Average Error: 20.1 → 0.1
Time: 4.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7821873237821501 \lor \neg \left(z \le 210024179.78601301\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\\ \end{array}\]
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 -7821873237821501 \lor \neg \left(z \le 210024179.78601301\right):\\
\;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\

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

\end{array}
double f(double x, double y, double z) {
        double r354210 = x;
        double r354211 = y;
        double r354212 = z;
        double r354213 = 0.0692910599291889;
        double r354214 = r354212 * r354213;
        double r354215 = 0.4917317610505968;
        double r354216 = r354214 + r354215;
        double r354217 = r354216 * r354212;
        double r354218 = 0.279195317918525;
        double r354219 = r354217 + r354218;
        double r354220 = r354211 * r354219;
        double r354221 = 6.012459259764103;
        double r354222 = r354212 + r354221;
        double r354223 = r354222 * r354212;
        double r354224 = 3.350343815022304;
        double r354225 = r354223 + r354224;
        double r354226 = r354220 / r354225;
        double r354227 = r354210 + r354226;
        return r354227;
}
↓
double f(double x, double y, double z) {
        double r354228 = z;
        double r354229 = -7821873237821501.0;
        bool r354230 = r354228 <= r354229;
        double r354231 = 210024179.786013;
        bool r354232 = r354228 <= r354231;
        double r354233 = !r354232;
        bool r354234 = r354230 || r354233;
        double r354235 = 0.07512208616047561;
        double r354236 = y;
        double r354237 = r354236 / r354228;
        double r354238 = 0.0692910599291889;
        double r354239 = r354238 * r354236;
        double r354240 = fma(r354235, r354237, r354239);
        double r354241 = x;
        double r354242 = r354240 + r354241;
        double r354243 = 6.012459259764103;
        double r354244 = r354228 + r354243;
        double r354245 = 3.350343815022304;
        double r354246 = fma(r354244, r354228, r354245);
        double r354247 = r354236 / r354246;
        double r354248 = 0.4917317610505968;
        double r354249 = fma(r354228, r354238, r354248);
        double r354250 = 0.279195317918525;
        double r354251 = fma(r354249, r354228, r354250);
        double r354252 = fma(r354247, r354251, r354241);
        double r354253 = r354234 ? r354242 : r354252;
        return r354253;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.1
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 < -7821873237821501.0 or 210024179.786013 < z

    1. Initial program 42.2

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

    2. Simplified34.8

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

    4. Applied add-sqr-sqrt34.8

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

    5. Applied *-un-lft-identity34.8

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

    6. Applied times-frac34.8

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

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

    if -7821873237821501.0 < z < 210024179.786013

    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. 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. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7821873237821501 \lor \neg \left(z \le 210024179.78601301\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, 0.0692910599291888946 \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))