Jmat.Real.gamma, branch z less than 0.5

Average Error: 2.3 → 0.6

Time: 1.3min

Precision: binary64

\[z \leq 0.5\]

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

↓

\[\begin{array}{l} t_0 := \frac{\pi}{\sin \left(z \cdot \pi\right)}\\ t_1 := \left(1 - z\right) - 1\\ t_2 := \frac{12.507343278686905}{t_1 + 5}\\ t_3 := \frac{-0.13857109526572012}{t_1 + 6}\\ t_4 := t_1 + 7\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{t_4}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{t_1 + 8}\\ \mathbf{if}\;z \leq -5.804343082721865 \cdot 10^{-12}:\\ \;\;\;\;t_0 \cdot \left(e^{\left(z + -7.5\right) + \log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + t_1}\right) + \frac{-1259.1392167224028}{2 + t_1}\right) + \frac{771.3234287776531}{t_1 + 3}\right) + \frac{-176.6150291621406}{t_1 + 4}\right) + t_2\right) + t_3\right) + t_5\right) + t_6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{e^{z + -7.5} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(7.5 - z\right)}^{\left(1 - z\right)}\right)}{{\left(0.5 + t_4\right)}^{0.5}} \cdot \left(t_6 + \left(t_5 + \left(t_3 + \left(t_2 + \mathsf{log1p}\left(\mathsf{expm1}\left(47.95075976068351 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, 519.1279660315847, 361.7355639412844\right), \frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (*
     (sqrt (* PI 2.0))
     (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5)))
    (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5))))
   (+
    (+
     (+
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0)))
          (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0)))
         (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0)))
        (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0)))
       (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0)))
      (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0)))
     (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0)))
    (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* z PI))))
        (t_1 (- (- 1.0 z) 1.0))
        (t_2 (/ 12.507343278686905 (+ t_1 5.0)))
        (t_3 (/ -0.13857109526572012 (+ t_1 6.0)))
        (t_4 (+ t_1 7.0))
        (t_5 (/ 9.984369578019572e-6 t_4))
        (t_6 (/ 1.5056327351493116e-7 (+ t_1 8.0))))
   (if (<= z -5.804343082721865e-12)
     (*
      t_0
      (*
       (exp
        (+ (+ z -7.5) (log (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0))))))
       (+
        (+
         (+
          (+
           (+
            (+
             (+
              (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 t_1)))
              (/ -1259.1392167224028 (+ 2.0 t_1)))
             (/ 771.3234287776531 (+ t_1 3.0)))
            (/ -176.6150291621406 (+ t_1 4.0)))
           t_2)
          t_3)
         t_5)
        t_6)))
     (*
      t_0
      (*
       (/
        (*
         (exp (+ z -7.5))
         (* (* (sqrt 2.0) (sqrt PI)) (pow (- 7.5 z) (- 1.0 z))))
        (pow (+ 0.5 t_4) 0.5))
       (+
        t_6
        (+
         t_5
         (+
          t_3
          (+
           t_2
           (log1p
            (expm1
             (+
              47.95075976068351
              (fma
               z
               (fma z 519.1279660315847 361.7355639412844)
               (+
                (/ 771.3234287776531 (- 3.0 z))
                (/ -176.6150291621406 (- 4.0 z))))))))))))))))

C

double code(double z) {
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_PI) * 2.0) * pow(((((1.0 - z) - 1.0) + 7.0) + 0.5), (((1.0 - z) - 1.0) + 0.5))) * exp(-((((1.0 - z) - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (((1.0 - z) - 1.0) + 1.0))) + (-1259.1392167224028 / (((1.0 - z) - 1.0) + 2.0))) + (771.3234287776531 / (((1.0 - z) - 1.0) + 3.0))) + (-176.6150291621406 / (((1.0 - z) - 1.0) + 4.0))) + (12.507343278686905 / (((1.0 - z) - 1.0) + 5.0))) + (-0.13857109526572012 / (((1.0 - z) - 1.0) + 6.0))) + (9.984369578019572e-6 / (((1.0 - z) - 1.0) + 7.0))) + (1.5056327351493116e-7 / (((1.0 - z) - 1.0) + 8.0))));
}

