Average Error: 29.7 → 1.3
Time: 18.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.9368894499979585 \cdot 10^{37} \lor \neg \left(z \le 5056003.575426906\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}, z, a\right), z, b\right), x\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.9368894499979585 \cdot 10^{37} \lor \neg \left(z \le 5056003.575426906\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}, 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 r364053 = x;
        double r364054 = y;
        double r364055 = z;
        double r364056 = 3.13060547623;
        double r364057 = r364055 * r364056;
        double r364058 = 11.1667541262;
        double r364059 = r364057 + r364058;
        double r364060 = r364059 * r364055;
        double r364061 = t;
        double r364062 = r364060 + r364061;
        double r364063 = r364062 * r364055;
        double r364064 = a;
        double r364065 = r364063 + r364064;
        double r364066 = r364065 * r364055;
        double r364067 = b;
        double r364068 = r364066 + r364067;
        double r364069 = r364054 * r364068;
        double r364070 = 15.234687407;
        double r364071 = r364055 + r364070;
        double r364072 = r364071 * r364055;
        double r364073 = 31.4690115749;
        double r364074 = r364072 + r364073;
        double r364075 = r364074 * r364055;
        double r364076 = 11.9400905721;
        double r364077 = r364075 + r364076;
        double r364078 = r364077 * r364055;
        double r364079 = 0.607771387771;
        double r364080 = r364078 + r364079;
        double r364081 = r364069 / r364080;
        double r364082 = r364053 + r364081;
        return r364082;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r364083 = z;
        double r364084 = -1.9368894499979585e+37;
        bool r364085 = r364083 <= r364084;
        double r364086 = 5056003.575426906;
        bool r364087 = r364083 <= r364086;
        double r364088 = !r364087;
        bool r364089 = r364085 || r364088;
        double r364090 = t;
        double r364091 = 2.0;
        double r364092 = pow(r364083, r364091);
        double r364093 = r364090 / r364092;
        double r364094 = y;
        double r364095 = 3.13060547623;
        double r364096 = x;
        double r364097 = fma(r364095, r364094, r364096);
        double r364098 = fma(r364093, r364094, r364097);
        double r364099 = 15.234687407;
        double r364100 = r364083 + r364099;
        double r364101 = 31.4690115749;
        double r364102 = fma(r364100, r364083, r364101);
        double r364103 = 11.9400905721;
        double r364104 = fma(r364102, r364083, r364103);
        double r364105 = 0.607771387771;
        double r364106 = fma(r364104, r364083, r364105);
        double r364107 = r364094 / r364106;
        double r364108 = 11.1667541262;
        double r364109 = fma(r364083, r364095, r364108);
        double r364110 = fma(r364109, r364083, r364090);
        double r364111 = cbrt(r364110);
        double r364112 = r364111 * r364111;
        double r364113 = r364112 * r364111;
        double r364114 = a;
        double r364115 = fma(r364113, r364083, r364114);
        double r364116 = b;
        double r364117 = fma(r364115, r364083, r364116);
        double r364118 = fma(r364107, r364117, r364096);
        double r364119 = r364089 ? r364098 : r364118;
        return r364119;
}

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

Original29.7
Target1.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.9368894499979585e+37 or 5056003.575426906 < z

    1. Initial program 57.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified55.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 9.1

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547622999996 \cdot y\right)}\]

    4. Simplified2.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)}\]

    if -1.9368894499979585e+37 < z < 5056003.575426906

    1. Initial program 1.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.6

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}}, z, a\right), z, b\right), x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.9368894499979585 \cdot 10^{37} \lor \neg \left(z \le 5056003.575426906\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right)}, z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))))