Average Error: 1.7 → 0.5
Time: 2.4min
Precision: binary64
\[z \leq 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
↓
\[\begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := t_0 + -0.9999999999998099\\ t_2 := \left(1 - z\right) - 1\\ t_3 := \frac{676.5203681218851}{1 + t_2}\\ t_4 := \sqrt[3]{t_2}\\ t_5 := t_2 + 7\\ t_6 := t_0 \cdot t_1\\ t_7 := \mathsf{fma}\left(t_6, \mathsf{fma}\left(t_0, t_1, -0.9999999999996197\right), 0.9999999999992396\right)\\ t_8 := 0.9999999999988594 + \left({t_0}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(t_5 + 0.5\right)}^{\left(t_2 + 0.5\right)}\right) \cdot e^{-0.5 - \left(7 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(t_7, \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({t_0}^{3}\right)}^{3}\right), t_8 \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {t_6}^{3}\right)\right)\right)}{t_7 \cdot t_8}}{\left(0.9999999999996197 + \left(t_3 \cdot t_3 - 0.9999999999998099 \cdot t_3\right)\right) \cdot \left(2 + t_4 \cdot \left(t_4 \cdot t_4\right)\right)} + \frac{771.3234287776531}{t_2 + 3}\right) + \frac{-176.6150291621406}{t_2 + 4}\right) + \frac{12.507343278686905}{t_2 + 5}\right) + \frac{-0.13857109526572012}{t_2 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_5}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_2 + 8}\right)\right) \end{array} \]
\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 := t_0 + -0.9999999999998099\\
t_2 := \left(1 - z\right) - 1\\
t_3 := \frac{676.5203681218851}{1 + t_2}\\
t_4 := \sqrt[3]{t_2}\\
t_5 := t_2 + 7\\
t_6 := t_0 \cdot t_1\\
t_7 := \mathsf{fma}\left(t_6, \mathsf{fma}\left(t_0, t_1, -0.9999999999996197\right), 0.9999999999992396\right)\\
t_8 := 0.9999999999988594 + \left({t_0}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\\
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(t_5 + 0.5\right)}^{\left(t_2 + 0.5\right)}\right) \cdot e^{-0.5 - \left(7 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(t_7, \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({t_0}^{3}\right)}^{3}\right), t_8 \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {t_6}^{3}\right)\right)\right)}{t_7 \cdot t_8}}{\left(0.9999999999996197 + \left(t_3 \cdot t_3 - 0.9999999999998099 \cdot t_3\right)\right) \cdot \left(2 + t_4 \cdot \left(t_4 \cdot t_4\right)\right)} + \frac{771.3234287776531}{t_2 + 3}\right) + \frac{-176.6150291621406}{t_2 + 4}\right) + \frac{12.507343278686905}{t_2 + 5}\right) + \frac{-0.13857109526572012}{t_2 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_5}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_2 + 8}\right)\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 (+ t_0 -0.9999999999998099))
        (t_2 (- (- 1.0 z) 1.0))
        (t_3 (/ 676.5203681218851 (+ 1.0 t_2)))
        (t_4 (cbrt t_2))
        (t_5 (+ t_2 7.0))
        (t_6 (* t_0 t_1))
        (t_7 (fma t_6 (fma t_0 t_1 -0.9999999999996197) 0.9999999999992396))
        (t_8
         (+
          0.9999999999988594
          (- (pow t_0 6.0) (/ 309629712.517218 (pow (- 1.0 z) 3.0))))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (sqrt (* PI 2.0)) (pow (+ t_5 0.5) (+ t_2 0.5)))
      (exp (- -0.5 (- 7.0 z))))
     (+
      (+
       (+
        (+
         (+
          (+
           (/
            (/
             (fma
              t_7
              (*
               (+ 1.0 (- 1.0 z))
               (+ 0.9999999999982891 (pow (pow t_0 3.0) 3.0)))
              (*
               t_8
               (* -1259.1392167224028 (+ 0.9999999999988594 (pow t_6 3.0)))))
             (* t_7 t_8))
            (*
             (+ 0.9999999999996197 (- (* t_3 t_3) (* 0.9999999999998099 t_3)))
             (+ 2.0 (* t_4 (* t_4 t_4)))))
           (/ 771.3234287776531 (+ t_2 3.0)))
          (/ -176.6150291621406 (+ t_2 4.0)))
         (/ 12.507343278686905 (+ t_2 5.0)))
        (/ -0.13857109526572012 (+ t_2 6.0)))
       (/ 9.984369578019572e-6 t_5))
      (/ 1.5056327351493116e-7 (+ t_2 8.0)))))))
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 = t_0 + -0.9999999999998099;
	double t_2 = (1.0 - z) - 1.0;
	double t_3 = 676.5203681218851 / (1.0 + t_2);
	double t_4 = cbrt(t_2);
	double t_5 = t_2 + 7.0;
	double t_6 = t_0 * t_1;
	double t_7 = fma(t_6, fma(t_0, t_1, -0.9999999999996197), 0.9999999999992396);
	double t_8 = 0.9999999999988594 + (pow(t_0, 6.0) - (309629712.517218 / pow((1.0 - z), 3.0)));
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_PI) * 2.0) * pow((t_5 + 0.5), (t_2 + 0.5))) * exp(-0.5 - (7.0 - z))) * ((((((((fma(t_7, ((1.0 + (1.0 - z)) * (0.9999999999982891 + pow(pow(t_0, 3.0), 3.0))), (t_8 * (-1259.1392167224028 * (0.9999999999988594 + pow(t_6, 3.0))))) / (t_7 * t_8)) / ((0.9999999999996197 + ((t_3 * t_3) - (0.9999999999998099 * t_3))) * (2.0 + (t_4 * (t_4 * t_4))))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))) + (-0.13857109526572012 / (t_2 + 6.0))) + (9.984369578019572e-6 / t_5)) + (1.5056327351493116e-7 / (t_2 + 8.0))));
}

