Average Error: 1.8 → 0.7
Time: 2.7m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)} + \left(\frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right)\right) + \left(\left(\left(\mathsf{fma}\left(\frac{1}{\sqrt{3 + \left(-z\right)}}, \frac{771.3234287776531346025876700878143310547}{\sqrt{3 + \left(-z\right)}}, 0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4}\right)\right) \cdot \left(\frac{{\left(\left(0.5 + 7\right) + \left(-z\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{\left(0.5 + 7\right) + \left(-z\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)\]
\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(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\left(\left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)} + \left(\frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right)\right) + \left(\left(\left(\mathsf{fma}\left(\frac{1}{\sqrt{3 + \left(-z\right)}}, \frac{771.3234287776531346025876700878143310547}{\sqrt{3 + \left(-z\right)}}, 0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4}\right)\right) \cdot \left(\frac{{\left(\left(0.5 + 7\right) + \left(-z\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{\left(0.5 + 7\right) + \left(-z\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)
double f(double z) {
        double r6307187 = atan2(1.0, 0.0);
        double r6307188 = z;
        double r6307189 = r6307187 * r6307188;
        double r6307190 = sin(r6307189);
        double r6307191 = r6307187 / r6307190;
        double r6307192 = 2.0;
        double r6307193 = r6307187 * r6307192;
        double r6307194 = sqrt(r6307193);
        double r6307195 = 1.0;
        double r6307196 = r6307195 - r6307188;
        double r6307197 = r6307196 - r6307195;
        double r6307198 = 7.0;
        double r6307199 = r6307197 + r6307198;
        double r6307200 = 0.5;
        double r6307201 = r6307199 + r6307200;
        double r6307202 = r6307197 + r6307200;
        double r6307203 = pow(r6307201, r6307202);
        double r6307204 = r6307194 * r6307203;
        double r6307205 = -r6307201;
        double r6307206 = exp(r6307205);
        double r6307207 = r6307204 * r6307206;
        double r6307208 = 0.9999999999998099;
        double r6307209 = 676.5203681218851;
        double r6307210 = r6307197 + r6307195;
        double r6307211 = r6307209 / r6307210;
        double r6307212 = r6307208 + r6307211;
        double r6307213 = -1259.1392167224028;
        double r6307214 = r6307197 + r6307192;
        double r6307215 = r6307213 / r6307214;
        double r6307216 = r6307212 + r6307215;
        double r6307217 = 771.3234287776531;
        double r6307218 = 3.0;
        double r6307219 = r6307197 + r6307218;
        double r6307220 = r6307217 / r6307219;
        double r6307221 = r6307216 + r6307220;
        double r6307222 = -176.6150291621406;
        double r6307223 = 4.0;
        double r6307224 = r6307197 + r6307223;
        double r6307225 = r6307222 / r6307224;
        double r6307226 = r6307221 + r6307225;
        double r6307227 = 12.507343278686905;
        double r6307228 = 5.0;
        double r6307229 = r6307197 + r6307228;
        double r6307230 = r6307227 / r6307229;
        double r6307231 = r6307226 + r6307230;
        double r6307232 = -0.13857109526572012;
        double r6307233 = 6.0;
        double r6307234 = r6307197 + r6307233;
