Average Error: 19.8 → 0.3
Time: 12.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.6510917084793137 \cdot 10^{26} \lor \neg \left(z \le 0.200318100483606681\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -2.6510917084793137 \cdot 10^{26} \lor \neg \left(z \le 0.200318100483606681\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \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 r400180 = x;
        double r400181 = y;
        double r400182 = z;
        double r400183 = 0.0692910599291889;
        double r400184 = r400182 * r400183;
        double r400185 = 0.4917317610505968;
        double r400186 = r400184 + r400185;
        double r400187 = r400186 * r400182;
        double r400188 = 0.279195317918525;
        double r400189 = r400187 + r400188;
        double r400190 = r400181 * r400189;
        double r400191 = 6.012459259764103;
        double r400192 = r400182 + r400191;
        double r400193 = r400192 * r400182;
        double r400194 = 3.350343815022304;
        double r400195 = r400193 + r400194;
        double r400196 = r400190 / r400195;
        double r400197 = r400180 + r400196;
        return r400197;
}
↓
double f(double x, double y, double z) {
        double r400198 = z;
        double r400199 = -2.6510917084793137e+26;
        bool r400200 = r400198 <= r400199;
        double r400201 = 0.20031810048360668;
        bool r400202 = r400198 <= r400201;
        double r400203 = !r400202;
        bool r400204 = r400200 || r400203;
        double r400205 = y;
        double r400206 = r400205 / r400198;
        double r400207 = 0.07512208616047561;
        double r400208 = 0.0692910599291889;
        double r400209 = x;
        double r400210 = fma(r400205, r400208, r400209);
        double r400211 = fma(r400206, r400207, r400210);
        double r400212 = r400198 * r400208;
        double r400213 = 0.4917317610505968;
        double r400214 = r400212 + r400213;
        double r400215 = r400214 * r400198;
        double r400216 = 0.279195317918525;
        double r400217 = r400215 + r400216;
        double r400218 = r400205 * r400217;
        double r400219 = 6.012459259764103;
        double r400220 = r400198 + r400219;
        double r400221 = r400220 * r400198;
        double r400222 = 3.350343815022304;
        double r400223 = r400221 + r400222;
        double r400224 = r400218 / r400223;
        double r400225 = r400209 + r400224;
        double r400226 = r400204 ? r400211 : r400225;
        return r400226;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.8
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.6510917084793137e+26 or 0.20031810048360668 < z

    1. Initial program 41.6

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

    2. Simplified35.0

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

    if -2.6510917084793137e+26 < z < 0.20031810048360668

    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 simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.6510917084793137 \cdot 10^{26} \lor \neg \left(z \le 0.200318100483606681\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))))