Average Error: 19.7 → 0.3
Time: 4.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.2676172898621406 \cdot 10^{53} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 -6.2676172898621406 \cdot 10^{53} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\
\;\;\;\;x + y \cdot \left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r289035 = x;
        double r289036 = y;
        double r289037 = z;
        double r289038 = 0.0692910599291889;
        double r289039 = r289037 * r289038;
        double r289040 = 0.4917317610505968;
        double r289041 = r289039 + r289040;
        double r289042 = r289041 * r289037;
        double r289043 = 0.279195317918525;
        double r289044 = r289042 + r289043;
        double r289045 = r289036 * r289044;
        double r289046 = 6.012459259764103;
        double r289047 = r289037 + r289046;
        double r289048 = r289047 * r289037;
        double r289049 = 3.350343815022304;
        double r289050 = r289048 + r289049;
        double r289051 = r289045 / r289050;
        double r289052 = r289035 + r289051;
        return r289052;
}
↓
double f(double x, double y, double z) {
        double r289053 = z;
        double r289054 = -6.267617289862141e+53;
        bool r289055 = r289053 <= r289054;
        double r289056 = 4.469388652929308e-07;
        bool r289057 = r289053 <= r289056;
        double r289058 = !r289057;
        bool r289059 = r289055 || r289058;
        double r289060 = x;
        double r289061 = y;
        double r289062 = 0.07512208616047561;
        double r289063 = 1.0;
        double r289064 = r289063 / r289053;
        double r289065 = r289062 * r289064;
        double r289066 = 0.0692910599291889;
        double r289067 = r289065 + r289066;
        double r289068 = 0.40462203869992125;
        double r289069 = 2.0;
        double r289070 = pow(r289053, r289069);
        double r289071 = r289063 / r289070;
        double r289072 = r289068 * r289071;
        double r289073 = r289067 - r289072;
        double r289074 = r289061 * r289073;
        double r289075 = r289060 + r289074;
        double r289076 = r289053 * r289066;
        double r289077 = 0.4917317610505968;
        double r289078 = r289076 + r289077;
        double r289079 = r289078 * r289053;
        double r289080 = 0.279195317918525;
        double r289081 = r289079 + r289080;
        double r289082 = 6.012459259764103;
        double r289083 = r289053 + r289082;
        double r289084 = r289083 * r289053;
        double r289085 = 3.350343815022304;
        double r289086 = r289084 + r289085;
        double r289087 = r289081 / r289086;
        double r289088 = r289061 * r289087;
        double r289089 = r289060 + r289088;
        double r289090 = r289059 ? r289075 : r289089;
        return r289090;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
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 < -6.267617289862141e+53 or 4.469388652929308e-07 < z

    1. Initial program 42.7

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

    3. Applied *-un-lft-identity42.7

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

    4. Applied times-frac34.5

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

    5. Simplified34.5

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

    6. Taylor expanded around inf 0.6

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)}\]

    if -6.267617289862141e+53 < z < 4.469388652929308e-07

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.2676172898621406 \cdot 10^{53} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291888946\right) - 0.404622038699921249 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))