Average Error: 1.8 → 1.8
Time: 52.9s
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 r126000 = atan2(1.0, 0.0);
        double r126001 = z;
        double r126002 = r126000 * r126001;
        double r126003 = sin(r126002);
        double r126004 = r126000 / r126003;
        double r126005 = 2.0;
        double r126006 = r126000 * r126005;
        double r126007 = sqrt(r126006);
        double r126008 = 1.0;
        double r126009 = r126008 - r126001;
        double r126010 = r126009 - r126008;
        double r126011 = 7.0;
        double r126012 = r126010 + r126011;
        double r126013 = 0.5;
        double r126014 = r126012 + r126013;
        double r126015 = r126010 + r126013;
        double r126016 = pow(r126014, r126015);
        double r126017 = r126007 * r126016;
        double r126018 = -r126014;
        double r126019 = exp(r126018);
        double r126020 = r126017 * r126019;
        double r126021 = 0.9999999999998099;
        double r126022 = 676.5203681218851;
        double r126023 = r126010 + r126008;
        double r126024 = r126022 / r126023;
        double r126025 = r126021 + r126024;
        double r126026 = -1259.1392167224028;
        double r126027 = r126010 + r126005;
        double r126028 = r126026 / r126027;
        double r126029 = r126025 + r126028;
        double r126030 = 771.3234287776531;
        double r126031 = 3.0;
        double r126032 = r126010 + r126031;
        double r126033 = r126030 / r126032;
        double r126034 = r126029 + r126033;
        double r126035 = -176.6150291621406;
        double r126036 = 4.0;
        double r126037 = r126010 + r126036;
        double r126038 = r126035 / r126037;
        double r126039 = r126034 + r126038;
        double r126040 = 12.507343278686905;
        double r126041 = 5.0;
        double r126042 = r126010 + r126041;
        double r126043 = r126040 / r126042;
        double r126044 = r126039 + r126043;
        double r126045 = -0.13857109526572012;
        double r126046 = 6.0;
        double r126047 = r126010 + r126046;
        double r126048 = r126045 / r126047;
        double r126049 = r126044 + r126048;
        double r126050 = 9.984369578019572e-06;
        double r126051 = r126050 / r126012;
        double r126052 = r126049 + r126051;
        double r126053 = 1.5056327351493116e-07;
        double r126054 = 8.0;
        double r126055 = r126010 + r126054;
        double r126056 = r126053 / r126055;
        double r126057 = r126052 + r126056;
        double r126058 = r126020 * r126057;
        double r126059 = r126004 * r126058;
        return r126059;
}
↓
double f(double z) {
        double r126060 = atan2(1.0, 0.0);
        double r126061 = z;
        double r126062 = r126060 * r126061;
        double r126063 = sin(r126062);
        double r126064 = r126060 / r126063;
        double r126065 = 2.0;
        double r126066 = r126060 * r126065;
        double r126067 = sqrt(r126066);
        double r126068 = 1.0;
        double r126069 = r126068 - r126061;
        double r126070 = r126069 - r126068;
        double r126071 = 7.0;
        double r126072 = r126070 + r126071;
        double r126073 = 0.5;
        double r126074 = r126072 + r126073;
        double r126075 = r126070 + r126073;
        double r126076 = pow(r126074, r126075);
        double r126077 = r126067 * r126076;
        double r126078 = -r126074;
        double r126079 = exp(r126078);
        double r126080 = r126077 * r126079;
        double r126081 = 0.9999999999998099;
        double r126082 = 676.5203681218851;
        double r126083 = r126070 + r126068;
        double r126084 = r126082 / r126083;
        double r126085 = r126081 + r126084;
        double r126086 = -1259.1392167224028;
        double r126087 = r126070 + r126065;
        double r126088 = r126086 / r126087;
        double r126089 = r126085 + r126088;
        double r126090 = 771.3234287776531;
        double r126091 = 3.0;
        double r126092 = r126070 + r126091;
        double r126093 = r126090 / r126092;
        double r126094 = r126089 + r126093;
        double r126095 = -176.6150291621406;
        double r126096 = 4.0;
        double r126097 = r126070 + r126096;
        double r126098 = r126095 / r126097;
        double r126099 = r126094 + r126098;
        double r126100 = 12.507343278686905;
        double r126101 = 5.0;
        double r126102 = r126070 + r126101;
        double r126103 = r126100 / r126102;
        double r126104 = r126099 + r126103;
        double r126105 = -0.13857109526572012;
        double r126106 = 6.0;
        double r126107 = r126070 + r126106;
        double r126108 = r126105 / r126107;
        double r126109 = r126104 + r126108;
        double r126110 = 9.984369578019572e-06;
        double r126111 = r126110 / r126072;
        double r126112 = r126109 + r126111;
        double r126113 = 1.5056327351493116e-07;
        double r126114 = 8.0;
        double r126115 = r126070 + r126114;
        double r126116 = r126113 / r126115;
        double r126117 = r126112 + r126116;
        double r126118 = r126080 * r126117;
        double r126119 = r126064 * r126118;
        return r126119;
}

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 2019362 +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))))))