Average Error: 1.8 → 0.5
Time: 3.0m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\frac{{\left(0.5 + \left(7 - z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}}{e^{0.5 + \left(7 - z\right)}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \sqrt[3]{\left(\left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{1 + \left(2 + \left(1 - z\right)\right)}\right) \cdot \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{1 + \left(2 + \left(1 - z\right)\right)}\right)\right) \cdot \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{1 + \left(2 + \left(1 - z\right)\right)}\right)}\right) + \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \sqrt{2 \cdot \pi}\]
\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(\frac{{\left(0.5 + \left(7 - z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}}{e^{0.5 + \left(7 - z\right)}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \sqrt[3]{\left(\left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{1 + \left(2 + \left(1 - z\right)\right)}\right) \cdot \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{1 + \left(2 + \left(1 - z\right)\right)}\right)\right) \cdot \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{1 + \left(2 + \left(1 - z\right)\right)}\right)}\right) + \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \sqrt{2 \cdot \pi}
double f(double z) {
        double r8333996 = atan2(1.0, 0.0);
        double r8333997 = z;
        double r8333998 = r8333996 * r8333997;
        double r8333999 = sin(r8333998);
        double r8334000 = r8333996 / r8333999;
        double r8334001 = 2.0;
        double r8334002 = r8333996 * r8334001;
        double r8334003 = sqrt(r8334002);
        double r8334004 = 1.0;
        double r8334005 = r8334004 - r8333997;
        double r8334006 = r8334005 - r8334004;
        double r8334007 = 7.0;
        double r8334008 = r8334006 + r8334007;
        double r8334009 = 0.5;
        double r8334010 = r8334008 + r8334009;
        double r8334011 = r8334006 + r8334009;
        double r8334012 = pow(r8334010, r8334011);
        double r8334013 = r8334003 * r8334012;
        double r8334014 = -r8334010;
        double r8334015 = exp(r8334014);
        double r8334016 = r8334013 * r8334015;
        double r8334017 = 0.9999999999998099;
        double r8334018 = 676.5203681218851;
        double r8334019 = r8334006 + r8334004;
        double r8334020 = r8334018 / r8334019;
        double r8334021 = r8334017 + r8334020;
        double r8334022 = -1259.1392167224028;
        double r8334023 = r8334006 + r8334001;
        double r8334024 = r8334022 / r8334023;
        double r8334025 = r8334021 + r8334024;
        double r8334026 = 771.3234287776531;
        double r8334027 = 3.0;
        double r8334028 = r8334006 + r8334027;
        double r8334029 = r8334026 / r8334028;
        double r8334030 = r8334025 + r8334029;
        double r8334031 = -176.6150291621406;
        double r8334032 = 4.0;
        double r8334033 = r8334006 + r8334032;
        double r8334034 = r8334031 / r8334033;
        double r8334035 = r8334030 + r8334034;
        double r8334036 = 12.507343278686905;
        double r8334037 = 5.0;
        double r8334038 = r8334006 + r8334037;
        double r8334039 = r8334036 / r8334038;
        double r8334040 = r8334035 + r8334039;
        double r8334041 = -0.13857109526572012;
        double r8334042 = 6.0;
        double r8334043 = r8334006 + r8334042;
        double r8334044 = r8334041 / r8334043;
        double r8334045 = r8334040 + r8334044;
        double r8334046 = 9.984369578019572e-06;
        double r8334047 = r8334046 / r8334008;
        double r8334048 = r8334045 + r8334047;
        double r8334049 = 1.5056327351493116e-07;
        double r8334050 = 8.0;
        double r8334051 = r8334006 + r8334050;
        double r8334052 = r8334049 / r8334051;
        double r8334053 = r8334048 + r8334052;
        double r8334054 = r8334016 * r8334053;
        double r8334055 = r8334000 * r8334054;
        return r8334055;
}
↓
double f(double z) {
        double r8334056 = 0.5;
        double r8334057 = 7.0;
        double r8334058 = z;
        double r8334059 = r8334057 - r8334058;
        double r8334060 = r8334056 + r8334059;
        double r8334061 = -r8334058;
        double r8334062 = r8334056 + r8334061;
        double r8334063 = pow(r8334060, r8334062);
        double r8334064 = exp(r8334060);
        double r8334065 = r8334063 / r8334064;
        double r8334066 = 1.5056327351493116e-07;
        double r8334067 = 8.0;
        double r8334068 = r8334067 - r8334058;
        double r8334069 = r8334066 / r8334068;
        double r8334070 = 9.984369578019572e-06;
        double r8334071 = r8334070 / r8334059;
        double r8334072 = r8334069 + r8334071;
        double r8334073 = -0.13857109526572012;
        double r8334074 = 6.0;
        double r8334075 = r8334074 - r8334058;
        double r8334076 = r8334073 / r8334075;
        double r8334077 = 771.3234287776531;
        double r8334078 = 2.0;
        double r8334079 = 1.0;
        double r8334080 = r8334079 - r8334058;
        double r8334081 = r8334078 + r8334080;
        double r8334082 = r8334077 / r8334081;
        double r8334083 = -176.6150291621406;
        double r8334084 = r8334079 + r8334081;
        double r8334085 = r8334083 / r8334084;
        double r8334086 = r8334082 + r8334085;
        double r8334087 = r8334086 * r8334086;
        double r8334088 = r8334087 * r8334086;
        double r8334089 = cbrt(r8334088);
        double r8334090 = r8334076 + r8334089;
        double r8334091 = 0.9999999999998099;
        double r8334092 = -1259.1392167224028;
        double r8334093 = r8334078 - r8334058;
        double r8334094 = r8334092 / r8334093;
        double r8334095 = 676.5203681218851;
        double r8334096 = r8334095 / r8334080;
        double r8334097 = r8334094 + r8334096;
        double r8334098 = r8334091 + r8334097;
        double r8334099 = r8334090 + r8334098;
        double r8334100 = r8334072 + r8334099;
        double r8334101 = 12.507343278686905;
        double r8334102 = 5.0;
        double r8334103 = r8334102 - r8334058;
        double r8334104 = r8334101 / r8334103;
        double r8334105 = r8334100 + r8334104;
        double r8334106 = r8334065 * r8334105;
        double r8334107 = atan2(1.0, 0.0);
        double r8334108 = r8334058 * r8334107;
        double r8334109 = sin(r8334108);
        double r8334110 = r8334107 / r8334109;
        double r8334111 = r8334106 * r8334110;
        double r8334112 = r8334078 * r8334107;
        double r8334113 = sqrt(r8334112);
        double r8334114 = r8334111 * r8334113;
        return r8334114;
}

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

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

  2. Simplified0.5

    \[\leadsto \color{blue}{\sqrt{2 \cdot \pi} \cdot \left(\left(\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 - z\right) + 0.5}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.5

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 - z\right) + 0.5}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \color{blue}{\sqrt[3]{\left(\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right) \cdot \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) \cdot \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)}}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\]

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(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))))))