Average Error: 29.2 → 1.1
Time: 24.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.038261497315181869126238031565883904229 \cdot 10^{47} \lor \neg \left(z \le 177470294739602374656\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right) \cdot z + t, z, a\right), z, b\right), x\right)\\ \end{array}\]
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 -7.038261497315181869126238031565883904229 \cdot 10^{47} \lor \neg \left(z \le 177470294739602374656\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, 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(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right) \cdot z + t, 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 r241358 = x;
        double r241359 = y;
        double r241360 = z;
        double r241361 = 3.13060547623;
        double r241362 = r241360 * r241361;
        double r241363 = 11.1667541262;
        double r241364 = r241362 + r241363;
        double r241365 = r241364 * r241360;
        double r241366 = t;
        double r241367 = r241365 + r241366;
        double r241368 = r241367 * r241360;
        double r241369 = a;
        double r241370 = r241368 + r241369;
        double r241371 = r241370 * r241360;
        double r241372 = b;
        double r241373 = r241371 + r241372;
        double r241374 = r241359 * r241373;
        double r241375 = 15.234687407;
        double r241376 = r241360 + r241375;
        double r241377 = r241376 * r241360;
        double r241378 = 31.4690115749;
        double r241379 = r241377 + r241378;
        double r241380 = r241379 * r241360;
        double r241381 = 11.9400905721;
        double r241382 = r241380 + r241381;
        double r241383 = r241382 * r241360;
        double r241384 = 0.607771387771;
        double r241385 = r241383 + r241384;
        double r241386 = r241374 / r241385;
        double r241387 = r241358 + r241386;
        return r241387;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r241388 = z;
        double r241389 = -7.038261497315182e+47;
        bool r241390 = r241388 <= r241389;
        double r241391 = 1.7747029473960237e+20;
        bool r241392 = r241388 <= r241391;
        double r241393 = !r241392;
        bool r241394 = r241390 || r241393;
        double r241395 = y;
        double r241396 = t;
        double r241397 = r241388 * r241388;
        double r241398 = r241396 / r241397;
        double r241399 = 3.13060547623;
        double r241400 = r241398 + r241399;
        double r241401 = x;
        double r241402 = fma(r241395, r241400, r241401);
        double r241403 = 15.234687407;
        double r241404 = r241388 + r241403;
        double r241405 = 31.4690115749;
        double r241406 = fma(r241404, r241388, r241405);
        double r241407 = 11.9400905721;
        double r241408 = fma(r241406, r241388, r241407);
        double r241409 = 0.607771387771;
        double r241410 = fma(r241408, r241388, r241409);
        double r241411 = r241395 / r241410;
        double r241412 = 11.1667541262;
        double r241413 = fma(r241388, r241399, r241412);
        double r241414 = r241413 * r241388;
        double r241415 = r241414 + r241396;
        double r241416 = a;
        double r241417 = fma(r241415, r241388, r241416);
        double r241418 = b;
        double r241419 = fma(r241417, r241388, r241418);
        double r241420 = fma(r241411, r241419, r241401);
        double r241421 = r241394 ? r241402 : r241420;
        return r241421;
}

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

Original29.2
Target0.9
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 < -7.038261497315182e+47 or 1.7747029473960237e+20 < z

    1. Initial program 59.5

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

    2. Simplified57.6

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

    4. Applied fma-udef57.6

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

    5. Taylor expanded around inf 9.2

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

    6. Simplified1.6

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

    if -7.038261497315182e+47 < z < 1.7747029473960237e+20

    1. Initial program 1.2

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

    2. Simplified0.8

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

    4. Applied fma-udef0.8

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.038261497315181869126238031565883904229 \cdot 10^{47} \lor \neg \left(z \le 177470294739602374656\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right) \cdot z + t, z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))