Jmat.Real.gamma, branch z greater than 0.5

Average Error: 59.8 → 0.8

Time: 3.7m

Precision: 64

• could not determine a ground truth for program body

  1. z = 5.03993005917793e+140

Math

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

↓

\[\left(\left(\left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right)\right) \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) + \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{-1259.1392167224028}{z - -1} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851}{z}\right)\right)\right)\right) \cdot \sqrt{\sqrt{\pi}}\right)\]

C

double f(double z) {
        double r10901002 = atan2(1.0, 0.0);
        double r10901003 = 2.0;
        double r10901004 = r10901002 * r10901003;
        double r10901005 = sqrt(r10901004);
        double r10901006 = z;
        double r10901007 = 1.0;
        double r10901008 = r10901006 - r10901007;
        double r10901009 = 7.0;
        double r10901010 = r10901008 + r10901009;
        double r10901011 = 0.5;
        double r10901012 = r10901010 + r10901011;
        double r10901013 = r10901008 + r10901011;
        double r10901014 = pow(r10901012, r10901013);
        double r10901015 = r10901005 * r10901014;
        double r10901016 = -r10901012;
        double r10901017 = exp(r10901016);
        double r10901018 = r10901015 * r10901017;
        double r10901019 = 0.9999999999998099;
        double r10901020 = 676.5203681218851;
        double r10901021 = r10901008 + r10901007;
        double r10901022 = r10901020 / r10901021;
        double r10901023 = r10901019 + r10901022;
        double r10901024 = -1259.1392167224028;
        double r10901025 = r10901008 + r10901003;
        double r10901026 = r10901024 / r10901025;
        double r10901027 = r10901023 + r10901026;
        double r10901028 = 771.3234287776531;
        double r10901029 = 3.0;
        double r10901030 = r10901008 + r10901029;
        double r10901031 = r10901028 / r10901030;
        double r10901032 = r10901027 + r10901031;
        double r10901033 = -176.6150291621406;
        double r10901034 = 4.0;
        double r10901035 = r10901008 + r10901034;
        double r10901036 = r10901033 / r10901035;
        double r10901037 = r10901032 + r10901036;
        double r10901038 = 12.507343278686905;
        double r10901039 = 5.0;
        double r10901040 = r10901008 + r10901039;
        double r10901041 = r10901038 / r10901040;
        double r10901042 = r10901037 + r10901041;
        double r10901043 = -0.13857109526572012;
        double r10901044 = 6.0;
        double r10901045 = r10901008 + r10901044;
        double r10901046 = r10901043 / r10901045;
        double r10901047 = r10901042 + r10901046;
        double r10901048 = 9.984369578019572e-06;
        double r10901049 = r10901048 / r10901010;
        double r10901050 = r10901047 + r10901049;
        double r10901051 = 1.5056327351493116e-07;
        double r10901052 = 8.0;
        double r10901053 = r10901008 + r10901052;
        double r10901054 = r10901051 / r10901053;
        double r10901055 = r10901050 + r10901054;
        double r10901056 = r10901018 * r10901055;
        return r10901056;
}

↓

double f(double z) {
        double r10901057 = -0.13857109526572012;
        double r10901058 = -1.0;
        double r10901059 = z;
        double r10901060 = -6.0;
        double r10901061 = r10901059 - r10901060;
        double r10901062 = r10901058 + r10901061;
        double r10901063 = r10901057 / r10901062;
        double r10901064 = 12.507343278686905;
        double r10901065 = 4.0;
        double r10901066 = r10901059 + r10901065;
        double r10901067 = r10901064 / r10901066;
        double r10901068 = r10901063 + r10901067;
        double r10901069 = 9.984369578019572e-06;
        double r10901070 = r10901069 / r10901061;
        double r10901071 = 1.5056327351493116e-07;
        double r10901072 = 7.0;
        double r10901073 = r10901059 + r10901072;
        double r10901074 = r10901071 / r10901073;
        double r10901075 = r10901070 + r10901074;
        double r10901076 = r10901068 + r10901075;
        double r10901077 = atan2(1.0, 0.0);
        double r10901078 = sqrt(r10901077);
        double r10901079 = sqrt(r10901078);
        double r10901080 = r10901076 * r10901079;
        double r10901081 = 6.5;
        double r10901082 = r10901059 + r10901081;
        double r10901083 = log(r10901082);
        double r10901084 = 0.5;
        double r10901085 = r10901084 - r10901059;
        double r10901086 = r10901083 * r10901085;
        double r10901087 = -r10901086;
        double r10901088 = exp(r10901087);
        double r10901089 = 2.0;
        double r10901090 = sqrt(r10901089);
        double r10901091 = r10901088 * r10901090;
        double r10901092 = exp(r10901082);
        double r10901093 = r10901091 / r10901092;
        double r10901094 = r10901093 * r10901079;
        double r10901095 = r10901080 * r10901094;
        double r10901096 = -176.6150291621406;
        double r10901097 = 3.0;
        double r10901098 = r10901059 + r10901097;
        double r10901099 = r10901096 / r10901098;
        double r10901100 = -1259.1392167224028;
        double r10901101 = r10901059 - r10901058;
        double r10901102 = r10901100 / r10901101;
        double r10901103 = 0.9999999999998099;
        double r10901104 = 771.3234287776531;
        double r10901105 = r10901059 + r10901089;
        double r10901106 = r10901104 / r10901105;
        double r10901107 = 676.5203681218851;
        double r10901108 = r10901107 / r10901059;
        double r10901109 = r10901106 + r10901108;
        double r10901110 = r10901103 + r10901109;
        double r10901111 = r10901102 + r10901110;
        double r10901112 = r10901099 + r10901111;
        double r10901113 = r10901112 * r10901079;
        double r10901114 = r10901094 * r10901113;
        double r10901115 = r10901095 + r10901114;
        return r10901115;
}

Error

Bits error versus z

Derivation

1. Initial program 59.8

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

2. Simplified 1.1

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

3. Taylor expanded around -inf 0.8

   \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\]
4. Using strategy rm

5. Applied add-sqr-sqrt 0.8

   \[\leadsto \left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\]

6. Applied sqrt-prod 1.1

   \[\leadsto \left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \color{blue}{\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\]

7. Applied associate-*r* 0.8

   \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\sqrt{\pi}}\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\]
8. Using strategy rm

9. Applied associate-*l* 0.8

   \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\right)}\]
10. Using strategy rm

11. Applied distribute-lft-in 0.8

    \[\leadsto \left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt{\pi}} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \sqrt{\sqrt{\pi}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)}\]

12. Applied distribute-lft-in 0.8

    \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)}\]

13. Final simplification 0.8

    \[\leadsto \left(\left(\left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right)\right) \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) + \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{z + 6.5}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{-1259.1392167224028}{z - -1} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851}{z}\right)\right)\right)\right) \cdot \sqrt{\sqrt{\pi}}\right)\]

Reproduce

herbie shell --seed 2019151 +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)))))