        double r6307235 = r6307232 / r6307234;
        double r6307236 = r6307231 + r6307235;
        double r6307237 = 9.984369578019572e-06;
        double r6307238 = r6307237 / r6307199;
        double r6307239 = r6307236 + r6307238;
        double r6307240 = 1.5056327351493116e-07;
        double r6307241 = 8.0;
        double r6307242 = r6307197 + r6307241;
        double r6307243 = r6307240 / r6307242;
        double r6307244 = r6307239 + r6307243;
        double r6307245 = r6307207 * r6307244;
        double r6307246 = r6307191 * r6307245;
        return r6307246;
}
↓
double f(double z) {
        double r6307247 = 9.984369578019572e-06;
        double r6307248 = 7.0;
        double r6307249 = z;
        double r6307250 = -r6307249;
        double r6307251 = r6307248 + r6307250;
        double r6307252 = r6307247 / r6307251;
        double r6307253 = -0.13857109526572012;
        double r6307254 = 6.0;
        double r6307255 = r6307250 + r6307254;
        double r6307256 = r6307253 / r6307255;
        double r6307257 = 1.5056327351493116e-07;
        double r6307258 = 8.0;
        double r6307259 = r6307258 + r6307250;
        double r6307260 = r6307257 / r6307259;
        double r6307261 = r6307256 + r6307260;
        double r6307262 = r6307252 + r6307261;
        double r6307263 = 1.0;
        double r6307264 = 3.0;
        double r6307265 = r6307264 + r6307250;
        double r6307266 = sqrt(r6307265);
        double r6307267 = r6307263 / r6307266;
        double r6307268 = 771.3234287776531;
        double r6307269 = r6307268 / r6307266;
        double r6307270 = 0.9999999999998099;
        double r6307271 = 676.5203681218851;
        double r6307272 = 1.0;
        double r6307273 = r6307272 - r6307249;
        double r6307274 = r6307271 / r6307273;
        double r6307275 = r6307270 + r6307274;
        double r6307276 = fma(r6307267, r6307269, r6307275);
        double r6307277 = -1259.1392167224028;
        double r6307278 = 2.0;
        double r6307279 = r6307250 + r6307278;
        double r6307280 = r6307277 / r6307279;
        double r6307281 = r6307276 + r6307280;
        double r6307282 = 12.507343278686905;
        double r6307283 = 5.0;
        double r6307284 = r6307250 + r6307283;
        double r6307285 = r6307282 / r6307284;
        double r6307286 = r6307281 + r6307285;
        double r6307287 = -176.6150291621406;
        double r6307288 = 4.0;
        double r6307289 = r6307250 + r6307288;
        double r6307290 = r6307287 / r6307289;
        double r6307291 = r6307286 + r6307290;
        double r6307292 = r6307262 + r6307291;
        double r6307293 = 0.5;
        double r6307294 = r6307293 + r6307248;
        double r6307295 = r6307294 + r6307250;
        double r6307296 = r6307250 + r6307293;
        double r6307297 = pow(r6307295, r6307296);
        double r6307298 = exp(r6307295);
        double r6307299 = r6307297 / r6307298;
        double r6307300 = atan2(1.0, 0.0);
        double r6307301 = r6307300 * r6307249;
        double r6307302 = sin(r6307301);
        double r6307303 = r6307300 / r6307302;
        double r6307304 = r6307278 * r6307300;
        double r6307305 = sqrt(r6307304);
        double r6307306 = r6307303 * r6307305;
        double r6307307 = r6307299 * r6307306;
        double r6307308 = r6307292 * r6307307;
        return r6307308;
}

