Average Error: 19.9 → 0.1
Time: 20.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -78986841760531753205760:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{elif}\;z \le 25491854467815592:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \end{array}\]
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 -78986841760531753205760:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\

\mathbf{elif}\;z \le 25491854467815592:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{\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), x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r13350066 = x;
        double r13350067 = y;
        double r13350068 = z;
        double r13350069 = 0.0692910599291889;
        double r13350070 = r13350068 * r13350069;
        double r13350071 = 0.4917317610505968;
        double r13350072 = r13350070 + r13350071;
        double r13350073 = r13350072 * r13350068;
        double r13350074 = 0.279195317918525;
        double r13350075 = r13350073 + r13350074;
        double r13350076 = r13350067 * r13350075;
        double r13350077 = 6.012459259764103;
        double r13350078 = r13350068 + r13350077;
        double r13350079 = r13350078 * r13350068;
        double r13350080 = 3.350343815022304;
        double r13350081 = r13350079 + r13350080;
        double r13350082 = r13350076 / r13350081;
        double r13350083 = r13350066 + r13350082;
        return r13350083;
}
↓
double f(double x, double y, double z) {
        double r13350084 = z;
        double r13350085 = -7.898684176053175e+22;
        bool r13350086 = r13350084 <= r13350085;
        double r13350087 = y;
        double r13350088 = r13350087 / r13350084;
        double r13350089 = 0.07512208616047561;
        double r13350090 = 0.0692910599291889;
        double r13350091 = x;
        double r13350092 = fma(r13350090, r13350087, r13350091);
        double r13350093 = fma(r13350088, r13350089, r13350092);
        double r13350094 = 2.549185446781559e+16;
        bool r13350095 = r13350084 <= r13350094;
        double r13350096 = 6.012459259764103;
        double r13350097 = r13350096 + r13350084;
        double r13350098 = 3.350343815022304;
        double r13350099 = fma(r13350097, r13350084, r13350098);
        double r13350100 = r13350087 / r13350099;
        double r13350101 = 0.4917317610505968;
        double r13350102 = fma(r13350090, r13350084, r13350101);
        double r13350103 = 0.279195317918525;
        double r13350104 = fma(r13350102, r13350084, r13350103);
        double r13350105 = fma(r13350100, r13350104, r13350091);
        double r13350106 = r13350095 ? r13350105 : r13350093;
        double r13350107 = r13350086 ? r13350093 : r13350106;
        return r13350107;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.9
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 < -7.898684176053175e+22 or 2.549185446781559e+16 < z

    1. Initial program 42.3

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

    2. Simplified35.2

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

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)}\]

    if -7.898684176053175e+22 < z < 2.549185446781559e+16

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -78986841760531753205760:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{elif}\;z \le 25491854467815592:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(6.012459259764103336465268512256443500519 + z, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.06929105992918889456166908757950295694172, z, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))