Average Error: 59.9 → 0.9
Time: 2.5m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z - -5} + \left(\left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right) + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\right) \cdot \left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z} \cdot \sqrt{2 \cdot \pi}}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}\right) + \left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(0.5 + \left(z - 1\right)\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right)\]
\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)
↓
\left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z - -5} + \left(\left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right) + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\right) \cdot \left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z} \cdot \sqrt{2 \cdot \pi}}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}\right) + \left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(0.5 + \left(z - 1\right)\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right)
double f(double z) {
        double r8171972 = atan2(1.0, 0.0);
        double r8171973 = 2.0;
        double r8171974 = r8171972 * r8171973;
        double r8171975 = sqrt(r8171974);
        double r8171976 = z;
        double r8171977 = 1.0;
        double r8171978 = r8171976 - r8171977;
        double r8171979 = 7.0;
        double r8171980 = r8171978 + r8171979;
        double r8171981 = 0.5;
        double r8171982 = r8171980 + r8171981;
        double r8171983 = r8171978 + r8171981;
        double r8171984 = pow(r8171982, r8171983);
        double r8171985 = r8171975 * r8171984;
        double r8171986 = -r8171982;
        double r8171987 = exp(r8171986);
        double r8171988 = r8171985 * r8171987;
        double r8171989 = 0.9999999999998099;
        double r8171990 = 676.5203681218851;
        double r8171991 = r8171978 + r8171977;
        double r8171992 = r8171990 / r8171991;
        double r8171993 = r8171989 + r8171992;
        double r8171994 = -1259.1392167224028;
        double r8171995 = r8171978 + r8171973;
        double r8171996 = r8171994 / r8171995;
        double r8171997 = r8171993 + r8171996;
        double r8171998 = 771.3234287776531;
        double r8171999 = 3.0;
        double r8172000 = r8171978 + r8171999;
        double r8172001 = r8171998 / r8172000;
        double r8172002 = r8171997 + r8172001;
        double r8172003 = -176.6150291621406;
        double r8172004 = 4.0;
        double r8172005 = r8171978 + r8172004;
        double r8172006 = r8172003 / r8172005;
        double r8172007 = r8172002 + r8172006;
        double r8172008 = 12.507343278686905;
        double r8172009 = 5.0;
        double r8172010 = r8171978 + r8172009;
        double r8172011 = r8172008 / r8172010;
        double r8172012 = r8172007 + r8172011;
        double r8172013 = -0.13857109526572012;
        double r8172014 = 6.0;
        double r8172015 = r8171978 + r8172014;
        double r8172016 = r8172013 / r8172015;
        double r8172017 = r8172012 + r8172016;
        double r8172018 = 9.984369578019572e-06;
        double r8172019 = r8172018 / r8171980;
        double r8172020 = r8172017 + r8172019;
        double r8172021 = 1.5056327351493116e-07;
        double r8172022 = 8.0;
        double r8172023 = r8171978 + r8172022;
        double r8172024 = r8172021 / r8172023;
        double r8172025 = r8172020 + r8172024;
        double r8172026 = r8171988 * r8172025;
        return r8172026;
}
↓
double f(double z) {
        double r8172027 = 12.507343278686905;
        double r8172028 = z;
        double r8172029 = 4.0;
        double r8172030 = r8172028 + r8172029;
        double r8172031 = r8172027 / r8172030;
        double r8172032 = -0.13857109526572012;
        double r8172033 = -5.0;
        double r8172034 = r8172028 - r8172033;
        double r8172035 = r8172032 / r8172034;
        double r8172036 = 676.5203681218851;
        double r8172037 = r8172036 / r8172028;
        double r8172038 = 0.9999999999998099;
        double r8172039 = -1259.1392167224028;
        double r8172040 = -1.0;
        double r8172041 = r8172028 - r8172040;
        double r8172042 = r8172039 / r8172041;
        double r8172043 = r8172038 + r8172042;
        double r8172044 = r8172037 + r8172043;
        double r8172045 = 771.3234287776531;
        double r8172046 = 2.0;
        double r8172047 = r8172028 + r8172046;
        double r8172048 = r8172045 / r8172047;
        double r8172049 = r8172044 + r8172048;
        double r8172050 = -176.6150291621406;
        double r8172051 = 3.0;
        double r8172052 = r8172051 + r8172028;
        double r8172053 = r8172050 / r8172052;
        double r8172054 = r8172049 + r8172053;
        double r8172055 = r8172035 + r8172054;
        double r8172056 = r8172031 + r8172055;
        double r8172057 = 1.0;
        double r8172058 = r8172028 - r8172057;
        double r8172059 = 7.0;
        double r8172060 = r8172058 + r8172059;
        double r8172061 = 0.5;
        double r8172062 = r8172060 + r8172061;
        double r8172063 = -r8172062;
        double r8172064 = exp(r8172063);
        double r8172065 = pow(r8172062, r8172028);
        double r8172066 = atan2(1.0, 0.0);
        double r8172067 = r8172046 * r8172066;
        double r8172068 = sqrt(r8172067);
        double r8172069 = r8172065 * r8172068;
        double r8172070 = r8172057 - r8172061;
        double r8172071 = pow(r8172062, r8172070);
        double r8172072 = r8172069 / r8172071;
        double r8172073 = r8172064 * r8172072;
        double r8172074 = r8172056 * r8172073;
        double r8172075 = r8172061 + r8172058;
        double r8172076 = pow(r8172062, r8172075);
        double r8172077 = r8172068 * r8172076;
        double r8172078 = r8172077 * r8172064;
        double r8172079 = 1.5056327351493116e-07;
        double r8172080 = r8172028 + r8172059;
        double r8172081 = r8172079 / r8172080;
        double r8172082 = 9.984369578019572e-06;
        double r8172083 = r8172082 / r8172060;
        double r8172084 = r8172081 + r8172083;
        double r8172085 = r8172078 * r8172084;
        double r8172086 = r8172074 + r8172085;
        return r8172086;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.9

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

  2. Simplified0.9

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

  4. Applied associate-+l-0.9

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

  5. Applied pow-sub0.9

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

  6. Applied associate-*l/0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- z 1) 7) 0.5) (+ (- z 1) 0.5))) (exp (- (+ (+ (- z 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1) 1))) (/ -1259.1392167224028 (+ (- z 1) 2))) (/ 771.3234287776531 (+ (- z 1) 3))) (/ -176.6150291621406 (+ (- z 1) 4))) (/ 12.507343278686905 (+ (- z 1) 5))) (/ -0.13857109526572012 (+ (- z 1) 6))) (/ 9.984369578019572e-06 (+ (- z 1) 7))) (/ 1.5056327351493116e-07 (+ (- z 1) 8)))))