Average Error: 29.3 → 1.4
Time: 28.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.692733421424862403420484325153801185463 \cdot 10^{67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509, x\right)\right)\\ \mathbf{elif}\;z \le 2356527327781351729875910656000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509, x\right)\right)\\ \end{array}\]
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 -1.692733421424862403420484325153801185463 \cdot 10^{67}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509, x\right)\right)\\

\mathbf{elif}\;z \le 2356527327781351729875910656000:\\
\;\;\;\;\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)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509, x\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r47384402 = x;
        double r47384403 = y;
        double r47384404 = z;
        double r47384405 = 3.13060547623;
        double r47384406 = r47384404 * r47384405;
        double r47384407 = 11.1667541262;
        double r47384408 = r47384406 + r47384407;
        double r47384409 = r47384408 * r47384404;
        double r47384410 = t;
        double r47384411 = r47384409 + r47384410;
        double r47384412 = r47384411 * r47384404;
        double r47384413 = a;
        double r47384414 = r47384412 + r47384413;
        double r47384415 = r47384414 * r47384404;
        double r47384416 = b;
        double r47384417 = r47384415 + r47384416;
        double r47384418 = r47384403 * r47384417;
        double r47384419 = 15.234687407;
        double r47384420 = r47384404 + r47384419;
        double r47384421 = r47384420 * r47384404;
        double r47384422 = 31.4690115749;
        double r47384423 = r47384421 + r47384422;
        double r47384424 = r47384423 * r47384404;
        double r47384425 = 11.9400905721;
        double r47384426 = r47384424 + r47384425;
        double r47384427 = r47384426 * r47384404;
        double r47384428 = 0.607771387771;
        double r47384429 = r47384427 + r47384428;
        double r47384430 = r47384418 / r47384429;
        double r47384431 = r47384402 + r47384430;
        return r47384431;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r47384432 = z;
        double r47384433 = -1.6927334214248624e+67;
        bool r47384434 = r47384432 <= r47384433;
        double r47384435 = t;
        double r47384436 = r47384435 / r47384432;
        double r47384437 = y;
        double r47384438 = r47384437 / r47384432;
        double r47384439 = 3.13060547623;
        double r47384440 = x;
        double r47384441 = fma(r47384437, r47384439, r47384440);
        double r47384442 = fma(r47384436, r47384438, r47384441);
        double r47384443 = 2.3565273277813517e+30;
        bool r47384444 = r47384432 <= r47384443;
        double r47384445 = 15.234687407;
        double r47384446 = r47384432 + r47384445;
        double r47384447 = 31.4690115749;
        double r47384448 = fma(r47384446, r47384432, r47384447);
        double r47384449 = 11.9400905721;
        double r47384450 = fma(r47384448, r47384432, r47384449);
        double r47384451 = 0.607771387771;
        double r47384452 = fma(r47384450, r47384432, r47384451);
        double r47384453 = r47384437 / r47384452;
        double r47384454 = 11.1667541262;
        double r47384455 = fma(r47384432, r47384439, r47384454);
        double r47384456 = fma(r47384455, r47384432, r47384435);
        double r47384457 = a;
        double r47384458 = fma(r47384456, r47384432, r47384457);
        double r47384459 = b;
        double r47384460 = fma(r47384458, r47384432, r47384459);
        double r47384461 = fma(r47384453, r47384460, r47384440);
        double r47384462 = r47384444 ? r47384461 : r47384442;
        double r47384463 = r47384434 ? r47384442 : r47384462;
        return r47384463;
}

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

Original29.3
Target1.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.6927334214248624e+67 or 2.3565273277813517e+30 < z

    1. Initial program 60.9

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

    2. Simplified59.2

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified1.4

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

    if -1.6927334214248624e+67 < z < 2.3565273277813517e+30

    1. Initial program 2.7

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

    2. Simplified1.5

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.692733421424862403420484325153801185463 \cdot 10^{67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509, x\right)\right)\\ \mathbf{elif}\;z \le 2356527327781351729875910656000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (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 1.0)))))

  (+ 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))))