Average Error: 1.8 → 0.7
Time: 4.8m
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 r7784400 = atan2(1.0, 0.0);
        double r7784401 = z;
        double r7784402 = r7784400 * r7784401;
        double r7784403 = sin(r7784402);
        double r7784404 = r7784400 / r7784403;
        double r7784405 = 2.0;
        double r7784406 = r7784400 * r7784405;
        double r7784407 = sqrt(r7784406);
        double r7784408 = 1.0;
        double r7784409 = r7784408 - r7784401;
        double r7784410 = r7784409 - r7784408;
        double r7784411 = 7.0;
        double r7784412 = r7784410 + r7784411;
        double r7784413 = 0.5;
        double r7784414 = r7784412 + r7784413;
        double r7784415 = r7784410 + r7784413;
        double r7784416 = pow(r7784414, r7784415);
        double r7784417 = r7784407 * r7784416;
        double r7784418 = -r7784414;
        double r7784419 = exp(r7784418);
        double r7784420 = r7784417 * r7784419;
        double r7784421 = 0.9999999999998099;
        double r7784422 = 676.5203681218851;
        double r7784423 = r7784410 + r7784408;
        double r7784424 = r7784422 / r7784423;
        double r7784425 = r7784421 + r7784424;
        double r7784426 = -1259.1392167224028;
        double r7784427 = r7784410 + r7784405;
        double r7784428 = r7784426 / r7784427;
        double r7784429 = r7784425 + r7784428;
        double r7784430 = 771.3234287776531;
        double r7784431 = 3.0;
        double r7784432 = r7784410 + r7784431;
        double r7784433 = r7784430 / r7784432;
        double r7784434 = r7784429 + r7784433;
        double r7784435 = -176.6150291621406;
        double r7784436 = 4.0;
        double r7784437 = r7784410 + r7784436;
        double r7784438 = r7784435 / r7784437;
        double r7784439 = r7784434 + r7784438;
        double r7784440 = 12.507343278686905;
        double r7784441 = 5.0;
        double r7784442 = r7784410 + r7784441;
        double r7784443 = r7784440 / r7784442;
        double r7784444 = r7784439 + r7784443;
        double r7784445 = -0.13857109526572012;
        double r7784446 = 6.0;
        double r7784447 = r7784410 + r7784446;
        double r7784448 = r7784445 / r7784447;
        double r7784449 = r7784444 + r7784448;
        double r7784450 = 9.984369578019572e-06;
        double r7784451 = r7784450 / r7784412;
        double r7784452 = r7784449 + r7784451;
        double r7784453 = 1.5056327351493116e-07;
        double r7784454 = 8.0;
        double r7784455 = r7784410 + r7784454;
        double r7784456 = r7784453 / r7784455;
        double r7784457 = r7784452 + r7784456;
        double r7784458 = r7784420 * r7784457;
        double r7784459 = r7784404 * r7784458;
        return r7784459;
}
↓
double f(double z) {
        double r7784460 = 9.984369578019572e-06;
        double r7784461 = 7.0;
        double r7784462 = z;
        double r7784463 = -r7784462;
        double r7784464 = r7784461 + r7784463;
        double r7784465 = r7784460 / r7784464;
        double r7784466 = -0.13857109526572012;
        double r7784467 = 6.0;
        double r7784468 = r7784463 + r7784467;
        double r7784469 = r7784466 / r7784468;
        double r7784470 = 1.5056327351493116e-07;
        double r7784471 = 8.0;
        double r7784472 = r7784471 + r7784463;
        double r7784473 = r7784470 / r7784472;
        double r7784474 = r7784469 + r7784473;
        double r7784475 = r7784465 + r7784474;
        double r7784476 = 1.0;
        double r7784477 = 3.0;
        double r7784478 = r7784477 + r7784463;
        double r7784479 = sqrt(r7784478);
        double r7784480 = r7784476 / r7784479;
        double r7784481 = 771.3234287776531;
        double r7784482 = r7784481 / r7784479;
        double r7784483 = 0.9999999999998099;
        double r7784484 = 676.5203681218851;
        double r7784485 = 1.0;
        double r7784486 = r7784485 - r7784462;
        double r7784487 = r7784484 / r7784486;
        double r7784488 = r7784483 + r7784487;
        double r7784489 = fma(r7784480, r7784482, r7784488);
        double r7784490 = -1259.1392167224028;
        double r7784491 = 2.0;
        double r7784492 = r7784463 + r7784491;
        double r7784493 = r7784490 / r7784492;
        double r7784494 = r7784489 + r7784493;
        double r7784495 = 12.507343278686905;
        double r7784496 = 5.0;
        double r7784497 = r7784463 + r7784496;
        double r7784498 = r7784495 / r7784497;
        double r7784499 = r7784494 + r7784498;
        double r7784500 = -176.6150291621406;
        double r7784501 = 4.0;
        double r7784502 = r7784463 + r7784501;
        double r7784503 = r7784500 / r7784502;
        double r7784504 = r7784499 + r7784503;
        double r7784505 = r7784475 + r7784504;
        double r7784506 = 0.5;
        double r7784507 = r7784506 + r7784461;
        double r7784508 = r7784507 + r7784463;
        double r7784509 = r7784463 + r7784506;
        double r7784510 = pow(r7784508, r7784509);
        double r7784511 = exp(r7784508);
        double r7784512 = r7784510 / r7784511;
        double r7784513 = atan2(1.0, 0.0);
        double r7784514 = r7784513 * r7784462;
        double r7784515 = sin(r7784514);
        double r7784516 = r7784513 / r7784515;
        double r7784517 = r7784491 * r7784513;
        double r7784518 = sqrt(r7784517);
        double r7784519 = r7784516 * r7784518;
        double r7784520 = r7784512 * r7784519;
        double r7784521 = r7784505 * r7784520;
        return r7784521;
}

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))))))