Average Error: 1.8 → 1.8
Time: 53.1s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\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(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\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(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
double f(double z) {
        double r123989 = atan2(1.0, 0.0);
        double r123990 = z;
        double r123991 = r123989 * r123990;
        double r123992 = sin(r123991);
        double r123993 = r123989 / r123992;
        double r123994 = 2.0;
        double r123995 = r123989 * r123994;
        double r123996 = sqrt(r123995);
        double r123997 = 1.0;
        double r123998 = r123997 - r123990;
        double r123999 = r123998 - r123997;
        double r124000 = 7.0;
        double r124001 = r123999 + r124000;
        double r124002 = 0.5;
        double r124003 = r124001 + r124002;
        double r124004 = r123999 + r124002;
        double r124005 = pow(r124003, r124004);
        double r124006 = r123996 * r124005;
        double r124007 = -r124003;
        double r124008 = exp(r124007);
        double r124009 = r124006 * r124008;
        double r124010 = 0.9999999999998099;
        double r124011 = 676.5203681218851;
        double r124012 = r123999 + r123997;
        double r124013 = r124011 / r124012;
        double r124014 = r124010 + r124013;
        double r124015 = -1259.1392167224028;
        double r124016 = r123999 + r123994;
        double r124017 = r124015 / r124016;
        double r124018 = r124014 + r124017;
        double r124019 = 771.3234287776531;
        double r124020 = 3.0;
        double r124021 = r123999 + r124020;
        double r124022 = r124019 / r124021;
        double r124023 = r124018 + r124022;
        double r124024 = -176.6150291621406;
        double r124025 = 4.0;
        double r124026 = r123999 + r124025;
        double r124027 = r124024 / r124026;
        double r124028 = r124023 + r124027;
        double r124029 = 12.507343278686905;
        double r124030 = 5.0;
        double r124031 = r123999 + r124030;
        double r124032 = r124029 / r124031;
        double r124033 = r124028 + r124032;
        double r124034 = -0.13857109526572012;
        double r124035 = 6.0;
        double r124036 = r123999 + r124035;
        double r124037 = r124034 / r124036;
        double r124038 = r124033 + r124037;
        double r124039 = 9.984369578019572e-06;
        double r124040 = r124039 / r124001;
        double r124041 = r124038 + r124040;
        double r124042 = 1.5056327351493116e-07;
        double r124043 = 8.0;
        double r124044 = r123999 + r124043;
        double r124045 = r124042 / r124044;
        double r124046 = r124041 + r124045;
        double r124047 = r124009 * r124046;
        double r124048 = r123993 * r124047;
        return r124048;
}
↓
double f(double z) {
        double r124049 = atan2(1.0, 0.0);
        double r124050 = z;
        double r124051 = r124049 * r124050;
        double r124052 = sin(r124051);
        double r124053 = r124049 / r124052;
        double r124054 = 2.0;
        double r124055 = r124049 * r124054;
        double r124056 = sqrt(r124055);
        double r124057 = 1.0;
        double r124058 = r124057 - r124050;
        double r124059 = r124058 - r124057;
        double r124060 = 7.0;
        double r124061 = r124059 + r124060;
        double r124062 = 0.5;
        double r124063 = r124061 + r124062;
        double r124064 = r124059 + r124062;
        double r124065 = pow(r124063, r124064);
        double r124066 = r124056 * r124065;
        double r124067 = -r124063;
        double r124068 = exp(r124067);
        double r124069 = r124066 * r124068;
        double r124070 = 0.9999999999998099;
        double r124071 = 676.5203681218851;
        double r124072 = r124059 + r124057;
        double r124073 = r124071 / r124072;
        double r124074 = r124070 + r124073;
        double r124075 = -1259.1392167224028;
        double r124076 = r124059 + r124054;
        double r124077 = r124075 / r124076;
        double r124078 = r124074 + r124077;
        double r124079 = 771.3234287776531;
        double r124080 = 3.0;
        double r124081 = r124059 + r124080;
        double r124082 = r124079 / r124081;
        double r124083 = r124078 + r124082;
        double r124084 = -176.6150291621406;
        double r124085 = 4.0;
        double r124086 = r124059 + r124085;
        double r124087 = r124084 / r124086;
        double r124088 = r124083 + r124087;
        double r124089 = 12.507343278686905;
        double r124090 = 5.0;
        double r124091 = r124059 + r124090;
        double r124092 = r124089 / r124091;
        double r124093 = r124088 + r124092;
        double r124094 = -0.13857109526572012;
        double r124095 = 6.0;
        double r124096 = r124059 + r124095;
        double r124097 = r124094 / r124096;
        double r124098 = r124093 + r124097;
        double r124099 = 9.984369578019572e-06;
        double r124100 = r124099 / r124061;
        double r124101 = r124098 + r124100;
        double r124102 = 1.5056327351493116e-07;
        double r124103 = 8.0;
        double r124104 = r124059 + r124103;
        double r124105 = r124102 / r124104;
        double r124106 = r124101 + r124105;
        double r124107 = r124069 * r124106;
        double r124108 = r124053 * r124107;
        return r124108;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.8

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Final simplification1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

Reproduce

herbie shell --seed 2020001 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1 z) 1) 1))) (/ -1259.1392167224028 (+ (- (- 1 z) 1) 2))) (/ 771.3234287776531 (+ (- (- 1 z) 1) 3))) (/ -176.6150291621406 (+ (- (- 1 z) 1) 4))) (/ 12.507343278686905 (+ (- (- 1 z) 1) 5))) (/ -0.13857109526572012 (+ (- (- 1 z) 1) 6))) (/ 9.984369578019572e-06 (+ (- (- 1 z) 1) 7))) (/ 1.5056327351493116e-07 (+ (- (- 1 z) 1) 8))))))