Average Error: 19.6 → 0.1
Time: 6.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1731582918361017216925696 \lor \neg \left(z \le 447984676.597118556499481201171875\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{y}} + x\\ \end{array}\]
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 -1731582918361017216925696 \lor \neg \left(z \le 447984676.597118556499481201171875\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r295632 = x;
        double r295633 = y;
        double r295634 = z;
        double r295635 = 0.0692910599291889;
        double r295636 = r295634 * r295635;
        double r295637 = 0.4917317610505968;
        double r295638 = r295636 + r295637;
        double r295639 = r295638 * r295634;
        double r295640 = 0.279195317918525;
        double r295641 = r295639 + r295640;
        double r295642 = r295633 * r295641;
        double r295643 = 6.012459259764103;
        double r295644 = r295634 + r295643;
        double r295645 = r295644 * r295634;
        double r295646 = 3.350343815022304;
        double r295647 = r295645 + r295646;
        double r295648 = r295642 / r295647;
        double r295649 = r295632 + r295648;
        return r295649;
}
↓
double f(double x, double y, double z) {
        double r295650 = z;
        double r295651 = -1.7315829183610172e+24;
        bool r295652 = r295650 <= r295651;
        double r295653 = 447984676.59711856;
        bool r295654 = r295650 <= r295653;
        double r295655 = !r295654;
        bool r295656 = r295652 || r295655;
        double r295657 = 0.07512208616047561;
        double r295658 = r295657 / r295650;
        double r295659 = y;
        double r295660 = 0.0692910599291889;
        double r295661 = x;
        double r295662 = fma(r295659, r295660, r295661);
        double r295663 = fma(r295658, r295659, r295662);
        double r295664 = 0.4917317610505968;
        double r295665 = fma(r295650, r295660, r295664);
        double r295666 = 0.279195317918525;
        double r295667 = fma(r295665, r295650, r295666);
        double r295668 = 6.012459259764103;
        double r295669 = r295650 + r295668;
        double r295670 = 3.350343815022304;
        double r295671 = fma(r295669, r295650, r295670);
        double r295672 = r295671 / r295659;
        double r295673 = r295667 / r295672;
        double r295674 = r295673 + r295661;
        double r295675 = r295656 ? r295663 : r295674;
        return r295675;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.6
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 < -1.7315829183610172e+24 or 447984676.59711856 < z

    1. Initial program 41.2

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

    2. Simplified34.3

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

    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 -1.7315829183610172e+24 < z < 447984676.59711856

    1. Initial program 0.3

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

    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. Using strategy rm

    4. Applied clear-num0.2

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

    6. Applied fma-udef0.2

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1731582918361017216925696 \lor \neg \left(z \le 447984676.597118556499481201171875\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))))