Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.8 → 0.9

Time: 3.4m

Precision: 64

• could not determine a ground truth for program body

  1. z = 6.80656641367954e+59

Math

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

↓

\[\left(\left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{1}{\sqrt{e^{\left(z - 1\right) + 7}}}\right) \cdot \frac{\sqrt{\pi \cdot 2}}{\sqrt{e^{0.5}}}\right) \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

C

double f(double z) {
        double r253521 = atan2(1.0, 0.0);
        double r253522 = 2.0;
        double r253523 = r253521 * r253522;
        double r253524 = sqrt(r253523);
        double r253525 = z;
        double r253526 = 1.0;
        double r253527 = r253525 - r253526;
        double r253528 = 7.0;
        double r253529 = r253527 + r253528;
        double r253530 = 0.5;
        double r253531 = r253529 + r253530;
        double r253532 = r253527 + r253530;
        double r253533 = pow(r253531, r253532);
        double r253534 = r253524 * r253533;
        double r253535 = -r253531;
        double r253536 = exp(r253535);
        double r253537 = r253534 * r253536;
        double r253538 = 0.9999999999998099;
        double r253539 = 676.5203681218851;
        double r253540 = r253527 + r253526;
        double r253541 = r253539 / r253540;
        double r253542 = r253538 + r253541;
        double r253543 = -1259.1392167224028;
        double r253544 = r253527 + r253522;
        double r253545 = r253543 / r253544;
        double r253546 = r253542 + r253545;
        double r253547 = 771.3234287776531;
        double r253548 = 3.0;
        double r253549 = r253527 + r253548;
        double r253550 = r253547 / r253549;
        double r253551 = r253546 + r253550;
        double r253552 = -176.6150291621406;
        double r253553 = 4.0;
        double r253554 = r253527 + r253553;
        double r253555 = r253552 / r253554;
        double r253556 = r253551 + r253555;
        double r253557 = 12.507343278686905;
        double r253558 = 5.0;
        double r253559 = r253527 + r253558;
        double r253560 = r253557 / r253559;
        double r253561 = r253556 + r253560;
        double r253562 = -0.13857109526572012;
        double r253563 = 6.0;
        double r253564 = r253527 + r253563;
        double r253565 = r253562 / r253564;
        double r253566 = r253561 + r253565;
        double r253567 = 9.984369578019572e-06;
        double r253568 = r253567 / r253529;
        double r253569 = r253566 + r253568;
        double r253570 = 1.5056327351493116e-07;
        double r253571 = 8.0;
        double r253572 = r253527 + r253571;
        double r253573 = r253570 / r253572;
        double r253574 = r253569 + r253573;
        double r253575 = r253537 * r253574;
        return r253575;
}

