Average Error: 20.1 → 0.5
Time: 4.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -92387344.1766572 \lor \neg \left(z \le 1.257025260484713 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -92387344.1766572 \lor \neg \left(z \le 1.257025260484713 \cdot 10^{-12}\right):\\
\;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\

\mathbf{else}:\\
\;\;\;\;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}\\

\end{array}
double f(double x, double y, double z) {
        double r415825 = x;
        double r415826 = y;
        double r415827 = z;
        double r415828 = 0.0692910599291889;
        double r415829 = r415827 * r415828;
        double r415830 = 0.4917317610505968;
        double r415831 = r415829 + r415830;
        double r415832 = r415831 * r415827;
        double r415833 = 0.279195317918525;
        double r415834 = r415832 + r415833;
        double r415835 = r415826 * r415834;
        double r415836 = 6.012459259764103;
        double r415837 = r415827 + r415836;
        double r415838 = r415837 * r415827;
        double r415839 = 3.350343815022304;
        double r415840 = r415838 + r415839;
        double r415841 = r415835 / r415840;
        double r415842 = r415825 + r415841;
        return r415842;
}
↓
double f(double x, double y, double z) {
        double r415843 = z;
        double r415844 = -92387344.1766572;
        bool r415845 = r415843 <= r415844;
        double r415846 = 1.257025260484713e-12;
        bool r415847 = r415843 <= r415846;
        double r415848 = !r415847;
        bool r415849 = r415845 || r415848;
        double r415850 = 0.07512208616047561;
        double r415851 = y;
        double r415852 = r415851 / r415843;
        double r415853 = 0.0692910599291889;
        double r415854 = r415853 * r415851;
        double r415855 = fma(r415850, r415852, r415854);
        double r415856 = x;
        double r415857 = r415855 + r415856;
        double r415858 = r415843 * r415853;
        double r415859 = 0.4917317610505968;
        double r415860 = r415858 + r415859;
        double r415861 = r415860 * r415843;
        double r415862 = 0.279195317918525;
        double r415863 = r415861 + r415862;
        double r415864 = r415851 * r415863;
        double r415865 = 6.012459259764103;
        double r415866 = r415843 + r415865;
        double r415867 = r415866 * r415843;
        double r415868 = 3.350343815022304;
        double r415869 = r415867 + r415868;
        double r415870 = r415864 / r415869;
        double r415871 = r415856 + r415870;
        double r415872 = r415849 ? r415857 : r415871;
        return r415872;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.1
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -92387344.1766572 or 1.257025260484713e-12 < z

    1. Initial program 39.3

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

    2. Simplified32.5

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

    4. Applied clear-num32.6

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

    5. Taylor expanded around inf 0.9

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

    6. Simplified0.9

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

    if -92387344.1766572 < z < 1.257025260484713e-12

    1. Initial program 0.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -92387344.1766572 \lor \neg \left(z \le 1.257025260484713 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))