Average Error: 28.9 → 1.0
Time: 23.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -11377676592436386882634106867972293791840000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}, x\right)\\ \mathbf{elif}\;z \le 854771490603964610616754176:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}, x\right)\\ \end{array}\]
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 -11377676592436386882634106867972293791840000:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}, x\right)\\

\mathbf{elif}\;z \le 854771490603964610616754176:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}{\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)}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r15299042 = x;
        double r15299043 = y;
        double r15299044 = z;
        double r15299045 = 3.13060547623;
        double r15299046 = r15299044 * r15299045;
        double r15299047 = 11.1667541262;
        double r15299048 = r15299046 + r15299047;
        double r15299049 = r15299048 * r15299044;
        double r15299050 = t;
        double r15299051 = r15299049 + r15299050;
        double r15299052 = r15299051 * r15299044;
        double r15299053 = a;
        double r15299054 = r15299052 + r15299053;
        double r15299055 = r15299054 * r15299044;
        double r15299056 = b;
        double r15299057 = r15299055 + r15299056;
        double r15299058 = r15299043 * r15299057;
        double r15299059 = 15.234687407;
        double r15299060 = r15299044 + r15299059;
        double r15299061 = r15299060 * r15299044;
        double r15299062 = 31.4690115749;
        double r15299063 = r15299061 + r15299062;
        double r15299064 = r15299063 * r15299044;
        double r15299065 = 11.9400905721;
        double r15299066 = r15299064 + r15299065;
        double r15299067 = r15299066 * r15299044;
        double r15299068 = 0.607771387771;
        double r15299069 = r15299067 + r15299068;
        double r15299070 = r15299058 / r15299069;
        double r15299071 = r15299042 + r15299070;
        return r15299071;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r15299072 = z;
        double r15299073 = -1.1377676592436387e+43;
        bool r15299074 = r15299072 <= r15299073;
        double r15299075 = y;
        double r15299076 = t;
        double r15299077 = r15299072 * r15299072;
        double r15299078 = r15299076 / r15299077;
        double r15299079 = 3.13060547623;
        double r15299080 = r15299078 + r15299079;
        double r15299081 = 36.527041698806414;
        double r15299082 = r15299081 / r15299072;
        double r15299083 = r15299080 - r15299082;
        double r15299084 = x;
        double r15299085 = fma(r15299075, r15299083, r15299084);
        double r15299086 = 8.547714906039646e+26;
        bool r15299087 = r15299072 <= r15299086;
        double r15299088 = 1.0;
        double r15299089 = 15.234687407;
        double r15299090 = r15299089 + r15299072;
        double r15299091 = 31.4690115749;
        double r15299092 = fma(r15299072, r15299090, r15299091);
        double r15299093 = 11.9400905721;
        double r15299094 = fma(r15299072, r15299092, r15299093);
        double r15299095 = 0.607771387771;
        double r15299096 = fma(r15299072, r15299094, r15299095);
        double r15299097 = 11.1667541262;
        double r15299098 = fma(r15299079, r15299072, r15299097);
        double r15299099 = fma(r15299098, r15299072, r15299076);
        double r15299100 = a;
        double r15299101 = fma(r15299072, r15299099, r15299100);
        double r15299102 = b;
        double r15299103 = fma(r15299101, r15299072, r15299102);
        double r15299104 = r15299096 / r15299103;
        double r15299105 = r15299088 / r15299104;
        double r15299106 = fma(r15299075, r15299105, r15299084);
        double r15299107 = r15299087 ? r15299106 : r15299085;
        double r15299108 = r15299074 ? r15299085 : r15299107;
        return r15299108;
}

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

Original28.9
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 < -1.1377676592436387e+43 or 8.547714906039646e+26 < z

    1. Initial program 59.2

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

    2. Simplified56.4

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

    4. Applied clear-num56.4

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

    5. Taylor expanded around inf 1.5

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

    6. Simplified1.5

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

    if -1.1377676592436387e+43 < z < 8.547714906039646e+26

    1. Initial program 1.3

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

    2. Simplified0.5

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

    4. Applied clear-num0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -11377676592436386882634106867972293791840000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}, x\right)\\ \mathbf{elif}\;z \le 854771490603964610616754176:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))))