Average Error: 1.8 → 1.8
Time: 1.9m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]
\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(0.9999999999998099 + \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(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \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)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}
double f(double z) {
        double r1885318 = atan2(1.0, 0.0);
        double r1885319 = z;
        double r1885320 = r1885318 * r1885319;
        double r1885321 = sin(r1885320);
        double r1885322 = r1885318 / r1885321;
        double r1885323 = 2.0;
        double r1885324 = r1885318 * r1885323;
        double r1885325 = sqrt(r1885324);
        double r1885326 = 1.0;
        double r1885327 = r1885326 - r1885319;
        double r1885328 = r1885327 - r1885326;
        double r1885329 = 7.0;
        double r1885330 = r1885328 + r1885329;
        double r1885331 = 0.5;
        double r1885332 = r1885330 + r1885331;
        double r1885333 = r1885328 + r1885331;
        double r1885334 = pow(r1885332, r1885333);
        double r1885335 = r1885325 * r1885334;
        double r1885336 = -r1885332;
        double r1885337 = exp(r1885336);
        double r1885338 = r1885335 * r1885337;
        double r1885339 = 0.9999999999998099;
        double r1885340 = 676.5203681218851;
        double r1885341 = r1885328 + r1885326;
        double r1885342 = r1885340 / r1885341;
        double r1885343 = r1885339 + r1885342;
        double r1885344 = -1259.1392167224028;
        double r1885345 = r1885328 + r1885323;
        double r1885346 = r1885344 / r1885345;
        double r1885347 = r1885343 + r1885346;
        double r1885348 = 771.3234287776531;
        double r1885349 = 3.0;
        double r1885350 = r1885328 + r1885349;
        double r1885351 = r1885348 / r1885350;
        double r1885352 = r1885347 + r1885351;
        double r1885353 = -176.6150291621406;
        double r1885354 = 4.0;
        double r1885355 = r1885328 + r1885354;
        double r1885356 = r1885353 / r1885355;
        double r1885357 = r1885352 + r1885356;
        double r1885358 = 12.507343278686905;
        double r1885359 = 5.0;
        double r1885360 = r1885328 + r1885359;
        double r1885361 = r1885358 / r1885360;
        double r1885362 = r1885357 + r1885361;
        double r1885363 = -0.13857109526572012;
        double r1885364 = 6.0;
        double r1885365 = r1885328 + r1885364;
        double r1885366 = r1885363 / r1885365;
        double r1885367 = r1885362 + r1885366;
        double r1885368 = 9.984369578019572e-06;
        double r1885369 = r1885368 / r1885330;
        double r1885370 = r1885367 + r1885369;
        double r1885371 = 1.5056327351493116e-07;
        double r1885372 = 8.0;
        double r1885373 = r1885328 + r1885372;
        double r1885374 = r1885371 / r1885373;
        double r1885375 = r1885370 + r1885374;
        double r1885376 = r1885338 * r1885375;
        double r1885377 = r1885322 * r1885376;
        return r1885377;
}
↓
double f(double z) {
        double r1885378 = 2.0;
        double r1885379 = atan2(1.0, 0.0);
        double r1885380 = r1885378 * r1885379;
        double r1885381 = sqrt(r1885380);
        double r1885382 = 7.0;
        double r1885383 = 1.0;
        double r1885384 = z;
        double r1885385 = r1885383 - r1885384;
        double r1885386 = r1885385 - r1885383;
        double r1885387 = r1885382 + r1885386;
        double r1885388 = 0.5;
        double r1885389 = r1885387 + r1885388;
        double r1885390 = r1885388 + r1885386;
        double r1885391 = pow(r1885389, r1885390);
        double r1885392 = r1885381 * r1885391;
        double r1885393 = -r1885389;
        double r1885394 = exp(r1885393);
        double r1885395 = r1885392 * r1885394;
        double r1885396 = 1.5056327351493116e-07;
        double r1885397 = 8.0;
        double r1885398 = r1885397 + r1885386;
        double r1885399 = r1885396 / r1885398;
        double r1885400 = 9.984369578019572e-06;
        double r1885401 = r1885400 / r1885387;
        double r1885402 = -176.6150291621406;
        double r1885403 = 4.0;
        double r1885404 = r1885386 + r1885403;
        double r1885405 = r1885402 / r1885404;
        double r1885406 = 771.3234287776531;
        double r1885407 = 3.0;
        double r1885408 = r1885407 + r1885386;
        double r1885409 = r1885406 / r1885408;
        double r1885410 = 0.9999999999998099;
        double r1885411 = 676.5203681218851;
        double r1885412 = r1885386 + r1885383;
        double r1885413 = r1885411 / r1885412;
        double r1885414 = r1885410 + r1885413;
        double r1885415 = -1259.1392167224028;
        double r1885416 = r1885386 + r1885378;
        double r1885417 = r1885415 / r1885416;
        double r1885418 = r1885414 + r1885417;
        double r1885419 = r1885409 + r1885418;
        double r1885420 = r1885405 + r1885419;
        double r1885421 = 12.507343278686905;
        double r1885422 = 5.0;
        double r1885423 = r1885386 + r1885422;
        double r1885424 = r1885421 / r1885423;
        double r1885425 = r1885420 + r1885424;
        double r1885426 = -0.13857109526572012;
        double r1885427 = 6.0;
        double r1885428 = r1885427 + r1885386;
        double r1885429 = r1885426 / r1885428;
        double r1885430 = r1885425 + r1885429;
        double r1885431 = r1885401 + r1885430;
        double r1885432 = r1885399 + r1885431;
        double r1885433 = r1885395 * r1885432;
        double r1885434 = r1885379 * r1885384;
        double r1885435 = sin(r1885434);
        double r1885436 = r1885379 / r1885435;
        double r1885437 = r1885433 * r1885436;
        return r1885437;
}

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. Final simplification1.8

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

Reproduce

herbie shell --seed 2019153 
(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))))))