Average Error: 1.7 → 0.5
Time: 1.1min
Precision: binary64
\[z \leq 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
↓
\[\begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := 0.9999999999998099 - t_0\\ t_2 := \left(2 - z\right) \cdot t_1\\ t_3 := t_2 \cdot \left(3 - z\right)\\ t_4 := \left(4 - z\right) \cdot t_3\\ t_5 := \left(5 - z\right) \cdot t_4\\ t_6 := \left(6 - z\right) \cdot t_5\\ t_7 := \left(7 - z\right) \cdot t_6\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \mathsf{fma}\left(1.5056327351493116 \cdot 10^{-7}, t_7, \left(8 - z\right) \cdot \mathsf{fma}\left(9.984369578019572 \cdot 10^{-6}, t_6, \left(7 - z\right) \cdot \mathsf{fma}\left(-0.13857109526572012, t_5, \left(6 - z\right) \cdot \mathsf{fma}\left(12.507343278686905, t_4, \left(5 - z\right) \cdot \mathsf{fma}\left(-176.6150291621406, t_3, \left(4 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-1259.1392167224028, t_1, \left(2 - z\right) \cdot \left(0.9999999999996197 - t_0 \cdot t_0\right)\right), 3 - z, t_2 \cdot 771.3234287776531\right)\right)\right)\right)\right)\right)}{t_7 \cdot \left(8 - z\right)} \end{array} \]
\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{676.5203681218851}{1 - z}\\
t_1 := 0.9999999999998099 - t_0\\
t_2 := \left(2 - z\right) \cdot t_1\\
t_3 := t_2 \cdot \left(3 - z\right)\\
t_4 := \left(4 - z\right) \cdot t_3\\
t_5 := \left(5 - z\right) \cdot t_4\\
t_6 := \left(6 - z\right) \cdot t_5\\
t_7 := \left(7 - z\right) \cdot t_6\\
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \mathsf{fma}\left(1.5056327351493116 \cdot 10^{-7}, t_7, \left(8 - z\right) \cdot \mathsf{fma}\left(9.984369578019572 \cdot 10^{-6}, t_6, \left(7 - z\right) \cdot \mathsf{fma}\left(-0.13857109526572012, t_5, \left(6 - z\right) \cdot \mathsf{fma}\left(12.507343278686905, t_4, \left(5 - z\right) \cdot \mathsf{fma}\left(-176.6150291621406, t_3, \left(4 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-1259.1392167224028, t_1, \left(2 - z\right) \cdot \left(0.9999999999996197 - t_0 \cdot t_0\right)\right), 3 - z, t_2 \cdot 771.3234287776531\right)\right)\right)\right)\right)\right)}{t_7 \cdot \left(8 - z\right)}
\end{array}

