Average Error: 20.2 → 0.1
Time: 4.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.37136155034139111 \cdot 10^{27} \lor \neg \left(z \le 1879276473453759.75\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -3.37136155034139111 \cdot 10^{27} \lor \neg \left(z \le 1879276473453759.75\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\

\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 r300152 = x;
        double r300153 = y;
        double r300154 = z;
        double r300155 = 0.0692910599291889;
        double r300156 = r300154 * r300155;
        double r300157 = 0.4917317610505968;
        double r300158 = r300156 + r300157;
        double r300159 = r300158 * r300154;
        double r300160 = 0.279195317918525;
        double r300161 = r300159 + r300160;
        double r300162 = r300153 * r300161;
        double r300163 = 6.012459259764103;
        double r300164 = r300154 + r300163;
        double r300165 = r300164 * r300154;
        double r300166 = 3.350343815022304;
        double r300167 = r300165 + r300166;
        double r300168 = r300162 / r300167;
        double r300169 = r300152 + r300168;
        return r300169;
}
↓
double f(double x, double y, double z) {
        double r300170 = z;
        double r300171 = -3.371361550341391e+27;
        bool r300172 = r300170 <= r300171;
        double r300173 = 1879276473453759.8;
        bool r300174 = r300170 <= r300173;
        double r300175 = !r300174;
        bool r300176 = r300172 || r300175;
        double r300177 = 0.07512208616047561;
        double r300178 = r300177 / r300170;
        double r300179 = y;
        double r300180 = 0.0692910599291889;
        double r300181 = x;
        double r300182 = fma(r300179, r300180, r300181);
        double r300183 = fma(r300178, r300179, r300182);
        double r300184 = r300170 * r300180;
        double r300185 = 0.4917317610505968;
        double r300186 = r300184 + r300185;
        double r300187 = r300186 * r300170;
        double r300188 = 0.279195317918525;
        double r300189 = r300187 + r300188;
        double r300190 = r300179 * r300189;
        double r300191 = 6.012459259764103;
        double r300192 = r300170 + r300191;
        double r300193 = r300192 * r300170;
        double r300194 = 3.350343815022304;
        double r300195 = r300193 + r300194;
        double r300196 = r300190 / r300195;
        double r300197 = r300181 + r300196;
        double r300198 = r300176 ? r300183 : r300197;
        return r300198;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.2
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 < -3.371361550341391e+27 or 1879276473453759.8 < z

    1. Initial program 43.0

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

    2. Simplified36.1

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0

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

    if -3.371361550341391e+27 < z < 1879276473453759.8

    1. Initial program 0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.37136155034139111 \cdot 10^{27} \lor \neg \left(z \le 1879276473453759.75\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))))