Average Error: 19.5 → 0.6
Time: 20.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -248713679.72383326:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}
↓
\begin{array}{l}
\mathbf{if}\;z \le -248713679.72383326:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\
\;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r18046364 = x;
        double r18046365 = y;
        double r18046366 = z;
        double r18046367 = 0.0692910599291889;
        double r18046368 = r18046366 * r18046367;
        double r18046369 = 0.4917317610505968;
        double r18046370 = r18046368 + r18046369;
        double r18046371 = r18046370 * r18046366;
        double r18046372 = 0.279195317918525;
        double r18046373 = r18046371 + r18046372;
        double r18046374 = r18046365 * r18046373;
        double r18046375 = 6.012459259764103;
        double r18046376 = r18046366 + r18046375;
        double r18046377 = r18046376 * r18046366;
        double r18046378 = 3.350343815022304;
        double r18046379 = r18046377 + r18046378;
        double r18046380 = r18046374 / r18046379;
        double r18046381 = r18046364 + r18046380;
        return r18046381;
}
↓
double f(double x, double y, double z) {
        double r18046382 = z;
        double r18046383 = -248713679.72383326;
        bool r18046384 = r18046382 <= r18046383;
        double r18046385 = 0.0692910599291889;
        double r18046386 = y;
        double r18046387 = 0.07512208616047561;
        double r18046388 = r18046386 / r18046382;
        double r18046389 = x;
        double r18046390 = fma(r18046387, r18046388, r18046389);
        double r18046391 = fma(r18046385, r18046386, r18046390);
        double r18046392 = 8.728612712058619e-20;
        bool r18046393 = r18046382 <= r18046392;
        double r18046394 = r18046385 * r18046382;
        double r18046395 = 0.4917317610505968;
        double r18046396 = r18046394 + r18046395;
        double r18046397 = r18046382 * r18046396;
        double r18046398 = 0.279195317918525;
        double r18046399 = r18046397 + r18046398;
        double r18046400 = r18046386 * r18046399;
        double r18046401 = 3.350343815022304;
        double r18046402 = 6.012459259764103;
        double r18046403 = r18046382 + r18046402;
        double r18046404 = r18046382 * r18046403;
        double r18046405 = r18046401 + r18046404;
        double r18046406 = r18046400 / r18046405;
        double r18046407 = r18046389 + r18046406;
        double r18046408 = r18046393 ? r18046407 : r18046391;
        double r18046409 = r18046384 ? r18046391 : r18046408;
        return r18046409;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.5
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -248713679.72383326 or 8.728612712058619e-20 < z

    1. Initial program 38.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    2. Simplified31.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), x\right)}\]

    3. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291889 \cdot y\right)}\]

    4. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)}\]

    if -248713679.72383326 < z < 8.728612712058619e-20

    1. Initial program 0.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -248713679.72383326:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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 (+ (* (+ 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))))