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 r371695 = x;
        double r371696 = y;
        double r371697 = z;
        double r371698 = 0.0692910599291889;
        double r371699 = r371697 * r371698;
        double r371700 = 0.4917317610505968;
        double r371701 = r371699 + r371700;
        double r371702 = r371701 * r371697;
        double r371703 = 0.279195317918525;
        double r371704 = r371702 + r371703;
        double r371705 = r371696 * r371704;
        double r371706 = 6.012459259764103;
        double r371707 = r371697 + r371706;
        double r371708 = r371707 * r371697;
        double r371709 = 3.350343815022304;
        double r371710 = r371708 + r371709;
        double r371711 = r371705 / r371710;
        double r371712 = r371695 + r371711;
        return r371712;
}
↓
double f(double x, double y, double z) {
        double r371713 = z;
        double r371714 = -8.335532760039213e+25;
        bool r371715 = r371713 <= r371714;
        double r371716 = 147550388.83208096;
        bool r371717 = r371713 <= r371716;
        double r371718 = !r371717;
        bool r371719 = r371715 || r371718;
        double r371720 = 0.07512208616047561;
        double r371721 = r371720 / r371713;
        double r371722 = y;
        double r371723 = 0.0692910599291889;
        double r371724 = x;
        double r371725 = fma(r371722, r371723, r371724);
        double r371726 = fma(r371721, r371722, r371725);
        double r371727 = 0.4917317610505968;
        double r371728 = fma(r371713, r371723, r371727);
        double r371729 = 0.279195317918525;
        double r371730 = fma(r371728, r371713, r371729);
        double r371731 = r371722 * r371730;
        double r371732 = 6.012459259764103;
        double r371733 = 3.350343815022304;
        double r371734 = fma(r371713, r371713, r371733);
        double r371735 = fma(r371713, r371732, r371734);
        double r371736 = r371731 / r371735;
        double r371737 = r371736 + r371724;
        double r371738 = r371719 ? r371726 : r371737;
        return r371738;
}

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))))