Average Error: 1.8 → 1.2
Time: 2.5m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)} + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)} + \left(\left(\frac{-176.6150291621405870046146446838974952698}{4 + \left(-z\right)} + \frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6}\right) + \left(\left(0.9999999999998099298181841732002794742584 + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) + \frac{771.3234287776531346025876700878143310547}{\left(-z\right) + 3}\right)\right)\right)\right)\right) \cdot {\left(0.5 + \left(7 + \left(-z\right)\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \frac{\frac{\sqrt{2 \cdot \pi}}{\sqrt[3]{e^{0.5 + \left(7 + \left(-z\right)\right)}} \cdot \sqrt[3]{e^{0.5 + \left(7 + \left(-z\right)\right)}}}}{\sqrt[3]{e^{0.5 + \left(7 + \left(-z\right)\right)}}}\]
\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)
↓
\left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)} + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)} + \left(\left(\frac{-176.6150291621405870046146446838974952698}{4 + \left(-z\right)} + \frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6}\right) + \left(\left(0.9999999999998099298181841732002794742584 + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)} + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right) + \frac{771.3234287776531346025876700878143310547}{\left(-z\right) + 3}\right)\right)\right)\right)\right) \cdot {\left(0.5 + \left(7 + \left(-z\right)\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \frac{\frac{\sqrt{2 \cdot \pi}}{\sqrt[3]{e^{0.5 + \left(7 + \left(-z\right)\right)}} \cdot \sqrt[3]{e^{0.5 + \left(7 + \left(-z\right)\right)}}}}{\sqrt[3]{e^{0.5 + \left(7 + \left(-z\right)\right)}}}
double f(double z) {
        double r5841966 = atan2(1.0, 0.0);
        double r5841967 = z;
        double r5841968 = r5841966 * r5841967;
        double r5841969 = sin(r5841968);
        double r5841970 = r5841966 / r5841969;
        double r5841971 = 2.0;
        double r5841972 = r5841966 * r5841971;
        double r5841973 = sqrt(r5841972);
        double r5841974 = 1.0;
        double r5841975 = r5841974 - r5841967;
        double r5841976 = r5841975 - r5841974;
        double r5841977 = 7.0;
        double r5841978 = r5841976 + r5841977;
        double r5841979 = 0.5;
        double r5841980 = r5841978 + r5841979;
        double r5841981 = r5841976 + r5841979;
        double r5841982 = pow(r5841980, r5841981);
        double r5841983 = r5841973 * r5841982;
        double r5841984 = -r5841980;
        double r5841985 = exp(r5841984);
        double r5841986 = r5841983 * r5841985;
        double r5841987 = 0.9999999999998099;
        double r5841988 = 676.5203681218851;
        double r5841989 = r5841976 + r5841974;
        double r5841990 = r5841988 / r5841989;
        double r5841991 = r5841987 + r5841990;
        double r5841992 = -1259.1392167224028;
        double r5841993 = r5841976 + r5841971;
        double r5841994 = r5841992 / r5841993;
        double r5841995 = r5841991 + r5841994;
        double r5841996 = 771.3234287776531;
        double r5841997 = 3.0;
        double r5841998 = r5841976 + r5841997;
        double r5841999 = r5841996 / r5841998;
        double r5842000 = r5841995 + r5841999;
        double r5842001 = -176.6150291621406;
        double r5842002 = 4.0;
        double r5842003 = r5841976 + r5842002;
        double r5842004 = r5842001 / r5842003;
        double r5842005 = r5842000 + r5842004;
        double r5842006 = 12.507343278686905;
        double r5842007 = 5.0;
        double r5842008 = r5841976 + r5842007;
        double r5842009 = r5842006 / r5842008;
        double r5842010 = r5842005 + r5842009;
        double r5842011 = -0.13857109526572012;
        double r5842012 = 6.0;
        double r5842013 = r5841976 + r5842012;
        double r5842014 = r5842011 / r5842013;
        double r5842015 = r5842010 + r5842014;
        double r5842016 = 9.984369578019572e-06;
        double r5842017 = r5842016 / r5841978;
        double r5842018 = r5842015 + r5842017;
        double r5842019 = 1.5056327351493116e-07;
        double r5842020 = 8.0;
        double r5842021 = r5841976 + r5842020;
        double r5842022 = r5842019 / r5842021;
        double r5842023 = r5842018 + r5842022;
        double r5842024 = r5841986 * r5842023;
        double r5842025 = r5841970 * r5842024;
        return r5842025;
}
↓
double f(double z) {
        double r5842026 = atan2(1.0, 0.0);
        double r5842027 = z;
        double r5842028 = r5842027 * r5842026;
        double r5842029 = sin(r5842028);
        double r5842030 = r5842026 / r5842029;
        double r5842031 = 1.5056327351493116e-07;
        double r5842032 = 8.0;
        double r5842033 = -r5842027;
        double r5842034 = r5842032 + r5842033;
        double r5842035 = r5842031 / r5842034;
        double r5842036 = 12.507343278686905;
        double r5842037 = 5.0;
        double r5842038 = r5842033 + r5842037;
        double r5842039 = r5842036 / r5842038;
        double r5842040 = r5842035 + r5842039;
        double r5842041 = 9.984369578019572e-06;
        double r5842042 = 7.0;
        double r5842043 = r5842042 + r5842033;
        double r5842044 = r5842041 / r5842043;
        double r5842045 = -176.6150291621406;
        double r5842046 = 4.0;
        double r5842047 = r5842046 + r5842033;
        double r5842048 = r5842045 / r5842047;
        double r5842049 = -0.13857109526572012;
        double r5842050 = 6.0;
        double r5842051 = r5842033 + r5842050;
        double r5842052 = r5842049 / r5842051;
        double r5842053 = r5842048 + r5842052;
        double r5842054 = 0.9999999999998099;
        double r5842055 = -1259.1392167224028;
        double r5842056 = 2.0;
        double r5842057 = r5842056 + r5842033;
        double r5842058 = r5842055 / r5842057;
        double r5842059 = 676.5203681218851;
        double r5842060 = 1.0;
        double r5842061 = r5842060 - r5842027;
        double r5842062 = r5842059 / r5842061;
        double r5842063 = r5842058 + r5842062;
        double r5842064 = r5842054 + r5842063;
        double r5842065 = 771.3234287776531;
        double r5842066 = 3.0;
        double r5842067 = r5842033 + r5842066;
        double r5842068 = r5842065 / r5842067;
        double r5842069 = r5842064 + r5842068;
        double r5842070 = r5842053 + r5842069;
        double r5842071 = r5842044 + r5842070;
        double r5842072 = r5842040 + r5842071;
        double r5842073 = r5842030 * r5842072;
        double r5842074 = 0.5;
        double r5842075 = r5842074 + r5842043;
        double r5842076 = r5842033 + r5842074;
        double r5842077 = pow(r5842075, r5842076);
        double r5842078 = r5842073 * r5842077;
        double r5842079 = r5842056 * r5842026;
        double r5842080 = sqrt(r5842079);
        double r5842081 = exp(r5842075);
        double r5842082 = cbrt(r5842081);
        double r5842083 = r5842082 * r5842082;
        double r5842084 = r5842080 / r5842083;
        double r5842085 = r5842084 / r5842082;
        double r5842086 = r5842078 * r5842085;
        return r5842086;
}

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. Simplified1.2

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

  4. Applied add-cube-cbrt1.2

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

  5. Applied associate-/r*1.2

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

  6. Final simplification1.2

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

Reproduce

herbie shell --seed 2019172 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 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-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))