Average Error: 1.8 → 1.8
Time: 1.2m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\sqrt[3]{\sqrt[3]{\left(1 - z\right) - 1} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right)} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) + 6}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]
double f(double z) {
        double r7362931 = atan2(1.0, 0.0);
        double r7362932 = z;
        double r7362933 = r7362931 * r7362932;
        double r7362934 = sin(r7362933);
        double r7362935 = r7362931 / r7362934;
        double r7362936 = 2.0;
        double r7362937 = r7362931 * r7362936;
        double r7362938 = sqrt(r7362937);
        double r7362939 = 1.0;
        double r7362940 = r7362939 - r7362932;
        double r7362941 = r7362940 - r7362939;
        double r7362942 = 7.0;
        double r7362943 = r7362941 + r7362942;
        double r7362944 = 0.5;
        double r7362945 = r7362943 + r7362944;
        double r7362946 = r7362941 + r7362944;
        double r7362947 = pow(r7362945, r7362946);
        double r7362948 = r7362938 * r7362947;
        double r7362949 = -r7362945;
        double r7362950 = exp(r7362949);
        double r7362951 = r7362948 * r7362950;
        double r7362952 = 0.9999999999998099;
        double r7362953 = 676.5203681218851;
        double r7362954 = r7362941 + r7362939;
        double r7362955 = r7362953 / r7362954;
        double r7362956 = r7362952 + r7362955;
        double r7362957 = -1259.1392167224028;
        double r7362958 = r7362941 + r7362936;
        double r7362959 = r7362957 / r7362958;
        double r7362960 = r7362956 + r7362959;
        double r7362961 = 771.3234287776531;
        double r7362962 = 3.0;
        double r7362963 = r7362941 + r7362962;
        double r7362964 = r7362961 / r7362963;
        double r7362965 = r7362960 + r7362964;
        double r7362966 = -176.6150291621406;
        double r7362967 = 4.0;
        double r7362968 = r7362941 + r7362967;
        double r7362969 = r7362966 / r7362968;
        double r7362970 = r7362965 + r7362969;
        double r7362971 = 12.507343278686905;
        double r7362972 = 5.0;
        double r7362973 = r7362941 + r7362972;
        double r7362974 = r7362971 / r7362973;
        double r7362975 = r7362970 + r7362974;
        double r7362976 = -0.13857109526572012;
        double r7362977 = 6.0;
        double r7362978 = r7362941 + r7362977;
        double r7362979 = r7362976 / r7362978;
        double r7362980 = r7362975 + r7362979;
        double r7362981 = 9.984369578019572e-06;
        double r7362982 = r7362981 / r7362943;
        double r7362983 = r7362980 + r7362982;
        double r7362984 = 1.5056327351493116e-07;
        double r7362985 = 8.0;
        double r7362986 = r7362941 + r7362985;
        double r7362987 = r7362984 / r7362986;
        double r7362988 = r7362983 + r7362987;
        double r7362989 = r7362951 * r7362988;
        double r7362990 = r7362935 * r7362989;
        return r7362990;
}
↓
double f(double z) {
        double r7362991 = 2.0;
        double r7362992 = atan2(1.0, 0.0);
        double r7362993 = r7362991 * r7362992;
        double r7362994 = sqrt(r7362993);
        double r7362995 = 7.0;
        double r7362996 = 1.0;
        double r7362997 = z;
        double r7362998 = r7362996 - r7362997;
        double r7362999 = r7362998 - r7362996;
        double r7363000 = r7362995 + r7362999;
        double r7363001 = 0.5;
        double r7363002 = r7363000 + r7363001;
        double r7363003 = r7363001 + r7362999;
        double r7363004 = pow(r7363002, r7363003);
        double r7363005 = r7362994 * r7363004;
        double r7363006 = -r7363002;
        double r7363007 = exp(r7363006);
        double r7363008 = r7363005 * r7363007;
        double r7363009 = 1.5056327351493116e-07;
        double r7363010 = 8.0;
        double r7363011 = r7363010 + r7362999;
        double r7363012 = r7363009 / r7363011;
        double r7363013 = 9.984369578019572e-06;
        double r7363014 = r7363013 / r7363000;
        double r7363015 = -176.6150291621406;
        double r7363016 = 4.0;
        double r7363017 = r7362999 + r7363016;
        double r7363018 = r7363015 / r7363017;
        double r7363019 = 771.3234287776531;
        double r7363020 = 3.0;
        double r7363021 = r7363020 + r7362999;
        double r7363022 = r7363019 / r7363021;
        double r7363023 = 0.9999999999998099;
        double r7363024 = 676.5203681218851;
        double r7363025 = r7362999 + r7362996;
        double r7363026 = r7363024 / r7363025;
        double r7363027 = r7363023 + r7363026;
        double r7363028 = -1259.1392167224028;
        double r7363029 = r7362999 + r7362991;
        double r7363030 = r7363028 / r7363029;
        double r7363031 = r7363027 + r7363030;
        double r7363032 = r7363022 + r7363031;
        double r7363033 = r7363018 + r7363032;
        double r7363034 = 12.507343278686905;
        double r7363035 = 5.0;
        double r7363036 = r7362999 + r7363035;
        double r7363037 = r7363034 / r7363036;
        double r7363038 = r7363033 + r7363037;
        double r7363039 = -0.13857109526572012;
        double r7363040 = cbrt(r7362999);
        double r7363041 = r7363040 * r7363040;
        double r7363042 = r7363040 * r7363041;
        double r7363043 = cbrt(r7363042);
        double r7363044 = r7363043 * r7363041;
        double r7363045 = 6.0;
        double r7363046 = r7363044 + r7363045;
        double r7363047 = r7363039 / r7363046;
        double r7363048 = r7363038 + r7363047;
        double r7363049 = r7363014 + r7363048;
        double r7363050 = r7363012 + r7363049;
        double r7363051 = r7363008 * r7363050;
        double r7363052 = r7362992 * r7362997;
        double r7363053 = sin(r7363052);
        double r7363054 = r7362992 / r7363053;
        double r7363055 = r7363051 * r7363054;
        return r7363055;
}


\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^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \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)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\sqrt[3]{\sqrt[3]{\left(1 - z\right) - 1} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right)} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) + 6}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}

Error

Bits error versus z

Derivation

  1. Initial program 1.8

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\color{blue}{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1}} + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1}}} + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  6. Final simplification1.8

    \[\leadsto \left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\sqrt[3]{\sqrt[3]{\left(1 - z\right) - 1} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right)} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) + 6}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ 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))))))