Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.8 → 1.0

Time: 29.9s

Precision: 64

• could not determine a ground truth for program body

  1. z = 1.2797400683317594e+123

Math

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

↓

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

C

double f(double z) {
        double r222941 = atan2(1.0, 0.0);
        double r222942 = 2.0;
        double r222943 = r222941 * r222942;
        double r222944 = sqrt(r222943);
        double r222945 = z;
        double r222946 = 1.0;
        double r222947 = r222945 - r222946;
        double r222948 = 7.0;
        double r222949 = r222947 + r222948;
        double r222950 = 0.5;
        double r222951 = r222949 + r222950;
        double r222952 = r222947 + r222950;
        double r222953 = pow(r222951, r222952);
        double r222954 = r222944 * r222953;
        double r222955 = -r222951;
        double r222956 = exp(r222955);
        double r222957 = r222954 * r222956;
        double r222958 = 0.9999999999998099;
        double r222959 = 676.5203681218851;
        double r222960 = r222947 + r222946;
        double r222961 = r222959 / r222960;
        double r222962 = r222958 + r222961;
        double r222963 = -1259.1392167224028;
        double r222964 = r222947 + r222942;
        double r222965 = r222963 / r222964;
        double r222966 = r222962 + r222965;
        double r222967 = 771.3234287776531;
        double r222968 = 3.0;
        double r222969 = r222947 + r222968;
        double r222970 = r222967 / r222969;
        double r222971 = r222966 + r222970;
        double r222972 = -176.6150291621406;
        double r222973 = 4.0;
        double r222974 = r222947 + r222973;
        double r222975 = r222972 / r222974;
        double r222976 = r222971 + r222975;
        double r222977 = 12.507343278686905;
        double r222978 = 5.0;
        double r222979 = r222947 + r222978;
        double r222980 = r222977 / r222979;
        double r222981 = r222976 + r222980;
        double r222982 = -0.13857109526572012;
        double r222983 = 6.0;
        double r222984 = r222947 + r222983;
        double r222985 = r222982 / r222984;
        double r222986 = r222981 + r222985;
        double r222987 = 9.984369578019572e-06;
        double r222988 = r222987 / r222949;
        double r222989 = r222986 + r222988;
        double r222990 = 1.5056327351493116e-07;
        double r222991 = 8.0;
        double r222992 = r222947 + r222991;
        double r222993 = r222990 / r222992;
        double r222994 = r222989 + r222993;
        double r222995 = r222957 * r222994;
        return r222995;
}

↓

double f(double z) {
        double r222996 = atan2(1.0, 0.0);
        double r222997 = 2.0;
        double r222998 = r222996 * r222997;
        double r222999 = sqrt(r222998);
        double r223000 = z;
        double r223001 = 1.0;
        double r223002 = r223000 - r223001;
        double r223003 = 7.0;
        double r223004 = r223002 + r223003;
        double r223005 = 0.5;
        double r223006 = r223004 + r223005;
        double r223007 = r223002 + r223005;
        double r223008 = pow(r223006, r223007);
        double r223009 = r222999 * r223008;
        double r223010 = -r223006;
        double r223011 = exp(r223010);
        double r223012 = sqrt(r223011);
        double r223013 = r223009 * r223012;
        double r223014 = r223013 * r223012;
        double r223015 = 0.9999999999998099;
        double r223016 = 676.5203681218851;
        double r223017 = r223016 / r223000;
        double r223018 = r223015 + r223017;
        double r223019 = -1259.1392167224028;
        double r223020 = r223002 + r222997;
        double r223021 = r223019 / r223020;
        double r223022 = r223018 + r223021;
        double r223023 = 771.3234287776531;
        double r223024 = 3.0;
        double r223025 = r223002 + r223024;
        double r223026 = r223023 / r223025;
        double r223027 = r223022 + r223026;
        double r223028 = -176.6150291621406;
        double r223029 = 4.0;
        double r223030 = r223002 + r223029;
        double r223031 = r223028 / r223030;
        double r223032 = r223027 + r223031;
        double r223033 = 12.507343278686905;
        double r223034 = 5.0;
        double r223035 = r223002 + r223034;
        double r223036 = r223033 / r223035;
        double r223037 = r223032 + r223036;
        double r223038 = -0.13857109526572012;
        double r223039 = 6.0;
        double r223040 = r223002 + r223039;
        double r223041 = r223038 / r223040;
        double r223042 = r223037 + r223041;
        double r223043 = 9.984369578019572e-06;
        double r223044 = r223043 / r223004;
        double r223045 = r223042 + r223044;
        double r223046 = 1.5056327351493116e-07;
        double r223047 = 8.0;
        double r223048 = r223002 + r223047;
        double r223049 = r223046 / r223048;
        double r223050 = r223045 + r223049;
        double r223051 = r223014 * r223050;
        return r223051;
}

Error

Bits error versus z

Derivation

1. Initial program 61.8

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

2. Taylor expanded around 0 1.0

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

4. Applied add-sqr-sqrt 1.0

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

5. Applied associate-*r* 1.0

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

6. Final simplification 1.0

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

Reproduce

herbie shell --seed 2020059 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  (* (* (* (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)))))