Average Error: 1.7 → 1.1
Time: 1.3min
Precision: binary64
\[z \leq 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
↓
\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t_0 + 7\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \sqrt{e^{z + -7.5}}\right) \cdot \sqrt{e^{-0.5 - t_1}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{9 - z \cdot z} \cdot \left(z + 3\right)\right) + \frac{-176.6150291621406}{t_0 + 4}\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_0 + 8}\right)\right) \end{array} \]
\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 := \left(1 - z\right) - 1\\
t_1 := t_0 + 7\\
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \sqrt{e^{z + -7.5}}\right) \cdot \sqrt{e^{-0.5 - t_1}}\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}{9 - z \cdot z} \cdot \left(z + 3\right)\right) + \frac{-176.6150291621406}{t_0 + 4}\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_0 + 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 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (*
       (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
       (sqrt (exp (+ z -7.5))))
      (sqrt (exp (- -0.5 t_1))))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            47.95075976068351
            (fma z 361.7355639412844 (* 519.1279660315847 (* z z))))
           (* (/ 771.3234287776531 (- 9.0 (* z z))) (+ z 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 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 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	return (((double) M_PI) / sin(((double) M_PI) * z)) * ((((pow((7.5 - z), (0.5 - z)) * sqrt(((double) M_PI) * 2.0)) * sqrt(exp(z + -7.5))) * sqrt(exp(-0.5 - t_1))) * (((((((47.95075976068351 + fma(z, 361.7355639412844, (519.1279660315847 * (z * z)))) + ((771.3234287776531 / (9.0 - (z * z))) * (z + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 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. Taylor expanded in z around 0 1.6

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

  3. Simplified1.6

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

  4. Applied add-sqr-sqrt_binary641.1

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

  5. Applied associate-*r*_binary641.1

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

  6. Simplified1.1

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

  7. Applied add-sqr-sqrt_binary641.1

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

  8. Simplified1.1

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

  9. Simplified1.1

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

  10. Applied flip--_binary641.1

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

  11. Applied associate-/r/_binary641.1

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

  12. Applied sqrt-prod_binary641.1

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

  13. Applied flip--_binary641.1

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

  14. Applied associate-/r/_binary641.1

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

  15. Applied sqrt-prod_binary641.1

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

  16. Applied swap-sqr_binary642.2

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

  17. Simplified2.2

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt{e^{z + -7.5}}\right) \cdot \sqrt{e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \color{blue}{\frac{771.3234287776531}{9 - z \cdot z}} \cdot \left(\sqrt{3 + z} \cdot \sqrt{3 + z}\right)\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  18. Simplified1.1

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt{e^{z + -7.5}}\right) \cdot \sqrt{e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{9 - z \cdot z} \cdot \color{blue}{\left(z + 3\right)}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

  19. Final simplification1.1

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \sqrt{e^{z + -7.5}}\right) \cdot \sqrt{e^{-0.5 - \left(\left(\left(1 - z\right) - 1\right) + 7\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(47.95075976068351 + \mathsf{fma}\left(z, 361.7355639412844, 519.1279660315847 \cdot \left(z \cdot z\right)\right)\right) + \frac{771.3234287776531}{9 - z \cdot z} \cdot \left(z + 3\right)\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

Reproduce

herbie shell --seed 2022088 
(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))))))