704169880642 z) (/ 5864.8025282699045 (pow z 3.0)))))
      x)
     (if (<= z 111344112.01846091)
       (let* ((t_2
               (+
                0.607771387771
                (+
                 (* (pow z 3.0) (+ z 15.234687407))
                 (* z (+ 11.9400905721 (* z 31.4690115749)))))))
         (+
          (+
           (fma 3.13060547623 (/ (* y (pow z 4.0)) t_2) x)
           (* (/ y t_2) (fma a z b)))
          (fma
           11.1667541262
           (/ (* y (pow z 3.0)) t_2)
           (/ (* y (* t (* z z))) t_2))))
       (fma y (- t_1 (/ 36.52704169880642 z)) 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 t_1 = 3.13060547623 + ((t / (z * z)) + ((457.9610022158428 / (z * z)) + (a / pow(z, 3.0))));
	double tmp;
	if (z <= -1381235740.2720332) {
		tmp = fma(y, (t_1 - fma(15.234687407, (t / pow(z, 3.0)), ((36.52704169880642 / z) + (5864.8025282699045 / pow(z, 3.0))))), x);
	} else if (z <= 111344112.01846091) {
		double t_2 = 0.607771387771 + ((pow(z, 3.0) * (z + 15.234687407)) + (z * (11.9400905721 + (z * 31.4690115749))));
		tmp = (fma(3.13060547623, ((y * pow(z, 4.0)) / t_2), x) + ((y / t_2) * fma(a, z, b))) + fma(11.1667541262, ((y * pow(z, 3.0)) / t_2), ((y * (t * (z * z))) / t_2));
	} else {
		tmp = fma(y, (t_1 - (36.52704169880642 / z)), 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

Original29.7
Target1.1
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -1381235740.2720332

    1. Initial program 56.7

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

    2. Simplified53.3

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

    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

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

    if -1381235740.2720332 < z < 111344112.018460914

    1. Initial program 0.3

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

    2. Simplified0.3

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

    3. Taylor expanded in y around 0 0.6

      \[\leadsto \color{blue}{11.1667541262 \cdot \frac{y \cdot {z}^{3}}{0.607771387771 + \left(31.4690115749 \cdot {z}^{2} + \left({z}^{4} + \left(15.234687407 \cdot {z}^{3} + 11.9400905721 \cdot z\right)\right)\right)} + \left(\frac{y \cdot \left(t \cdot {z}^{2}\right)}{0.607771387771 + \left(31.4690115749 \cdot {z}^{2} + \left({z}^{4} + \left(15.234687407 \cdot {z}^{3} + 11.9400905721 \cdot z\right)\right)\right)} + \left(\frac{y \cdot b}{0.607771387771 + \left(31.4690115749 \cdot {z}^{2} + \left({z}^{4} + \left(15.234687407 \cdot {z}^{3} + 11.9400905721 \cdot z\right)\right)\right)} + \left(3.13060547623 \cdot \frac{y \cdot {z}^{4}}{0.607771387771 + \left(31.4690115749 \cdot {z}^{2} + \left({z}^{4} + \left(15.234687407 \cdot {z}^{3} + 11.9400905721 \cdot z\right)\right)\right)} + \left(\frac{y \cdot \left(a \cdot z\right)}{0.607771387771 + \left(31.4690115749 \cdot {z}^{2} + \left({z}^{4} + \left(15.234687407 \cdot {z}^{3} + 11.9400905721 \cdot z\right)\right)\right)} + x\right)\right)\right)\right)} \]

    4. Simplified0.6

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(3.13060547623, \frac{{z}^{4} \cdot y}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + 31.4690115749 \cdot z\right)\right)}, x\right) + \frac{y}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + 31.4690115749 \cdot z\right)\right)} \cdot \mathsf{fma}\left(a, z, b\right)\right) + \mathsf{fma}\left(11.1667541262, \frac{{z}^{3} \cdot y}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + 31.4690115749 \cdot z\right)\right)}, \frac{\left(t \cdot \left(z \cdot z\right)\right) \cdot y}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + 31.4690115749 \cdot z\right)\right)}\right)} \]

    if 111344112.018460914 < z

    1. Initial program 56.1

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

    2. Simplified53.5

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

    3. Taylor expanded in z around inf 1.1

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

    4. Simplified1.1

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

    5. Taylor expanded in z around inf 1.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1381235740.2720332:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\ \mathbf{elif}\;z \leq 111344112.01846091:\\ \;\;\;\;\left(\mathsf{fma}\left(3.13060547623, \frac{y \cdot {z}^{4}}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)\right)}, x\right) + \frac{y}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)\right)} \cdot \mathsf{fma}\left(a, z, b\right)\right) + \mathsf{fma}\left(11.1667541262, \frac{y \cdot {z}^{3}}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)\right)}, \frac{y \cdot \left(t \cdot \left(z \cdot z\right)\right)}{0.607771387771 + \left({z}^{3} \cdot \left(z + 15.234687407\right) + z \cdot \left(11.9400905721 + z \cdot 31.4690115749\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \frac{36.52704169880642}{z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))