Average Error: 20.1 → 0.1
Time: 4.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -83355327600392130835513344 \lor \neg \left(z \le 147550388.83208096027374267578125\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)} + x\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}
↓
\begin{array}{l}
\mathbf{if}\;z \le -83355327600392130835513344 \lor \neg \left(z \le 147550388.83208096027374267578125\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)} + x\\

\end{array}
double f(double x, double y, double z) {
        double r378210 = x;
        double r378211 = y;
        double r378212 = z;
        double r378213 = 0.0692910599291889;
        double r378214 = r378212 * r378213;
        double r378215 = 0.4917317610505968;
        double r378216 = r378214 + r378215;
        double r378217 = r378216 * r378212;
        double r378218 = 0.279195317918525;
        double r378219 = r378217 + r378218;
        double r378220 = r378211 * r378219;
        double r378221 = 6.012459259764103;
        double r378222 = r378212 + r378221;
        double r378223 = r378222 * r378212;
        double r378224 = 3.350343815022304;
        double r378225 = r378223 + r378224;
        double r378226 = r378220 / r378225;
        double r378227 = r378210 + r378226;
        return r378227;
}
↓
double f(double x, double y, double z) {
        double r378228 = z;
        double r378229 = -8.335532760039213e+25;
        bool r378230 = r378228 <= r378229;
        double r378231 = 147550388.83208096;
        bool r378232 = r378228 <= r378231;
        double r378233 = !r378232;
        bool r378234 = r378230 || r378233;
        double r378235 = 0.07512208616047561;
        double r378236 = r378235 / r378228;
        double r378237 = y;
        double r378238 = 0.0692910599291889;
        double r378239 = x;
        double r378240 = fma(r378237, r378238, r378239);
        double r378241 = fma(r378236, r378237, r378240);
        double r378242 = 0.4917317610505968;
        double r378243 = fma(r378228, r378238, r378242);
        double r378244 = 0.279195317918525;
        double r378245 = fma(r378243, r378228, r378244);
        double r378246 = r378237 * r378245;
        double r378247 = 6.012459259764103;
        double r378248 = 3.350343815022304;
        double r378249 = fma(r378228, r378228, r378248);
        double r378250 = fma(r378228, r378247, r378249);
        double r378251 = r378246 / r378250;
        double r378252 = r378251 + r378239;
        double r378253 = r378234 ? r378241 : r378252;
        return r378253;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.1
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.335532760039213e+25 or 147550388.83208096 < z

    1. Initial program 41.8

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

    2. Simplified34.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)}\]

    if -8.335532760039213e+25 < z < 147550388.83208096

    1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)}\]

    3. Taylor expanded around 0 0.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{6.012459259764103336465268512256443500519 \cdot z + \left({z}^{2} + 3.350343815022303939343828460550867021084\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

    4. Simplified0.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.6

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)}, \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}, x\right)\]
    7. Using strategy rm

    8. Applied fma-udef0.6

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)} \cdot \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}\right) + x}\]

    9. Simplified0.2

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -83355327600392130835513344 \lor \neg \left(z \le 147550388.83208096027374267578125\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))