Average Error: 20.1 → 0.3
Time: 1.0m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -853.3972873017340816659270785748958587646 \lor \neg \left(z \le 5.081171329344766007807265850715339183807\right):\\ \;\;\;\;x + y \cdot \frac{1}{\left(14.43187621926893804413793986896052956581 - \frac{15.64635683029203505611803848296403884888}{z}\right) + \frac{101.237333520038163214849191717803478241}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(7.93650581153375334064747903539682738483 \cdot 10^{-4} \cdot \left({z}^{2} \cdot y\right) + 0.0833333333333332315628894093606504611671 \cdot y\right) - 0.002777777777517226320824761387484613806009 \cdot \left(z \cdot y\right)\right)\\ \end{array}\]
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 -853.3972873017340816659270785748958587646 \lor \neg \left(z \le 5.081171329344766007807265850715339183807\right):\\
\;\;\;\;x + y \cdot \frac{1}{\left(14.43187621926893804413793986896052956581 - \frac{15.64635683029203505611803848296403884888}{z}\right) + \frac{101.237333520038163214849191717803478241}{z \cdot z}}\\

\mathbf{else}:\\
\;\;\;\;x + \left(\left(7.93650581153375334064747903539682738483 \cdot 10^{-4} \cdot \left({z}^{2} \cdot y\right) + 0.0833333333333332315628894093606504611671 \cdot y\right) - 0.002777777777517226320824761387484613806009 \cdot \left(z \cdot y\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r269767 = x;
        double r269768 = y;
        double r269769 = z;
        double r269770 = 0.0692910599291889;
        double r269771 = r269769 * r269770;
        double r269772 = 0.4917317610505968;
        double r269773 = r269771 + r269772;
        double r269774 = r269773 * r269769;
        double r269775 = 0.279195317918525;
        double r269776 = r269774 + r269775;
        double r269777 = r269768 * r269776;
        double r269778 = 6.012459259764103;
        double r269779 = r269769 + r269778;
        double r269780 = r269779 * r269769;
        double r269781 = 3.350343815022304;
        double r269782 = r269780 + r269781;
        double r269783 = r269777 / r269782;
        double r269784 = r269767 + r269783;
        return r269784;
}
↓
double f(double x, double y, double z) {
        double r269785 = z;
        double r269786 = -853.3972873017341;
        bool r269787 = r269785 <= r269786;
        double r269788 = 5.081171329344766;
        bool r269789 = r269785 <= r269788;
        double r269790 = !r269789;
        bool r269791 = r269787 || r269790;
        double r269792 = x;
        double r269793 = y;
        double r269794 = 1.0;
        double r269795 = 14.431876219268938;
        double r269796 = 15.646356830292035;
        double r269797 = r269796 / r269785;
        double r269798 = r269795 - r269797;
        double r269799 = 101.23733352003816;
        double r269800 = r269785 * r269785;
        double r269801 = r269799 / r269800;
        double r269802 = r269798 + r269801;
        double r269803 = r269794 / r269802;
        double r269804 = r269793 * r269803;
        double r269805 = r269792 + r269804;
        double r269806 = 0.0007936505811533753;
        double r269807 = 2.0;
        double r269808 = pow(r269785, r269807);
        double r269809 = r269808 * r269793;
        double r269810 = r269806 * r269809;
        double r269811 = 0.08333333333333323;
        double r269812 = r269811 * r269793;
        double r269813 = r269810 + r269812;
        double r269814 = 0.0027777777775172263;
        double r269815 = r269785 * r269793;
        double r269816 = r269814 * r269815;
        double r269817 = r269813 - r269816;
        double r269818 = r269792 + r269817;
        double r269819 = r269791 ? r269805 : r269818;
        return r269819;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -853.3972873017341 or 5.081171329344766 < 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. Using strategy rm

    3. Applied associate-/l*32.6

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.3

      \[\leadsto x + \frac{y}{\color{blue}{\left(14.43187621926893804413793986896052956581 - \frac{15.64635683029203505611803848296403884888}{z}\right) + \frac{101.237333520038163214849191717803478241}{z \cdot z}}}\]
    6. Using strategy rm

    7. Applied div-inv0.2

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

    if -853.3972873017341 < z < 5.081171329344766

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.5

      \[\leadsto x + \color{blue}{\left(\left(7.93650581153375334064747903539682738483 \cdot 10^{-4} \cdot \left({z}^{2} \cdot y\right) + 0.0833333333333332315628894093606504611671 \cdot y\right) - 0.002777777777517226320824761387484613806009 \cdot \left(z \cdot y\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -853.3972873017340816659270785748958587646 \lor \neg \left(z \le 5.081171329344766007807265850715339183807\right):\\ \;\;\;\;x + y \cdot \frac{1}{\left(14.43187621926893804413793986896052956581 - \frac{15.64635683029203505611803848296403884888}{z}\right) + \frac{101.237333520038163214849191717803478241}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(7.93650581153375334064747903539682738483 \cdot 10^{-4} \cdot \left({z}^{2} \cdot y\right) + 0.0833333333333332315628894093606504611671 \cdot y\right) - 0.002777777777517226320824761387484613806009 \cdot \left(z \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.6524566747) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (/ 1 (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394))))