Average Error: 29.3 → 1.4
Time: 59.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -10275167136366415889384921525911552:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 2533004793734.77001953125:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]
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 -10275167136366415889384921525911552:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\

\mathbf{elif}\;z \le 2533004793734.77001953125:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r13332218 = x;
        double r13332219 = y;
        double r13332220 = z;
        double r13332221 = 3.13060547623;
        double r13332222 = r13332220 * r13332221;
        double r13332223 = 11.1667541262;
        double r13332224 = r13332222 + r13332223;
        double r13332225 = r13332224 * r13332220;
        double r13332226 = t;
        double r13332227 = r13332225 + r13332226;
        double r13332228 = r13332227 * r13332220;
        double r13332229 = a;
        double r13332230 = r13332228 + r13332229;
        double r13332231 = r13332230 * r13332220;
        double r13332232 = b;
        double r13332233 = r13332231 + r13332232;
        double r13332234 = r13332219 * r13332233;
        double r13332235 = 15.234687407;
        double r13332236 = r13332220 + r13332235;
        double r13332237 = r13332236 * r13332220;
        double r13332238 = 31.4690115749;
        double r13332239 = r13332237 + r13332238;
        double r13332240 = r13332239 * r13332220;
        double r13332241 = 11.9400905721;
        double r13332242 = r13332240 + r13332241;
        double r13332243 = r13332242 * r13332220;
        double r13332244 = 0.607771387771;
        double r13332245 = r13332243 + r13332244;
        double r13332246 = r13332234 / r13332245;
        double r13332247 = r13332218 + r13332246;
        return r13332247;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r13332248 = z;
        double r13332249 = -1.0275167136366416e+34;
        bool r13332250 = r13332248 <= r13332249;
        double r13332251 = y;
        double r13332252 = t;
        double r13332253 = r13332252 / r13332248;
        double r13332254 = r13332253 / r13332248;
        double r13332255 = 3.13060547623;
        double r13332256 = r13332254 + r13332255;
        double r13332257 = x;
        double r13332258 = fma(r13332251, r13332256, r13332257);
        double r13332259 = 2533004793734.77;
        bool r13332260 = r13332248 <= r13332259;
        double r13332261 = 1.0;
        double r13332262 = 15.234687407;
        double r13332263 = r13332262 + r13332248;
        double r13332264 = 31.4690115749;
        double r13332265 = fma(r13332248, r13332263, r13332264);
        double r13332266 = 11.9400905721;
        double r13332267 = fma(r13332248, r13332265, r13332266);
        double r13332268 = 0.607771387771;
        double r13332269 = fma(r13332248, r13332267, r13332268);
        double r13332270 = sqrt(r13332269);
        double r13332271 = r13332261 / r13332270;
        double r13332272 = 11.1667541262;
        double r13332273 = fma(r13332255, r13332248, r13332272);
        double r13332274 = fma(r13332273, r13332248, r13332252);
        double r13332275 = a;
        double r13332276 = fma(r13332248, r13332274, r13332275);
        double r13332277 = b;
        double r13332278 = fma(r13332276, r13332248, r13332277);
        double r13332279 = r13332278 / r13332270;
        double r13332280 = r13332271 * r13332279;
        double r13332281 = fma(r13332251, r13332280, r13332257);
        double r13332282 = r13332260 ? r13332281 : r13332258;
        double r13332283 = r13332250 ? r13332258 : r13332282;
        return r13332283;
}

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
Target1.0
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 < -1.0275167136366416e+34 or 2533004793734.77 < z

    1. Initial program 58.1

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

    2. Simplified55.3

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

    3. Taylor expanded around inf 9.4

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

    4. Simplified2.0

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

    if -1.0275167136366416e+34 < z < 2533004793734.77

    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

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

    4. Applied add-sqr-sqrt0.9

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

    5. Applied *-un-lft-identity0.9

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\]

    6. Applied times-frac0.7

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -10275167136366415889384921525911552:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 2533004793734.77001953125:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))