Average Error: 29.5 → 1.7
Time: 5.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.03381827806414272 \cdot 10^{62}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{\frac{y}{\frac{z}{t}}}{{z}^{\left(\frac{2}{2}\right)}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 495619.95934221218:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + t \cdot \frac{y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.03381827806414272 \cdot 10^{62}:\\
\;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{\frac{y}{\frac{z}{t}}}{{z}^{\left(\frac{2}{2}\right)}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\

\mathbf{elif}\;z \le 495619.95934221218:\\
\;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\

\mathbf{else}:\\
\;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + t \cdot \frac{y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r316470 = x;
        double r316471 = y;
        double r316472 = z;
        double r316473 = 3.13060547623;
        double r316474 = r316472 * r316473;
        double r316475 = 11.1667541262;
        double r316476 = r316474 + r316475;
        double r316477 = r316476 * r316472;
        double r316478 = t;
        double r316479 = r316477 + r316478;
        double r316480 = r316479 * r316472;
        double r316481 = a;
        double r316482 = r316480 + r316481;
        double r316483 = r316482 * r316472;
        double r316484 = b;
        double r316485 = r316483 + r316484;
        double r316486 = r316471 * r316485;
        double r316487 = 15.234687407;
        double r316488 = r316472 + r316487;
        double r316489 = r316488 * r316472;
        double r316490 = 31.4690115749;
        double r316491 = r316489 + r316490;
        double r316492 = r316491 * r316472;
        double r316493 = 11.9400905721;
        double r316494 = r316492 + r316493;
        double r316495 = r316494 * r316472;
        double r316496 = 0.607771387771;
        double r316497 = r316495 + r316496;
        double r316498 = r316486 / r316497;
        double r316499 = r316470 + r316498;
        return r316499;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r316500 = z;
        double r316501 = -4.0338182780641427e+62;
        bool r316502 = r316500 <= r316501;
        double r316503 = x;
        double r316504 = 3.13060547623;
        double r316505 = y;
        double r316506 = r316504 * r316505;
        double r316507 = t;
        double r316508 = r316500 / r316507;
        double r316509 = r316505 / r316508;
        double r316510 = 2.0;
        double r316511 = r316510 / r316510;
        double r316512 = pow(r316500, r316511);
        double r316513 = r316509 / r316512;
        double r316514 = r316506 + r316513;
        double r316515 = 36.527041698806414;
        double r316516 = r316505 / r316500;
        double r316517 = r316515 * r316516;
        double r316518 = r316514 - r316517;
        double r316519 = r316503 + r316518;
        double r316520 = 495619.9593422122;
        bool r316521 = r316500 <= r316520;
        double r316522 = 15.234687407;
        double r316523 = r316500 + r316522;
        double r316524 = r316523 * r316500;
        double r316525 = 31.4690115749;
        double r316526 = r316524 + r316525;
        double r316527 = r316526 * r316500;
        double r316528 = 11.9400905721;
        double r316529 = r316527 + r316528;
        double r316530 = r316529 * r316500;
        double r316531 = 0.607771387771;
        double r316532 = r316530 + r316531;
        double r316533 = r316500 * r316504;
        double r316534 = 11.1667541262;
        double r316535 = r316533 + r316534;
        double r316536 = r316535 * r316500;
        double r316537 = r316536 + r316507;
        double r316538 = r316537 * r316500;
        double r316539 = a;
        double r316540 = r316538 + r316539;
        double r316541 = r316540 * r316500;
        double r316542 = b;
        double r316543 = r316541 + r316542;
        double r316544 = r316532 / r316543;
        double r316545 = r316505 / r316544;
        double r316546 = r316503 + r316545;
        double r316547 = pow(r316500, r316510);
        double r316548 = r316505 / r316547;
        double r316549 = r316507 * r316548;
        double r316550 = r316506 + r316549;
        double r316551 = r316550 - r316517;
        double r316552 = r316503 + r316551;
        double r316553 = r316521 ? r316546 : r316552;
        double r316554 = r316502 ? r316519 : r316553;
        return r316554;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target1.2
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.0338182780641427e+62

    1. Initial program 62.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Taylor expanded around inf 8.6

      \[\leadsto x + \color{blue}{\left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]
    3. Using strategy rm

    4. Applied sqr-pow8.6

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{\color{blue}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

    5. Applied associate-/r*8.6

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \color{blue}{\frac{\frac{t \cdot y}{{z}^{\left(\frac{2}{2}\right)}}}{{z}^{\left(\frac{2}{2}\right)}}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

    6. Simplified2.1

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \frac{\color{blue}{\frac{y}{\frac{z}{t}}}}{{z}^{\left(\frac{2}{2}\right)}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

    if -4.0338182780641427e+62 < z < 495619.9593422122

    1. Initial program 2.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
    2. Using strategy rm

    3. Applied associate-/l*0.8

      \[\leadsto x + \color{blue}{\frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}}\]

    if 495619.9593422122 < z

    1. Initial program 56.3

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Taylor expanded around inf 10.9

      \[\leadsto x + \color{blue}{\left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity10.9

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{\color{blue}{\left(1 \cdot z\right)}}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

    5. Applied unpow-prod-down10.9

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{\color{blue}{{1}^{2} \cdot {z}^{2}}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

    6. Applied times-frac3.4

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \color{blue}{\frac{t}{{1}^{2}} \cdot \frac{y}{{z}^{2}}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

    7. Simplified3.4

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \color{blue}{t} \cdot \frac{y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.03381827806414272 \cdot 10^{62}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{\frac{y}{\frac{z}{t}}}{{z}^{\left(\frac{2}{2}\right)}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 495619.95934221218:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + t \cdot \frac{y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))