Average Error: 29.8 → 1.8
Time: 20.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.076810882413635365417964376133599293418 \cdot 10^{56} \lor \neg \left(z \le 128535262688296208\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \end{array}\]
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 -3.076810882413635365417964376133599293418 \cdot 10^{56} \lor \neg \left(z \le 128535262688296208\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;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}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r212439 = x;
        double r212440 = y;
        double r212441 = z;
        double r212442 = 3.13060547623;
        double r212443 = r212441 * r212442;
        double r212444 = 11.1667541262;
        double r212445 = r212443 + r212444;
        double r212446 = r212445 * r212441;
        double r212447 = t;
        double r212448 = r212446 + r212447;
        double r212449 = r212448 * r212441;
        double r212450 = a;
        double r212451 = r212449 + r212450;
        double r212452 = r212451 * r212441;
        double r212453 = b;
        double r212454 = r212452 + r212453;
        double r212455 = r212440 * r212454;
        double r212456 = 15.234687407;
        double r212457 = r212441 + r212456;
        double r212458 = r212457 * r212441;
        double r212459 = 31.4690115749;
        double r212460 = r212458 + r212459;
        double r212461 = r212460 * r212441;
        double r212462 = 11.9400905721;
        double r212463 = r212461 + r212462;
        double r212464 = r212463 * r212441;
        double r212465 = 0.607771387771;
        double r212466 = r212464 + r212465;
        double r212467 = r212455 / r212466;
        double r212468 = r212439 + r212467;
        return r212468;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r212469 = z;
        double r212470 = -3.0768108824136354e+56;
        bool r212471 = r212469 <= r212470;
        double r212472 = 1.2853526268829621e+17;
        bool r212473 = r212469 <= r212472;
        double r212474 = !r212473;
        bool r212475 = r212471 || r212474;
        double r212476 = y;
        double r212477 = 3.13060547623;
        double r212478 = t;
        double r212479 = 2.0;
        double r212480 = pow(r212469, r212479);
        double r212481 = r212478 / r212480;
        double r212482 = r212477 + r212481;
        double r212483 = x;
        double r212484 = fma(r212476, r212482, r212483);
        double r212485 = r212469 * r212477;
        double r212486 = 11.1667541262;
        double r212487 = r212485 + r212486;
        double r212488 = r212487 * r212469;
        double r212489 = r212488 + r212478;
        double r212490 = r212489 * r212469;
        double r212491 = a;
        double r212492 = r212490 + r212491;
        double r212493 = r212492 * r212469;
        double r212494 = b;
        double r212495 = r212493 + r212494;
        double r212496 = r212476 * r212495;
        double r212497 = 15.234687407;
        double r212498 = r212469 + r212497;
        double r212499 = r212498 * r212469;
        double r212500 = 31.4690115749;
        double r212501 = r212499 + r212500;
        double r212502 = r212501 * r212469;
        double r212503 = 11.9400905721;
        double r212504 = r212502 + r212503;
        double r212505 = r212504 * r212469;
        double r212506 = 0.607771387771;
        double r212507 = r212505 + r212506;
        double r212508 = r212496 / r212507;
        double r212509 = r212483 + r212508;
        double r212510 = r212475 ? r212484 : r212509;
        return r212510;
}

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

Target

Original29.8
Target1.0
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.0768108824136354e+56 or 1.2853526268829621e+17 < z

    1. Initial program 59.6

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

    2. Simplified57.8

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

    3. Taylor expanded around inf 8.6

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

    4. Simplified1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)}\]

    if -3.0768108824136354e+56 < z < 1.2853526268829621e+17

    1. Initial program 1.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.076810882413635365417964376133599293418 \cdot 10^{56} \lor \neg \left(z \le 128535262688296208\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))