Average Error: 19.3 → 0.1
Time: 7.3s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -19637172304780185600 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}
↓
\begin{array}{l}
\mathbf{if}\;z \le -19637172304780185600 \lor \neg \left(z \le 63485.5636438174624\right):\\
\;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r371776 = x;
        double r371777 = y;
        double r371778 = z;
        double r371779 = 0.0692910599291889;
        double r371780 = r371778 * r371779;
        double r371781 = 0.4917317610505968;
        double r371782 = r371780 + r371781;
        double r371783 = r371782 * r371778;
        double r371784 = 0.279195317918525;
        double r371785 = r371783 + r371784;
        double r371786 = r371777 * r371785;
        double r371787 = 6.012459259764103;
        double r371788 = r371778 + r371787;
        double r371789 = r371788 * r371778;
        double r371790 = 3.350343815022304;
        double r371791 = r371789 + r371790;
        double r371792 = r371786 / r371791;
        double r371793 = r371776 + r371792;
        return r371793;
}
↓
double f(double x, double y, double z) {
        double r371794 = z;
        double r371795 = -1.9637172304780186e+19;
        bool r371796 = r371794 <= r371795;
        double r371797 = 63485.56364381746;
        bool r371798 = r371794 <= r371797;
        double r371799 = !r371798;
        bool r371800 = r371796 || r371799;
        double r371801 = 0.0692910599291889;
        double r371802 = y;
        double r371803 = 0.07512208616047561;
        double r371804 = r371802 / r371794;
        double r371805 = x;
        double r371806 = fma(r371803, r371804, r371805);
        double r371807 = fma(r371801, r371802, r371806);
        double r371808 = 0.4917317610505968;
        double r371809 = fma(r371794, r371801, r371808);
        double r371810 = 0.279195317918525;
        double r371811 = fma(r371809, r371794, r371810);
        double r371812 = 6.012459259764103;
        double r371813 = r371794 + r371812;
        double r371814 = 3.350343815022304;
        double r371815 = fma(r371813, r371794, r371814);
        double r371816 = r371811 / r371815;
        double r371817 = r371802 * r371816;
        double r371818 = r371805 + r371817;
        double r371819 = r371800 ? r371807 : r371818;
        return r371819;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.3
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.9637172304780186e+19 or 63485.56364381746 < z

    1. Initial program 40.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

    2. Simplified33.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)}\]

    if -1.9637172304780186e+19 < z < 63485.56364381746

    1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.2

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\color{blue}{1 \cdot \left(\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}}\]

    5. Simplified0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

    6. Simplified0.1

      \[\leadsto x + y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -19637172304780185600 \lor \neg \left(z \le 63485.5636438174624\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291888946, y, \mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))