↓

double f(double z) {
        double r253576 = z;
        double r253577 = 1.0;
        double r253578 = r253576 - r253577;
        double r253579 = 7.0;
        double r253580 = r253578 + r253579;
        double r253581 = 0.5;
        double r253582 = r253580 + r253581;
        double r253583 = r253578 + r253581;
        double r253584 = pow(r253582, r253583);
        double r253585 = exp(r253582);
        double r253586 = sqrt(r253585);
        double r253587 = r253584 / r253586;
        double r253588 = 1.0;
        double r253589 = exp(r253580);
        double r253590 = sqrt(r253589);
        double r253591 = r253588 / r253590;
        double r253592 = r253587 * r253591;
        double r253593 = atan2(1.0, 0.0);
        double r253594 = 2.0;
        double r253595 = r253593 * r253594;
        double r253596 = sqrt(r253595);
        double r253597 = exp(r253581);
        double r253598 = sqrt(r253597);
        double r253599 = r253596 / r253598;
        double r253600 = r253592 * r253599;
        double r253601 = 676.5203681218851;
        double r253602 = r253601 / r253576;
        double r253603 = 0.9999999999998099;
        double r253604 = r253602 + r253603;
        double r253605 = -1259.1392167224028;
        double r253606 = r253578 + r253594;
        double r253607 = r253605 / r253606;
        double r253608 = r253604 + r253607;
        double r253609 = 771.3234287776531;
        double r253610 = 3.0;
        double r253611 = r253578 + r253610;
        double r253612 = r253609 / r253611;
        double r253613 = -176.6150291621406;
        double r253614 = 4.0;
        double r253615 = r253578 + r253614;
        double r253616 = r253613 / r253615;
        double r253617 = r253612 + r253616;
        double r253618 = 12.507343278686905;
        double r253619 = 5.0;
        double r253620 = r253578 + r253619;
        double r253621 = r253618 / r253620;
        double r253622 = r253617 + r253621;
        double r253623 = -0.13857109526572012;
        double r253624 = 6.0;
        double r253625 = r253578 + r253624;
        double r253626 = r253623 / r253625;
        double r253627 = 9.984369578019572e-06;
        double r253628 = r253627 / r253580;
        double r253629 = r253626 + r253628;
        double r253630 = 1.5056327351493116e-07;
        double r253631 = 8.0;
        double r253632 = r253578 + r253631;
        double r253633 = r253630 / r253632;
        double r253634 = r253629 + r253633;
        double r253635 = r253622 + r253634;
        double r253636 = r253608 + r253635;
        double r253637 = r253600 * r253636;
        return r253637;
}

Error

Bits error versus z

Derivation

1. Initial program 61.8

   \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

2. Simplified 1.1

   \[\leadsto \color{blue}{\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{e^{\left(\left(z - 1\right) + 7\right) + 0.5}} \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 1.1

   \[\leadsto \frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{\color{blue}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}} \cdot \sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}}} \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

5. Applied times-frac 0.9

   \[\leadsto \color{blue}{\left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{\sqrt{\pi \cdot 2}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}}\right)} \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]
6. Using strategy rm

7. Applied exp-sum 0.9

   \[\leadsto \left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{\sqrt{\pi \cdot 2}}{\sqrt{\color{blue}{e^{\left(z - 1\right) + 7} \cdot e^{0.5}}}}\right) \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

8. Applied sqrt-prod 0.9

   \[\leadsto \left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{\sqrt{\pi \cdot 2}}{\color{blue}{\sqrt{e^{\left(z - 1\right) + 7}} \cdot \sqrt{e^{0.5}}}}\right) \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

9. Applied *-un-lft-identity 0.9

   \[\leadsto \left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{\color{blue}{1 \cdot \sqrt{\pi \cdot 2}}}{\sqrt{e^{\left(z - 1\right) + 7}} \cdot \sqrt{e^{0.5}}}\right) \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

10. Applied times-frac 1.1

    \[\leadsto \left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \color{blue}{\left(\frac{1}{\sqrt{e^{\left(z - 1\right) + 7}}} \cdot \frac{\sqrt{\pi \cdot 2}}{\sqrt{e^{0.5}}}\right)}\right) \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

11. Applied associate-*r* 0.9

    \[\leadsto \color{blue}{\left(\left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{1}{\sqrt{e^{\left(z - 1\right) + 7}}}\right) \cdot \frac{\sqrt{\pi \cdot 2}}{\sqrt{e^{0.5}}}\right)} \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

12. Final simplification 0.9

    \[\leadsto \left(\left(\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}}{\sqrt{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \frac{1}{\sqrt{e^{\left(z - 1\right) + 7}}}\right) \cdot \frac{\sqrt{\pi \cdot 2}}{\sqrt{e^{0.5}}}\right) \cdot \left(\left(\left(\frac{676.520368121885099}{z} + 0.99999999999980993\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \left(\left(\left(\frac{771.32342877765313}{\left(z - 1\right) + 3} + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \left(\left(\frac{-0.138571095265720118}{\left(z - 1\right) + 6} + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019198 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- z 1.0) 8.0)))))