Average Error: 20.0 → 0.1
Time: 8.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.36903654096377077 \cdot 10^{69} \lor \neg \left(z \le 2180725.6674509291\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \mathsf{fma}\left(0.0692910599291888946, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\ \end{array}\]
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 -3.36903654096377077 \cdot 10^{69} \lor \neg \left(z \le 2180725.6674509291\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \mathsf{fma}\left(0.0692910599291888946, y, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r356483 = x;
        double r356484 = y;
        double r356485 = z;
        double r356486 = 0.0692910599291889;
        double r356487 = r356485 * r356486;
        double r356488 = 0.4917317610505968;
        double r356489 = r356487 + r356488;
        double r356490 = r356489 * r356485;
        double r356491 = 0.279195317918525;
        double r356492 = r356490 + r356491;
        double r356493 = r356484 * r356492;
        double r356494 = 6.012459259764103;
        double r356495 = r356485 + r356494;
        double r356496 = r356495 * r356485;
        double r356497 = 3.350343815022304;
        double r356498 = r356496 + r356497;
        double r356499 = r356493 / r356498;
        double r356500 = r356483 + r356499;
        return r356500;
}
↓
double f(double x, double y, double z) {
        double r356501 = z;
        double r356502 = -3.3690365409637708e+69;
        bool r356503 = r356501 <= r356502;
        double r356504 = 2180725.667450929;
        bool r356505 = r356501 <= r356504;
        double r356506 = !r356505;
        bool r356507 = r356503 || r356506;
        double r356508 = y;
        double r356509 = r356508 / r356501;
        double r356510 = 0.07512208616047561;
        double r356511 = 0.0692910599291889;
        double r356512 = x;
        double r356513 = fma(r356511, r356508, r356512);
        double r356514 = fma(r356509, r356510, r356513);
        double r356515 = 0.4917317610505968;
        double r356516 = fma(r356501, r356511, r356515);
        double r356517 = 0.279195317918525;
        double r356518 = fma(r356516, r356501, r356517);
        double r356519 = 6.012459259764103;
        double r356520 = r356501 + r356519;
        double r356521 = 3.350343815022304;
        double r356522 = fma(r356520, r356501, r356521);
        double r356523 = r356518 / r356522;
        double r356524 = r356508 * r356523;
        double r356525 = r356512 + r356524;
        double r356526 = r356507 ? r356514 : r356525;
        return r356526;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
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 < -3.3690365409637708e+69 or 2180725.667450929 < z

    1. Initial program 45.1

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

    2. Simplified38.3

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

    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

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

    if -3.3690365409637708e+69 < z < 2180725.667450929

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.36903654096377077 \cdot 10^{69} \lor \neg \left(z \le 2180725.6674509291\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \mathsf{fma}\left(0.0692910599291888946, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))