Average Error: 19.3 → 0.1
Time: 18.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4347718231920.455078125 \lor \neg \left(z \le 14154540400533552\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)\\ \end{array}\]
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 -4347718231920.455078125 \lor \neg \left(z \le 14154540400533552\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r253705 = x;
        double r253706 = y;
        double r253707 = z;
        double r253708 = 0.0692910599291889;
        double r253709 = r253707 * r253708;
        double r253710 = 0.4917317610505968;
        double r253711 = r253709 + r253710;
        double r253712 = r253711 * r253707;
        double r253713 = 0.279195317918525;
        double r253714 = r253712 + r253713;
        double r253715 = r253706 * r253714;
        double r253716 = 6.012459259764103;
        double r253717 = r253707 + r253716;
        double r253718 = r253717 * r253707;
        double r253719 = 3.350343815022304;
        double r253720 = r253718 + r253719;
        double r253721 = r253715 / r253720;
        double r253722 = r253705 + r253721;
        return r253722;
}
↓
double f(double x, double y, double z) {
        double r253723 = z;
        double r253724 = -4347718231920.455;
        bool r253725 = r253723 <= r253724;
        double r253726 = 14154540400533552.0;
        bool r253727 = r253723 <= r253726;
        double r253728 = !r253727;
        bool r253729 = r253725 || r253728;
        double r253730 = 1.0;
        double r253731 = 101.23733352003816;
        double r253732 = r253723 * r253723;
        double r253733 = r253731 / r253732;
        double r253734 = 15.646356830292035;
        double r253735 = r253734 / r253723;
        double r253736 = r253733 - r253735;
        double r253737 = 14.431876219268938;
        double r253738 = r253736 + r253737;
        double r253739 = r253730 / r253738;
        double r253740 = y;
        double r253741 = x;
        double r253742 = fma(r253739, r253740, r253741);
        double r253743 = 6.012459259764103;
        double r253744 = r253743 + r253723;
        double r253745 = 3.350343815022304;
        double r253746 = fma(r253744, r253723, r253745);
        double r253747 = r253740 / r253746;
        double r253748 = 0.0692910599291889;
        double r253749 = 0.4917317610505968;
        double r253750 = fma(r253723, r253748, r253749);
        double r253751 = 0.279195317918525;
        double r253752 = fma(r253750, r253723, r253751);
        double r253753 = r253747 * r253752;
        double r253754 = r253741 + r253753;
        double r253755 = r253729 ? r253742 : r253754;
        return r253755;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.3
Target0.2
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 < -4347718231920.455 or 14154540400533552.0 < z

    1. Initial program 40.6

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

    2. Simplified32.6

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

    4. Applied clear-num32.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}, y, x\right)\]

    5. Simplified32.5

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}, y, x\right)\]

    6. Taylor expanded around inf 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\left(101.237333520038163214849191717803478241 \cdot \frac{1}{{z}^{2}} + 14.43187621926893804413793986896052956581\right) - 15.64635683029203505611803848296403884888 \cdot \frac{1}{z}}}, y, x\right)\]

    7. Simplified0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{14.43187621926893804413793986896052956581 + \left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right)}}, y, x\right)\]

    if -4347718231920.455 < z < 14154540400533552.0

    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

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

    4. Applied fma-udef0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4347718231920.455078125 \lor \neg \left(z \le 14154540400533552\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ 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))))