Average Error: 29.5 → 1.1
Time: 6.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -16178886472109744128 \lor \neg \left(z \le 226488841917468233033438467018719232\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{{z}^{\left(\frac{2}{2}\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]
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 -16178886472109744128 \lor \neg \left(z \le 226488841917468233033438467018719232\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{{z}^{\left(\frac{2}{2}\right)}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\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(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r326392 = x;
        double r326393 = y;
        double r326394 = z;
        double r326395 = 3.13060547623;
        double r326396 = r326394 * r326395;
        double r326397 = 11.1667541262;
        double r326398 = r326396 + r326397;
        double r326399 = r326398 * r326394;
        double r326400 = t;
        double r326401 = r326399 + r326400;
        double r326402 = r326401 * r326394;
        double r326403 = a;
        double r326404 = r326402 + r326403;
        double r326405 = r326404 * r326394;
        double r326406 = b;
        double r326407 = r326405 + r326406;
        double r326408 = r326393 * r326407;
        double r326409 = 15.234687407;
        double r326410 = r326394 + r326409;
        double r326411 = r326410 * r326394;
        double r326412 = 31.4690115749;
        double r326413 = r326411 + r326412;
        double r326414 = r326413 * r326394;
        double r326415 = 11.9400905721;
        double r326416 = r326414 + r326415;
        double r326417 = r326416 * r326394;
        double r326418 = 0.607771387771;
        double r326419 = r326417 + r326418;
        double r326420 = r326408 / r326419;
        double r326421 = r326392 + r326420;
        return r326421;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r326422 = z;
        double r326423 = -1.6178886472109744e+19;
        bool r326424 = r326422 <= r326423;
        double r326425 = 2.2648884191746823e+35;
        bool r326426 = r326422 <= r326425;
        double r326427 = !r326426;
        bool r326428 = r326424 || r326427;
        double r326429 = y;
        double r326430 = 3.13060547623;
        double r326431 = t;
        double r326432 = r326431 / r326422;
        double r326433 = 2.0;
        double r326434 = r326433 / r326433;
        double r326435 = pow(r326422, r326434);
        double r326436 = r326432 / r326435;
        double r326437 = r326430 + r326436;
        double r326438 = x;
        double r326439 = fma(r326429, r326437, r326438);
        double r326440 = 15.234687407;
        double r326441 = r326422 + r326440;
        double r326442 = 31.4690115749;
        double r326443 = fma(r326441, r326422, r326442);
        double r326444 = 11.9400905721;
        double r326445 = fma(r326443, r326422, r326444);
        double r326446 = 0.607771387771;
        double r326447 = fma(r326445, r326422, r326446);
        double r326448 = r326429 / r326447;
        double r326449 = 1.0;
        double r326450 = 11.1667541262;
        double r326451 = fma(r326422, r326430, r326450);
        double r326452 = fma(r326451, r326422, r326431);
        double r326453 = r326449 * r326452;
        double r326454 = a;
        double r326455 = fma(r326453, r326422, r326454);
        double r326456 = b;
        double r326457 = fma(r326455, r326422, r326456);
        double r326458 = fma(r326448, r326457, r326438);
        double r326459 = r326428 ? r326439 : r326458;
        return r326459;
}

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

Original29.5
Target1.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.6178886472109744e+19 or 2.2648884191746823e+35 < z

    1. Initial program 58.5

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

    2. Simplified56.4

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

    3. Taylor expanded around inf 8.5

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

    4. Simplified1.7

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

    6. Applied sqr-pow1.7

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

    7. Applied associate-/r*1.7

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

    8. Simplified1.7

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

    if -1.6178886472109744e+19 < z < 2.2648884191746823e+35

    1. Initial program 0.9

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

    2. Simplified0.4

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

    4. Applied *-un-lft-identity0.4

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right)}, z, a\right), z, b\right), x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -16178886472109744128 \lor \neg \left(z \le 226488841917468233033438467018719232\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{{z}^{\left(\frac{2}{2}\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))