Average Error: 19.7 → 0.1
Time: 1.3m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1029710412402.12939453125:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 186841422.6442936956882476806640625:\\ \;\;\;\;x + \frac{\left(\left(0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot z\right) \cdot z + 0.2791953179185249767080279070796677842736\right) \cdot y}{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]
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 -1029710412402.12939453125:\\
\;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\

\mathbf{elif}\;z \le 186841422.6442936956882476806640625:\\
\;\;\;\;x + \frac{\left(\left(0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot z\right) \cdot z + 0.2791953179185249767080279070796677842736\right) \cdot y}{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r15742407 = x;
        double r15742408 = y;
        double r15742409 = z;
        double r15742410 = 0.0692910599291889;
        double r15742411 = r15742409 * r15742410;
        double r15742412 = 0.4917317610505968;
        double r15742413 = r15742411 + r15742412;
        double r15742414 = r15742413 * r15742409;
        double r15742415 = 0.279195317918525;
        double r15742416 = r15742414 + r15742415;
        double r15742417 = r15742408 * r15742416;
        double r15742418 = 6.012459259764103;
        double r15742419 = r15742409 + r15742418;
        double r15742420 = r15742419 * r15742409;
        double r15742421 = 3.350343815022304;
        double r15742422 = r15742420 + r15742421;
        double r15742423 = r15742417 / r15742422;
        double r15742424 = r15742407 + r15742423;
        return r15742424;
}
↓
double f(double x, double y, double z) {
        double r15742425 = z;
        double r15742426 = -1029710412402.1294;
        bool r15742427 = r15742425 <= r15742426;
        double r15742428 = y;
        double r15742429 = 0.0692910599291889;
        double r15742430 = r15742428 / r15742425;
        double r15742431 = 0.07512208616047561;
        double r15742432 = x;
        double r15742433 = fma(r15742430, r15742431, r15742432);
        double r15742434 = fma(r15742428, r15742429, r15742433);
        double r15742435 = 186841422.6442937;
        bool r15742436 = r15742425 <= r15742435;
        double r15742437 = 0.4917317610505968;
        double r15742438 = r15742429 * r15742425;
        double r15742439 = r15742437 + r15742438;
        double r15742440 = r15742439 * r15742425;
        double r15742441 = 0.279195317918525;
        double r15742442 = r15742440 + r15742441;
        double r15742443 = r15742442 * r15742428;
        double r15742444 = 3.350343815022304;
        double r15742445 = 6.012459259764103;
        double r15742446 = r15742445 + r15742425;
        double r15742447 = r15742446 * r15742425;
        double r15742448 = r15742444 + r15742447;
        double r15742449 = r15742443 / r15742448;
        double r15742450 = r15742432 + r15742449;
        double r15742451 = r15742436 ? r15742450 : r15742434;
        double r15742452 = r15742427 ? r15742434 : r15742451;
        return r15742452;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.7
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 < -1029710412402.1294 or 186841422.6442937 < z

    1. Initial program 41.0

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

    2. Simplified33.7

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

    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

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

    if -1029710412402.1294 < z < 186841422.6442937

    1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1029710412402.12939453125:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 186841422.6442936956882476806640625:\\ \;\;\;\;x + \frac{\left(\left(0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot z\right) \cdot z + 0.2791953179185249767080279070796677842736\right) \cdot y}{3.350343815022303939343828460550867021084 + \left(6.012459259764103336465268512256443500519 + z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))