Average Error: 19.7 → 0.4
Time: 4.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2713952720505764 \cdot 10^{154} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}} + x\\ \end{array}\]
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 -1.2713952720505764 \cdot 10^{154} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}} + x\\

\end{array}
double f(double x, double y, double z) {
        double r378022 = x;
        double r378023 = y;
        double r378024 = z;
        double r378025 = 0.0692910599291889;
        double r378026 = r378024 * r378025;
        double r378027 = 0.4917317610505968;
        double r378028 = r378026 + r378027;
        double r378029 = r378028 * r378024;
        double r378030 = 0.279195317918525;
        double r378031 = r378029 + r378030;
        double r378032 = r378023 * r378031;
        double r378033 = 6.012459259764103;
        double r378034 = r378024 + r378033;
        double r378035 = r378034 * r378024;
        double r378036 = 3.350343815022304;
        double r378037 = r378035 + r378036;
        double r378038 = r378032 / r378037;
        double r378039 = r378022 + r378038;
        return r378039;
}
↓
double f(double x, double y, double z) {
        double r378040 = z;
        double r378041 = -1.2713952720505764e+154;
        bool r378042 = r378040 <= r378041;
        double r378043 = 4.469388652929308e-07;
        bool r378044 = r378040 <= r378043;
        double r378045 = !r378044;
        bool r378046 = r378042 || r378045;
        double r378047 = 0.07512208616047561;
        double r378048 = r378047 / r378040;
        double r378049 = y;
        double r378050 = 0.0692910599291889;
        double r378051 = x;
        double r378052 = fma(r378049, r378050, r378051);
        double r378053 = fma(r378048, r378049, r378052);
        double r378054 = 0.4917317610505968;
        double r378055 = fma(r378040, r378050, r378054);
        double r378056 = 0.279195317918525;
        double r378057 = fma(r378055, r378040, r378056);
        double r378058 = 6.012459259764103;
        double r378059 = r378040 + r378058;
        double r378060 = 3.350343815022304;
        double r378061 = fma(r378059, r378040, r378060);
        double r378062 = sqrt(r378061);
        double r378063 = r378057 / r378062;
        double r378064 = r378063 / r378062;
        double r378065 = r378049 * r378064;
        double r378066 = r378065 + r378051;
        double r378067 = r378046 ? r378053 : r378066;
        return r378067;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.7
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2713952720505764e+154 or 4.469388652929308e-07 < z

    1. Initial program 46.9

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

    2. Simplified42.6

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

    3. Taylor expanded around inf 0.7

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

    4. Simplified0.7

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

    if -1.2713952720505764e+154 < z < 4.469388652929308e-07

    1. Initial program 3.4

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

    2. Simplified0.7

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

    4. Applied fma-udef0.7

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

    6. Applied div-inv0.7

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

    7. Applied associate-*l*0.4

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

    8. Simplified0.1

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

    10. Applied add-sqr-sqrt0.4

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

    11. Applied associate-/r*0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2713952720505764 \cdot 10^{154} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z}, y, \mathsf{fma}\left(y, 0.0692910599291888946, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291888946, 0.49173176105059679\right), z, 0.279195317918524977\right)}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}}}{\sqrt{\mathsf{fma}\left(z + 6.0124592597641033, z, 3.35034381502230394\right)}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))