↓

double code(double z) {
	double t_0 = ((double) M_PI) / sin(z * ((double) M_PI));
	double t_1 = (1.0 - z) - 1.0;
	double t_2 = 12.507343278686905 / (t_1 + 5.0);
	double t_3 = -0.13857109526572012 / (t_1 + 6.0);
	double t_4 = t_1 + 7.0;
	double t_5 = 9.984369578019572e-6 / t_4;
	double t_6 = 1.5056327351493116e-7 / (t_1 + 8.0);
	double tmp;
	if (z <= -5.804343082721865e-12) {
		tmp = t_0 * (exp((z + -7.5) + log(pow((7.5 - z), (0.5 - z)) * sqrt(((double) M_PI) * 2.0))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_1))) + (-1259.1392167224028 / (2.0 + t_1))) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + t_2) + t_3) + t_5) + t_6));
	} else {
		tmp = t_0 * (((exp(z + -7.5) * ((sqrt(2.0) * sqrt((double) M_PI)) * pow((7.5 - z), (1.0 - z)))) / pow((0.5 + t_4), 0.5)) * (t_6 + (t_5 + (t_3 + (t_2 + log1p(expm1(47.95075976068351 + fma(z, fma(z, 519.1279660315847, 361.7355639412844), ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))))))));
	}
	return tmp;
}

Error

Bits error versus z

Derivation

1. Split input into 2 regimes

2. if z < -5.80434308272186515e-12

   1. Initial program 22.2

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. Applied pow-to-exp_binary64 22.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot \color{blue}{e^{\log \left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Applied add-exp-log_binary64 22.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{e^{\log \left(\sqrt{\pi \cdot 2}\right)}} \cdot e^{\log \left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. Applied prod-exp_binary64 22.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{e^{\log \left(\sqrt{\pi \cdot 2}\right) + \log \left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}} \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Applied prod-exp_binary64 1.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{e^{\left(\log \left(\sqrt{\pi \cdot 2}\right) + \log \left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 0.5\right)\right) + \left(-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Simplified 1.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{\color{blue}{\left(z + -7.5\right) + \log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -5.80434308272186515e-12 < z

   1. Initial program 1.7

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. Taylor expanded in z around 0 1.1

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(47.95075976068351 + \left(519.1279660315847 \cdot {z}^{2} + 361.7355639412844 \cdot z\right)\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 1.1

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. Applied associate-+l-_binary64 1.1

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\color{blue}{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Applied pow-sub_binary64 1.1

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\frac{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - z\right)}}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Applied associate-*r/_binary64 1.1

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - z\right)}}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}} \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. Applied associate-*l/_binary64 0.6

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - z\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}} \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Simplified 0.6

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{\color{blue}{e^{z + -7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(1 - z\right)}\right)}}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. Applied log1p-expm1-u_binary64 0.6

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{e^{z + -7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(1 - z\right)}\right)}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}} \cdot \left(\left(\left(\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. Simplified 0.6

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{e^{z + -7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(1 - z\right)}\right)}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}} \cdot \left(\left(\left(\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(47.95075976068351 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, 519.1279660315847, 361.7355639412844\right), \frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. Applied sqrt-prod_binary64 0.6

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{e^{z + -7.5} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\pi}\right)} \cdot {\left(7.5 - z\right)}^{\left(1 - z\right)}\right)}{{\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}} \cdot \left(\left(\left(\left(\mathsf{log1p}\left(\mathsf{expm1}\left(47.95075976068351 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, 519.1279660315847, 361.7355639412844\right), \frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.804343082721865 \cdot 10^{-12}:\\ \;\;\;\;\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(e^{\left(z + -7.5\right) + \log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(\left(1 - z\right) - 1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(\left(1 - z\right) - 1\right)}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\frac{e^{z + -7.5} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(7.5 - z\right)}^{\left(1 - z\right)}\right)}{{\left(0.5 + \left(\left(\left(1 - z\right) - 1\right) + 7\right)\right)}^{0.5}} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7} + \left(\frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6} + \left(\frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5} + \mathsf{log1p}\left(\mathsf{expm1}\left(47.95075976068351 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, 519.1279660315847, 361.7355639412844\right), \frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))