Average Error: 29.4 → 1.0
Time: 13.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.535870723425017265441515703700627878683 \cdot 10^{50} \lor \neg \left(z \le 1.053511161193009402933625668226348721359 \cdot 10^{55}\right):\\ \;\;\;\;x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t}{\frac{{z}^{2}}{y}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot z + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.535870723425017265441515703700627878683 \cdot 10^{50} \lor \neg \left(z \le 1.053511161193009402933625668226348721359 \cdot 10^{55}\right):\\
\;\;\;\;x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t}{\frac{{z}^{2}}{y}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot z + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r211377 = x;
        double r211378 = y;
        double r211379 = z;
        double r211380 = 3.13060547623;
        double r211381 = r211379 * r211380;
        double r211382 = 11.1667541262;
        double r211383 = r211381 + r211382;
        double r211384 = r211383 * r211379;
        double r211385 = t;
        double r211386 = r211384 + r211385;
        double r211387 = r211386 * r211379;
        double r211388 = a;
        double r211389 = r211387 + r211388;
        double r211390 = r211389 * r211379;
        double r211391 = b;
        double r211392 = r211390 + r211391;
        double r211393 = r211378 * r211392;
        double r211394 = 15.234687407;
        double r211395 = r211379 + r211394;
        double r211396 = r211395 * r211379;
        double r211397 = 31.4690115749;
        double r211398 = r211396 + r211397;
        double r211399 = r211398 * r211379;
        double r211400 = 11.9400905721;
        double r211401 = r211399 + r211400;
        double r211402 = r211401 * r211379;
        double r211403 = 0.607771387771;
        double r211404 = r211402 + r211403;
        double r211405 = r211393 / r211404;
        double r211406 = r211377 + r211405;
        return r211406;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r211407 = z;
        double r211408 = -1.5358707234250173e+50;
        bool r211409 = r211407 <= r211408;
        double r211410 = 1.0535111611930094e+55;
        bool r211411 = r211407 <= r211410;
        double r211412 = !r211411;
        bool r211413 = r211409 || r211412;
        double r211414 = x;
        double r211415 = 7049496828096731.0;
        double r211416 = 2251799813685248.0;
        double r211417 = r211415 / r211416;
        double r211418 = y;
        double r211419 = r211417 * r211418;
        double r211420 = t;
        double r211421 = 2.0;
        double r211422 = pow(r211407, r211421);
        double r211423 = r211422 / r211418;
        double r211424 = r211420 / r211423;
        double r211425 = r211419 + r211424;
        double r211426 = 1285181026435087.0;
        double r211427 = 35184372088832.0;
        double r211428 = r211426 / r211427;
        double r211429 = r211418 / r211407;
        double r211430 = r211428 * r211429;
        double r211431 = r211425 - r211430;
        double r211432 = r211414 + r211431;
        double r211433 = 4288183283079449.0;
        double r211434 = 281474976710656.0;
        double r211435 = r211433 / r211434;
        double r211436 = r211407 + r211435;
        double r211437 = r211436 * r211407;
        double r211438 = 4428869650076171.0;
        double r211439 = 140737488355328.0;
        double r211440 = r211438 / r211439;
        double r211441 = r211437 + r211440;
        double r211442 = r211441 * r211407;
        double r211443 = 3360836715704971.0;
        double r211444 = r211443 / r211434;
        double r211445 = r211442 + r211444;
        double r211446 = r211445 * r211407;
        double r211447 = 2737158995491925.0;
        double r211448 = 4503599627370496.0;
        double r211449 = r211447 / r211448;
        double r211450 = r211446 + r211449;
        double r211451 = r211407 * r211417;
        double r211452 = 3143161857605767.0;
        double r211453 = r211452 / r211434;
        double r211454 = r211451 + r211453;
        double r211455 = r211454 * r211407;
        double r211456 = r211455 + r211420;
        double r211457 = r211456 * r211407;
        double r211458 = a;
        double r211459 = r211457 + r211458;
        double r211460 = r211459 * r211407;
        double r211461 = b;
        double r211462 = r211460 + r211461;
        double r211463 = r211450 / r211462;
        double r211464 = r211418 / r211463;
        double r211465 = r211414 + r211464;
        double r211466 = r211413 ? r211432 : r211465;
        return r211466;
}

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

Original29.4
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.5358707234250173e+50 or 1.0535111611930094e+55 < z

    1. Initial program 61.7

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    2. Taylor expanded around inf 8.0

      \[\leadsto x + \color{blue}{\left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)}\]

    3. Simplified8.0

      \[\leadsto x + \color{blue}{\left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)}\]
    4. Using strategy rm

    5. Applied associate-/l*0.9

      \[\leadsto x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \color{blue}{\frac{t}{\frac{{z}^{2}}{y}}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\]

    if -1.5358707234250173e+50 < z < 1.0535111611930094e+55

    1. Initial program 2.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
    2. Using strategy rm

    3. Applied associate-/l*1.1

      \[\leadsto x + \color{blue}{\frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}}\]

    4. Simplified1.1

      \[\leadsto x + \frac{y}{\color{blue}{\frac{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot z + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.535870723425017265441515703700627878683 \cdot 10^{50} \lor \neg \left(z \le 1.053511161193009402933625668226348721359 \cdot 10^{55}\right):\\ \;\;\;\;x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t}{\frac{{z}^{2}}{y}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot z + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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 -6.4993449962526318e53) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.0669654369142868e59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004) (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004))))