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. Applied flip3-+_binary641.7

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

  3. Applied frac-add_binary641.0

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

  4. Applied flip3-+_binary641.0

    \[\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(\frac{\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right) + \color{blue}{\frac{{\left(0.9999999999998099 \cdot 0.9999999999998099\right)}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)}} \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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 associate-*l/_binary641.0

    \[\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(\frac{\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right) + \color{blue}{\frac{\left({\left(0.9999999999998099 \cdot 0.9999999999998099\right)}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)}}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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. Applied flip3-+_binary641.0

    \[\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(\frac{\color{blue}{\frac{{\left({0.9999999999998099}^{3}\right)}^{3} + {\left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)}^{3}}{{0.9999999999998099}^{3} \cdot {0.9999999999998099}^{3} + \left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} - {0.9999999999998099}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)}} \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right) + \frac{\left({\left(0.9999999999998099 \cdot 0.9999999999998099\right)}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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) \]

  7. Applied associate-*l/_binary641.0

    \[\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(\frac{\color{blue}{\frac{\left({\left({0.9999999999998099}^{3}\right)}^{3} + {\left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)}^{3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right)}{{0.9999999999998099}^{3} \cdot {0.9999999999998099}^{3} + \left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} - {0.9999999999998099}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)}} + \frac{\left({\left(0.9999999999998099 \cdot 0.9999999999998099\right)}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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) \]

  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(\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(\frac{\color{blue}{\frac{\left(\left({\left({0.9999999999998099}^{3}\right)}^{3} + {\left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)}^{3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 2\right)\right) \cdot \left(\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)\right) + \left({0.9999999999998099}^{3} \cdot {0.9999999999998099}^{3} + \left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} - {0.9999999999998099}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)\right) \cdot \left(\left({\left(0.9999999999998099 \cdot 0.9999999999998099\right)}^{3} + {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right) \cdot -1259.1392167224028\right)}{\left({0.9999999999998099}^{3} \cdot {0.9999999999998099}^{3} + \left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} - {0.9999999999998099}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)\right) \cdot \left(\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)\right)}}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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) \]

  9. Simplified0.5

    \[\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(\frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}}{\left({0.9999999999998099}^{3} \cdot {0.9999999999998099}^{3} + \left({\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3} - {0.9999999999998099}^{3} \cdot {\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}\right)\right) \cdot \left(\left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099\right) + \left(\left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) - \left(0.9999999999998099 \cdot 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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) \]

  10. Simplified0.5

    \[\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(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\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) \]

  11. Applied add-cube-cbrt_binary640.5

    \[\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(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1}} + 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) \]

  12. Applied *-un-lft-identity_binary640.5

    \[\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) - \color{blue}{1 \cdot 1}\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1} + 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) \]

  13. Applied *-un-lft-identity_binary640.5

    \[\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(\color{blue}{1 \cdot \left(1 - z\right)} - 1 \cdot 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1} + 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) \]

  14. Applied distribute-lft-out--_binary640.5

    \[\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(\color{blue}{1 \cdot \left(\left(1 - z\right) - 1\right)} + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1} + 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) \]

  15. Simplified0.5

    \[\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(1 \cdot \color{blue}{\left(-z\right)} + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) \cdot \left(\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1} + 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) \]

  16. Final simplification0.5

    \[\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^{-0.5 - \left(7 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right), \left(1 + \left(1 - z\right)\right) \cdot \left(0.9999999999982891 + {\left({\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)}^{3}\right), \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right) \cdot \left(-1259.1392167224028 \cdot \left(0.9999999999988594 + {\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right)\right)}^{3}\right)\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} + -0.9999999999998099\right), \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + -0.9999999999998099, -0.9999999999996197\right), 0.9999999999992396\right) \cdot \left(0.9999999999988594 + \left({\left(\frac{676.5203681218851}{1 - z}\right)}^{6} - \frac{309629712.517218}{{\left(1 - z\right)}^{3}}\right)\right)}}{\left(0.9999999999996197 + \left(\frac{676.5203681218851}{1 + \left(\left(1 - z\right) - 1\right)} \cdot \frac{676.5203681218851}{1 + \left(\left(1 - z\right) - 1\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 + \left(\left(1 - z\right) - 1\right)}\right)\right) \cdot \left(2 + \sqrt[3]{\left(1 - z\right) - 1} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\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) \]

Reproduce

herbie shell --seed 2022020 
(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))))))