Average Error: 20.1 → 0.1
Time: 4.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.21695302134859774 \cdot 10^{27} \lor \neg \left(z \le 318335141.94349515\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)} + x\\ \end{array}\]
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 -1.21695302134859774 \cdot 10^{27} \lor \neg \left(z \le 318335141.94349515\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;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)} + x\\

\end{array}
double f(double x, double y, double z) {
        double r334753 = x;
        double r334754 = y;
        double r334755 = z;
        double r334756 = 0.0692910599291889;
        double r334757 = r334755 * r334756;
        double r334758 = 0.4917317610505968;
        double r334759 = r334757 + r334758;
        double r334760 = r334759 * r334755;
        double r334761 = 0.279195317918525;
        double r334762 = r334760 + r334761;
        double r334763 = r334754 * r334762;
        double r334764 = 6.012459259764103;
        double r334765 = r334755 + r334764;
        double r334766 = r334765 * r334755;
        double r334767 = 3.350343815022304;
        double r334768 = r334766 + r334767;
        double r334769 = r334763 / r334768;
        double r334770 = r334753 + r334769;
        return r334770;
}
↓
double f(double x, double y, double z) {
        double r334771 = z;
        double r334772 = -1.2169530213485977e+27;
        bool r334773 = r334771 <= r334772;
        double r334774 = 318335141.94349515;
        bool r334775 = r334771 <= r334774;
        double r334776 = !r334775;
        bool r334777 = r334773 || r334776;
        double r334778 = 0.07512208616047561;
        double r334779 = r334778 / r334771;
        double r334780 = y;
        double r334781 = 0.0692910599291889;
        double r334782 = x;
        double r334783 = fma(r334780, r334781, r334782);
        double r334784 = fma(r334779, r334780, r334783);
        double r334785 = 0.4917317610505968;
        double r334786 = fma(r334771, r334781, r334785);
        double r334787 = 0.279195317918525;
        double r334788 = fma(r334786, r334771, r334787);
        double r334789 = 6.012459259764103;
        double r334790 = r334771 + r334789;
        double r334791 = 3.350343815022304;
        double r334792 = fma(r334790, r334771, r334791);
        double r334793 = r334788 / r334792;
        double r334794 = r334780 * r334793;
        double r334795 = r334794 + r334782;
        double r334796 = r334777 ? r334784 : r334795;
        return r334796;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.1
Target0.1
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 < -1.2169530213485977e+27 or 318335141.94349515 < z

    1. Initial program 42.3

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

    2. Simplified35.8

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

    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

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

    if -1.2169530213485977e+27 < z < 318335141.94349515

    1. Initial program 0.3

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

    2. Simplified0.1

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

    4. Applied fma-udef0.1

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

    6. Applied div-inv0.2

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

    7. Applied associate-*l*0.4

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.21695302134859774 \cdot 10^{27} \lor \neg \left(z \le 318335141.94349515\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))