Average Error: 30.2 → 1.0
Time: 28.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.807302090294030027556419953822894801612 \cdot 10^{70} \lor \neg \left(z \le 548437100344291305854952900722667028480\right):\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right) + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}\\ \end{array}\]
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 -3.807302090294030027556419953822894801612 \cdot 10^{70} \lor \neg \left(z \le 548437100344291305854952900722667028480\right):\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right) + 3.130605476229999961645944495103321969509, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r411086 = x;
        double r411087 = y;
        double r411088 = z;
        double r411089 = 3.13060547623;
        double r411090 = r411088 * r411089;
        double r411091 = 11.1667541262;
        double r411092 = r411090 + r411091;
        double r411093 = r411092 * r411088;
        double r411094 = t;
        double r411095 = r411093 + r411094;
        double r411096 = r411095 * r411088;
        double r411097 = a;
        double r411098 = r411096 + r411097;
        double r411099 = r411098 * r411088;
        double r411100 = b;
        double r411101 = r411099 + r411100;
        double r411102 = r411087 * r411101;
        double r411103 = 15.234687407;
        double r411104 = r411088 + r411103;
        double r411105 = r411104 * r411088;
        double r411106 = 31.4690115749;
        double r411107 = r411105 + r411106;
        double r411108 = r411107 * r411088;
        double r411109 = 11.9400905721;
        double r411110 = r411108 + r411109;
        double r411111 = r411110 * r411088;
        double r411112 = 0.607771387771;
        double r411113 = r411111 + r411112;
        double r411114 = r411102 / r411113;
        double r411115 = r411086 + r411114;
        return r411115;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r411116 = z;
        double r411117 = -3.80730209029403e+70;
        bool r411118 = r411116 <= r411117;
        double r411119 = 5.484371003442913e+38;
        bool r411120 = r411116 <= r411119;
        double r411121 = !r411120;
        bool r411122 = r411118 || r411121;
        double r411123 = y;
        double r411124 = t;
        double r411125 = r411124 / r411116;
        double r411126 = r411125 / r411116;
        double r411127 = 36.527041698806414;
        double r411128 = r411127 / r411116;
        double r411129 = r411126 - r411128;
        double r411130 = 3.13060547623;
        double r411131 = r411129 + r411130;
        double r411132 = x;
        double r411133 = fma(r411123, r411131, r411132);
        double r411134 = 11.1667541262;
        double r411135 = fma(r411130, r411116, r411134);
        double r411136 = fma(r411116, r411135, r411124);
        double r411137 = a;
        double r411138 = fma(r411136, r411116, r411137);
        double r411139 = b;
        double r411140 = fma(r411138, r411116, r411139);
        double r411141 = 15.234687407;
        double r411142 = r411141 + r411116;
        double r411143 = 31.4690115749;
        double r411144 = fma(r411116, r411142, r411143);
        double r411145 = 11.9400905721;
        double r411146 = fma(r411116, r411144, r411145);
        double r411147 = 0.607771387771;
        double r411148 = fma(r411116, r411146, r411147);
        double r411149 = r411140 / r411148;
        double r411150 = r411123 * r411149;
        double r411151 = r411132 + r411150;
        double r411152 = r411122 ? r411133 : r411151;
        return r411152;
}

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

Original30.2
Target0.8
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.80730209029403e+70 or 5.484371003442913e+38 < z

    1. Initial program 61.7

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

    2. Simplified60.5

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

    3. Taylor expanded around inf 0.7

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509\right) - 36.52704169880641416057187598198652267456 \cdot \frac{1}{z}}, x\right)\]

    4. Simplified0.7

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.130605476229999961645944495103321969509 + \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right)}, x\right)\]

    if -3.80730209029403e+70 < z < 5.484371003442913e+38

    1. Initial program 2.9

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

    2. Simplified1.3

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

    4. Applied fma-udef1.3

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

    5. Simplified1.3

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

    7. Applied *-un-lft-identity1.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.807302090294030027556419953822894801612 \cdot 10^{70} \lor \neg \left(z \le 548437100344291305854952900722667028480\right):\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right) + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))