Average Error: 20.2 → 0.1
Time: 3.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.37136155034139111 \cdot 10^{27} \lor \neg \left(z \le 1879276473453759.75\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 r346303 = x;
        double r346304 = y;
        double r346305 = z;
        double r346306 = 0.0692910599291889;
        double r346307 = r346305 * r346306;
        double r346308 = 0.4917317610505968;
        double r346309 = r346307 + r346308;
        double r346310 = r346309 * r346305;
        double r346311 = 0.279195317918525;
        double r346312 = r346310 + r346311;
        double r346313 = r346304 * r346312;
        double r346314 = 6.012459259764103;
        double r346315 = r346305 + r346314;
        double r346316 = r346315 * r346305;
        double r346317 = 3.350343815022304;
        double r346318 = r346316 + r346317;
        double r346319 = r346313 / r346318;
        double r346320 = r346303 + r346319;
        return r346320;
}
↓
double f(double x, double y, double z) {
        double r346321 = z;
        double r346322 = -3.371361550341391e+27;
        bool r346323 = r346321 <= r346322;
        double r346324 = 1879276473453759.8;
        bool r346325 = r346321 <= r346324;
        double r346326 = !r346325;
        bool r346327 = r346323 || r346326;
        double r346328 = 0.07512208616047561;
        double r346329 = r346328 / r346321;
        double r346330 = y;
        double r346331 = 0.0692910599291889;
        double r346332 = x;
        double r346333 = fma(r346330, r346331, r346332);
        double r346334 = fma(r346329, r346330, r346333);
        double r346335 = r346321 * r346331;
        double r346336 = 0.4917317610505968;
        double r346337 = r346335 + r346336;
        double r346338 = r346337 * r346321;
        double r346339 = 0.279195317918525;
        double r346340 = r346338 + r346339;
        double r346341 = r346330 * r346340;
        double r346342 = 6.012459259764103;
        double r346343 = r346321 + r346342;
        double r346344 = r346343 * r346321;
        double r346345 = 3.350343815022304;
        double r346346 = r346344 + r346345;
        double r346347 = r346341 / r346346;
        double r346348 = r346332 + r346347;
        double r346349 = r346327 ? r346334 : r346348;
        return r346349;
}

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))))