Average Error: 19.8 → 0.1
Time: 23.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -485777911.392265796661376953125:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \mathbf{elif}\;z \le 774965.757210233598016202449798583984375:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{\frac{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}} \cdot \sqrt{\frac{1}{\frac{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \end{array}\]
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 -485777911.392265796661376953125:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\

\mathbf{elif}\;z \le 774965.757210233598016202449798583984375:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{\frac{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}} \cdot \sqrt{\frac{1}{\frac{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\

\end{array}
double f(double x, double y, double z) {
        double r14733684 = x;
        double r14733685 = y;
        double r14733686 = z;
        double r14733687 = 0.0692910599291889;
        double r14733688 = r14733686 * r14733687;
        double r14733689 = 0.4917317610505968;
        double r14733690 = r14733688 + r14733689;
        double r14733691 = r14733690 * r14733686;
        double r14733692 = 0.279195317918525;
        double r14733693 = r14733691 + r14733692;
        double r14733694 = r14733685 * r14733693;
        double r14733695 = 6.012459259764103;
        double r14733696 = r14733686 + r14733695;
        double r14733697 = r14733696 * r14733686;
        double r14733698 = 3.350343815022304;
        double r14733699 = r14733697 + r14733698;
        double r14733700 = r14733694 / r14733699;
        double r14733701 = r14733684 + r14733700;
        return r14733701;
}
↓
double f(double x, double y, double z) {
        double r14733702 = z;
        double r14733703 = -485777911.3922658;
        bool r14733704 = r14733702 <= r14733703;
        double r14733705 = 1.0;
        double r14733706 = 101.23733352003816;
        double r14733707 = r14733702 * r14733702;
        double r14733708 = r14733706 / r14733707;
        double r14733709 = 15.646356830292035;
        double r14733710 = r14733709 / r14733702;
        double r14733711 = r14733708 - r14733710;
        double r14733712 = 14.431876219268938;
        double r14733713 = r14733711 + r14733712;
        double r14733714 = r14733705 / r14733713;
        double r14733715 = y;
        double r14733716 = x;
        double r14733717 = fma(r14733714, r14733715, r14733716);
        double r14733718 = 774965.7572102336;
        bool r14733719 = r14733702 <= r14733718;
        double r14733720 = 6.012459259764103;
        double r14733721 = r14733720 + r14733702;
        double r14733722 = 3.350343815022304;
        double r14733723 = fma(r14733721, r14733702, r14733722);
        double r14733724 = 0.0692910599291889;
        double r14733725 = 0.4917317610505968;
        double r14733726 = fma(r14733724, r14733702, r14733725);
        double r14733727 = 0.279195317918525;
        double r14733728 = fma(r14733726, r14733702, r14733727);
        double r14733729 = r14733723 / r14733728;
        double r14733730 = r14733705 / r14733729;
        double r14733731 = sqrt(r14733730);
        double r14733732 = r14733731 * r14733731;
        double r14733733 = fma(r14733732, r14733715, r14733716);
        double r14733734 = r14733719 ? r14733733 : r14733717;
        double r14733735 = r14733704 ? r14733717 : r14733734;
        return r14733735;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.8
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 < -485777911.3922658 or 774965.7572102336 < z

    1. Initial program 40.4

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

    2. Simplified32.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, y, x\right)}\]
    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. Taylor expanded around inf 0.0

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

    6. 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 -485777911.3922658 < z < 774965.7572102336

    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 clear-num0.4

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

    6. Applied add-sqr-sqrt0.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}} \cdot \sqrt{\frac{1}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}}, y, x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -485777911.392265796661376953125:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \mathbf{elif}\;z \le 774965.757210233598016202449798583984375:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{\frac{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}} \cdot \sqrt{\frac{1}{\frac{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}}}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(\frac{101.237333520038163214849191717803478241}{z \cdot z} - \frac{15.64635683029203505611803848296403884888}{z}\right) + 14.43187621926893804413793986896052956581}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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))))