Average Error: 60.1 → 0.8
Time: 3.8m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\frac{\left(\sqrt{2} \cdot {\left(6.5 + z\right)}^{\left(z - 0.5\right)}\right) \cdot \sqrt{\pi}}{\sqrt{e^{6.5 + z}}} \cdot \frac{\left(\left(\frac{-1259.1392167224028}{z - -1} + \left(\left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851}{z}\right) + 0.9999999999998099\right)\right) + \frac{-176.6150291621406}{3 + z}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{12.507343278686905}{4 + z} + \frac{-0.13857109526572012}{-1 + \left(z - -6\right)}\right)\right)}{\sqrt{e^{6.5 + z}}}\]
\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)
↓
\frac{\left(\sqrt{2} \cdot {\left(6.5 + z\right)}^{\left(z - 0.5\right)}\right) \cdot \sqrt{\pi}}{\sqrt{e^{6.5 + z}}} \cdot \frac{\left(\left(\frac{-1259.1392167224028}{z - -1} + \left(\left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851}{z}\right) + 0.9999999999998099\right)\right) + \frac{-176.6150291621406}{3 + z}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{12.507343278686905}{4 + z} + \frac{-0.13857109526572012}{-1 + \left(z - -6\right)}\right)\right)}{\sqrt{e^{6.5 + z}}}
double f(double z) {
        double r11142381 = atan2(1.0, 0.0);
        double r11142382 = 2.0;
        double r11142383 = r11142381 * r11142382;
        double r11142384 = sqrt(r11142383);
        double r11142385 = z;
        double r11142386 = 1.0;
        double r11142387 = r11142385 - r11142386;
        double r11142388 = 7.0;
        double r11142389 = r11142387 + r11142388;
        double r11142390 = 0.5;
        double r11142391 = r11142389 + r11142390;
        double r11142392 = r11142387 + r11142390;
        double r11142393 = pow(r11142391, r11142392);
        double r11142394 = r11142384 * r11142393;
        double r11142395 = -r11142391;
        double r11142396 = exp(r11142395);
        double r11142397 = r11142394 * r11142396;
        double r11142398 = 0.9999999999998099;
        double r11142399 = 676.5203681218851;
        double r11142400 = r11142387 + r11142386;
        double r11142401 = r11142399 / r11142400;
        double r11142402 = r11142398 + r11142401;
        double r11142403 = -1259.1392167224028;
        double r11142404 = r11142387 + r11142382;
        double r11142405 = r11142403 / r11142404;
        double r11142406 = r11142402 + r11142405;
        double r11142407 = 771.3234287776531;
        double r11142408 = 3.0;
        double r11142409 = r11142387 + r11142408;
        double r11142410 = r11142407 / r11142409;
        double r11142411 = r11142406 + r11142410;
        double r11142412 = -176.6150291621406;
        double r11142413 = 4.0;
        double r11142414 = r11142387 + r11142413;
        double r11142415 = r11142412 / r11142414;
        double r11142416 = r11142411 + r11142415;
        double r11142417 = 12.507343278686905;
        double r11142418 = 5.0;
        double r11142419 = r11142387 + r11142418;
        double r11142420 = r11142417 / r11142419;
        double r11142421 = r11142416 + r11142420;
        double r11142422 = -0.13857109526572012;
        double r11142423 = 6.0;
        double r11142424 = r11142387 + r11142423;
        double r11142425 = r11142422 / r11142424;
        double r11142426 = r11142421 + r11142425;
        double r11142427 = 9.984369578019572e-06;
        double r11142428 = r11142427 / r11142389;
        double r11142429 = r11142426 + r11142428;
        double r11142430 = 1.5056327351493116e-07;
        double r11142431 = 8.0;
        double r11142432 = r11142387 + r11142431;
        double r11142433 = r11142430 / r11142432;
        double r11142434 = r11142429 + r11142433;
        double r11142435 = r11142397 * r11142434;
        return r11142435;
}
↓
double f(double z) {
        double r11142436 = 2.0;
        double r11142437 = sqrt(r11142436);
        double r11142438 = 6.5;
        double r11142439 = z;
        double r11142440 = r11142438 + r11142439;
        double r11142441 = 0.5;
        double r11142442 = r11142439 - r11142441;
        double r11142443 = pow(r11142440, r11142442);
        double r11142444 = r11142437 * r11142443;
        double r11142445 = atan2(1.0, 0.0);
        double r11142446 = sqrt(r11142445);
        double r11142447 = r11142444 * r11142446;
        double r11142448 = exp(r11142440);
        double r11142449 = sqrt(r11142448);
        double r11142450 = r11142447 / r11142449;
        double r11142451 = -1259.1392167224028;
        double r11142452 = -1.0;
        double r11142453 = r11142439 - r11142452;
        double r11142454 = r11142451 / r11142453;
        double r11142455 = 771.3234287776531;
        double r11142456 = r11142436 + r11142439;
        double r11142457 = r11142455 / r11142456;
        double r11142458 = 676.5203681218851;
        double r11142459 = r11142458 / r11142439;
        double r11142460 = r11142457 + r11142459;
        double r11142461 = 0.9999999999998099;
        double r11142462 = r11142460 + r11142461;
        double r11142463 = r11142454 + r11142462;
        double r11142464 = -176.6150291621406;
        double r11142465 = 3.0;
        double r11142466 = r11142465 + r11142439;
        double r11142467 = r11142464 / r11142466;
        double r11142468 = r11142463 + r11142467;
        double r11142469 = 9.984369578019572e-06;
        double r11142470 = -6.0;
        double r11142471 = r11142439 - r11142470;
        double r11142472 = r11142469 / r11142471;
        double r11142473 = 1.5056327351493116e-07;
        double r11142474 = 7.0;
        double r11142475 = r11142474 + r11142439;
        double r11142476 = r11142473 / r11142475;
        double r11142477 = r11142472 + r11142476;
        double r11142478 = 12.507343278686905;
        double r11142479 = 4.0;
        double r11142480 = r11142479 + r11142439;
        double r11142481 = r11142478 / r11142480;
        double r11142482 = -0.13857109526572012;
        double r11142483 = r11142452 + r11142471;
        double r11142484 = r11142482 / r11142483;
        double r11142485 = r11142481 + r11142484;
        double r11142486 = r11142477 + r11142485;
        double r11142487 = r11142468 + r11142486;
        double r11142488 = r11142487 / r11142449;
        double r11142489 = r11142450 * r11142488;
        return r11142489;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 60.1

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

  2. Simplified1.1

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

  3. Taylor expanded around inf 0.8

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

  5. Applied associate-*r/1.1

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

  6. Applied associate-*l/0.8

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

  8. Applied add-sqr-sqrt0.9

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

  9. Applied times-frac0.8

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

  10. Final simplification0.8

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

Reproduce

herbie shell --seed 2019142 +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)))))