Average Error: 1.8 → 1.8
Time: 54.4s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\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(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \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(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
double f(double z) {
        double r140028 = atan2(1.0, 0.0);
        double r140029 = z;
        double r140030 = r140028 * r140029;
        double r140031 = sin(r140030);
        double r140032 = r140028 / r140031;
        double r140033 = 2.0;
        double r140034 = r140028 * r140033;
        double r140035 = sqrt(r140034);
        double r140036 = 1.0;
        double r140037 = r140036 - r140029;
        double r140038 = r140037 - r140036;
        double r140039 = 7.0;
        double r140040 = r140038 + r140039;
        double r140041 = 0.5;
        double r140042 = r140040 + r140041;
        double r140043 = r140038 + r140041;
        double r140044 = pow(r140042, r140043);
        double r140045 = r140035 * r140044;
        double r140046 = -r140042;
        double r140047 = exp(r140046);
        double r140048 = r140045 * r140047;
        double r140049 = 0.9999999999998099;
        double r140050 = 676.5203681218851;
        double r140051 = r140038 + r140036;
        double r140052 = r140050 / r140051;
        double r140053 = r140049 + r140052;
        double r140054 = -1259.1392167224028;
        double r140055 = r140038 + r140033;
        double r140056 = r140054 / r140055;
        double r140057 = r140053 + r140056;
        double r140058 = 771.3234287776531;
        double r140059 = 3.0;
        double r140060 = r140038 + r140059;
        double r140061 = r140058 / r140060;
        double r140062 = r140057 + r140061;
        double r140063 = -176.6150291621406;
        double r140064 = 4.0;
        double r140065 = r140038 + r140064;
        double r140066 = r140063 / r140065;
        double r140067 = r140062 + r140066;
        double r140068 = 12.507343278686905;
        double r140069 = 5.0;
        double r140070 = r140038 + r140069;
        double r140071 = r140068 / r140070;
        double r140072 = r140067 + r140071;
        double r140073 = -0.13857109526572012;
        double r140074 = 6.0;
        double r140075 = r140038 + r140074;
        double r140076 = r140073 / r140075;
        double r140077 = r140072 + r140076;
        double r140078 = 9.984369578019572e-06;
        double r140079 = r140078 / r140040;
        double r140080 = r140077 + r140079;
        double r140081 = 1.5056327351493116e-07;
        double r140082 = 8.0;
        double r140083 = r140038 + r140082;
        double r140084 = r140081 / r140083;
        double r140085 = r140080 + r140084;
        double r140086 = r140048 * r140085;
        double r140087 = r140032 * r140086;
        return r140087;
}
↓
double f(double z) {
        double r140088 = atan2(1.0, 0.0);
        double r140089 = z;
        double r140090 = r140088 * r140089;
        double r140091 = sin(r140090);
        double r140092 = r140088 / r140091;
        double r140093 = 2.0;
        double r140094 = r140088 * r140093;
        double r140095 = sqrt(r140094);
        double r140096 = 1.0;
        double r140097 = r140096 - r140089;
        double r140098 = r140097 - r140096;
        double r140099 = 7.0;
        double r140100 = r140098 + r140099;
        double r140101 = 0.5;
        double r140102 = r140100 + r140101;
        double r140103 = r140098 + r140101;
        double r140104 = pow(r140102, r140103);
        double r140105 = r140095 * r140104;
        double r140106 = -r140102;
        double r140107 = exp(r140106);
        double r140108 = r140105 * r140107;
        double r140109 = 0.9999999999998099;
        double r140110 = 676.5203681218851;
        double r140111 = r140098 + r140096;
        double r140112 = r140110 / r140111;
        double r140113 = r140109 + r140112;
        double r140114 = -1259.1392167224028;
        double r140115 = r140098 + r140093;
        double r140116 = r140114 / r140115;
        double r140117 = r140113 + r140116;
        double r140118 = 771.3234287776531;
        double r140119 = 3.0;
        double r140120 = r140098 + r140119;
        double r140121 = r140118 / r140120;
        double r140122 = r140117 + r140121;
        double r140123 = -176.6150291621406;
        double r140124 = 4.0;
        double r140125 = r140098 + r140124;
        double r140126 = r140123 / r140125;
        double r140127 = r140122 + r140126;
        double r140128 = 12.507343278686905;
        double r140129 = 5.0;
        double r140130 = r140098 + r140129;
        double r140131 = r140128 / r140130;
        double r140132 = r140127 + r140131;
        double r140133 = -0.13857109526572012;
        double r140134 = 6.0;
        double r140135 = r140098 + r140134;
        double r140136 = r140133 / r140135;
        double r140137 = r140132 + r140136;
        double r140138 = 9.984369578019572e-06;
        double r140139 = r140138 / r140100;
        double r140140 = r140137 + r140139;
        double r140141 = 1.5056327351493116e-07;
        double r140142 = 8.0;
        double r140143 = r140098 + r140142;
        double r140144 = r140141 / r140143;
        double r140145 = r140140 + r140144;
        double r140146 = r140108 * r140145;
        double r140147 = r140092 * r140146;
        return r140147;
}

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

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Final simplification1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))))))