(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 (/ 676.5203681218851 (- 1.0 z)))
        (t_1 (- 0.9999999999998099 t_0))
        (t_2 (* (- 2.0 z) t_1))
        (t_3 (* t_2 (- 3.0 z)))
        (t_4 (* (- 4.0 z) t_3))
        (t_5 (* (- 5.0 z) t_4))
        (t_6 (* (- 6.0 z) t_5))
        (t_7 (* (- 7.0 z) t_6)))
   (*
    (/ PI (sin (* PI z)))
    (/
     (*
      (* (exp (+ z -7.5)) (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0))))
      (fma
       1.5056327351493116e-7
       t_7
       (*
        (- 8.0 z)
        (fma
         9.984369578019572e-6
         t_6
         (*
          (- 7.0 z)
          (fma
           -0.13857109526572012
           t_5
           (*
            (- 6.0 z)
            (fma
             12.507343278686905
             t_4
             (*
              (- 5.0 z)
              (fma
               -176.6150291621406
               t_3
               (*
                (- 4.0 z)
                (fma
                 (fma
                  -1259.1392167224028
                  t_1
                  (* (- 2.0 z) (- 0.9999999999996197 (* t_0 t_0))))
                 (- 3.0 z)
                 (* t_2 771.3234287776531)))))))))))))
     (* t_7 (- 8.0 z))))))
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 = 676.5203681218851 / (1.0 - z);
	double t_1 = 0.9999999999998099 - t_0;
	double t_2 = (2.0 - z) * t_1;
	double t_3 = t_2 * (3.0 - z);
	double t_4 = (4.0 - z) * t_3;
	double t_5 = (5.0 - z) * t_4;
	double t_6 = (6.0 - z) * t_5;
	double t_7 = (7.0 - z) * t_6;
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((exp(z + -7.5) * (pow((7.5 - z), (0.5 - z)) * sqrt(((double) M_PI) * 2.0))) * fma(1.5056327351493116e-7, t_7, ((8.0 - z) * fma(9.984369578019572e-6, t_6, ((7.0 - z) * fma(-0.13857109526572012, t_5, ((6.0 - z) * fma(12.507343278686905, t_4, ((5.0 - z) * fma(-176.6150291621406, t_3, ((4.0 - z) * fma(fma(-1259.1392167224028, t_1, ((2.0 - z) * (0.9999999999996197 - (t_0 * t_0)))), (3.0 - z), (t_2 * 771.3234287776531))))))))))))) / (t_7 * (8.0 - z)));
}

Error

Bits error versus z

Derivation

  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. Simplified1.7

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

  3. Applied flip-+_binary641.7

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

  4. Applied frac-add_binary641.7

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

  5. Applied frac-add_binary641.7

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

  6. Applied frac-add_binary641.7

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

  7. Applied frac-add_binary641.0

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

  8. Applied frac-add_binary641.0

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

  9. Applied frac-add_binary641.0

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

  10. Applied frac-add_binary640.5

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

  11. Applied associate-*r/_binary640.5

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

  12. Simplified0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\color{blue}{\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \mathsf{fma}\left(1.5056327351493116 \cdot 10^{-7}, \left(7 - z\right) \cdot \left(\left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right)\right)\right), \left(8 - z\right) \cdot \mathsf{fma}\left(9.984369578019572 \cdot 10^{-6}, \left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right)\right), \left(7 - z\right) \cdot \mathsf{fma}\left(-0.13857109526572012, \left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right), \left(6 - z\right) \cdot \mathsf{fma}\left(12.507343278686905, \left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right), \left(5 - z\right) \cdot \mathsf{fma}\left(-176.6150291621406, \left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right), \left(4 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-1259.1392167224028, 0.9999999999998099 - \frac{676.5203681218851}{1 - z}, \left(2 - z\right) \cdot \left(0.9999999999996197 - \frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z}\right)\right), 3 - z, \left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot 771.3234287776531\right)\right)\right)\right)\right)\right)}}{\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot \left(8 - z\right)} \]

  13. Final simplification0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \mathsf{fma}\left(1.5056327351493116 \cdot 10^{-7}, \left(7 - z\right) \cdot \left(\left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right)\right)\right), \left(8 - z\right) \cdot \mathsf{fma}\left(9.984369578019572 \cdot 10^{-6}, \left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right)\right), \left(7 - z\right) \cdot \mathsf{fma}\left(-0.13857109526572012, \left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right), \left(6 - z\right) \cdot \mathsf{fma}\left(12.507343278686905, \left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right), \left(5 - z\right) \cdot \mathsf{fma}\left(-176.6150291621406, \left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right), \left(4 - z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-1259.1392167224028, 0.9999999999998099 - \frac{676.5203681218851}{1 - z}, \left(2 - z\right) \cdot \left(0.9999999999996197 - \frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z}\right)\right), 3 - z, \left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot 771.3234287776531\right)\right)\right)\right)\right)\right)}{\left(\left(7 - z\right) \cdot \left(\left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(3 - z\right)\right)\right)\right)\right)\right) \cdot \left(8 - z\right)} \]

Reproduce

herbie shell --seed 2022097 
(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))))))