Error

Bits error versus z

Derivation

  1. Initial program 1.8

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Simplified1.7

    \[\leadsto \color{blue}{\left(\left(\left(\frac{12.50734327868690520801919774385169148445}{5 + \left(0 - z\right)} + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(0 - z\right)} + \left(\frac{771.3234287776531346025876700878143310547}{3 + \left(0 - z\right)} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right)\right) + \frac{-176.6150291621405870046146446838974952698}{\left(0 - z\right) + 4}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(0 - z\right) + 7} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(0 - z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(0 - z\right) + 8}\right)\right)\right) \cdot \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{{\left(\left(7 + 0.5\right) + \left(0 - z\right)\right)}^{\left(\left(0 - z\right) + 0.5\right)}}{e^{\left(7 + 0.5\right) + \left(0 - z\right)}}\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt1.7

    \[\leadsto \left(\left(\left(\frac{12.50734327868690520801919774385169148445}{5 + \left(0 - z\right)} + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(0 - z\right)} + \left(\frac{771.3234287776531346025876700878143310547}{\color{blue}{\sqrt{3 + \left(0 - z\right)} \cdot \sqrt{3 + \left(0 - z\right)}}} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right)\right) + \frac{-176.6150291621405870046146446838974952698}{\left(0 - z\right) + 4}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(0 - z\right) + 7} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(0 - z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(0 - z\right) + 8}\right)\right)\right) \cdot \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{{\left(\left(7 + 0.5\right) + \left(0 - z\right)\right)}^{\left(\left(0 - z\right) + 0.5\right)}}{e^{\left(7 + 0.5\right) + \left(0 - z\right)}}\right)\]

  5. Applied *-un-lft-identity1.7

    \[\leadsto \left(\left(\left(\frac{12.50734327868690520801919774385169148445}{5 + \left(0 - z\right)} + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(0 - z\right)} + \left(\frac{\color{blue}{1 \cdot 771.3234287776531346025876700878143310547}}{\sqrt{3 + \left(0 - z\right)} \cdot \sqrt{3 + \left(0 - z\right)}} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right)\right) + \frac{-176.6150291621405870046146446838974952698}{\left(0 - z\right) + 4}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(0 - z\right) + 7} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(0 - z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(0 - z\right) + 8}\right)\right)\right) \cdot \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{{\left(\left(7 + 0.5\right) + \left(0 - z\right)\right)}^{\left(\left(0 - z\right) + 0.5\right)}}{e^{\left(7 + 0.5\right) + \left(0 - z\right)}}\right)\]

  6. Applied times-frac1.7

    \[\leadsto \left(\left(\left(\frac{12.50734327868690520801919774385169148445}{5 + \left(0 - z\right)} + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(0 - z\right)} + \left(\color{blue}{\frac{1}{\sqrt{3 + \left(0 - z\right)}} \cdot \frac{771.3234287776531346025876700878143310547}{\sqrt{3 + \left(0 - z\right)}}} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right)\right) + \frac{-176.6150291621405870046146446838974952698}{\left(0 - z\right) + 4}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(0 - z\right) + 7} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(0 - z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(0 - z\right) + 8}\right)\right)\right) \cdot \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{{\left(\left(7 + 0.5\right) + \left(0 - z\right)\right)}^{\left(\left(0 - z\right) + 0.5\right)}}{e^{\left(7 + 0.5\right) + \left(0 - z\right)}}\right)\]

  7. Applied fma-def0.7

    \[\leadsto \left(\left(\left(\frac{12.50734327868690520801919774385169148445}{5 + \left(0 - z\right)} + \left(\frac{-1259.139216722402807135949842631816864014}{2 + \left(0 - z\right)} + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{3 + \left(0 - z\right)}}, \frac{771.3234287776531346025876700878143310547}{\sqrt{3 + \left(0 - z\right)}}, 0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)}\right)\right) + \frac{-176.6150291621405870046146446838974952698}{\left(0 - z\right) + 4}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(0 - z\right) + 7} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(0 - z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(0 - z\right) + 8}\right)\right)\right) \cdot \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{{\left(\left(7 + 0.5\right) + \left(0 - z\right)\right)}^{\left(\left(0 - z\right) + 0.5\right)}}{e^{\left(7 + 0.5\right) + \left(0 - z\right)}}\right)\]

  8. Final simplification0.7

    \[\leadsto \left(\left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)} + \left(\frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right)\right) + \left(\left(\left(\mathsf{fma}\left(\frac{1}{\sqrt{3 + \left(-z\right)}}, \frac{771.3234287776531346025876700878143310547}{\sqrt{3 + \left(-z\right)}}, 0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4}\right)\right) \cdot \left(\frac{{\left(\left(0.5 + 7\right) + \left(-z\right)\right)}^{\left(\left(-z\right) + 0.5\right)}}{e^{\left(0.5 + 7\right) + \left(-z\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))