Average Error: 19.7 → 0.3
Time: 5.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.2676172898621406 \cdot 10^{53} \lor \neg \left(z \le 4.46938865292930797 \cdot 10^{-7}\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]
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 + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{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 r345175 = x;
        double r345176 = y;
        double r345177 = z;
        double r345178 = 0.0692910599291889;
        double r345179 = r345177 * r345178;
        double r345180 = 0.4917317610505968;
        double r345181 = r345179 + r345180;
        double r345182 = r345181 * r345177;
        double r345183 = 0.279195317918525;
        double r345184 = r345182 + r345183;
        double r345185 = r345176 * r345184;
        double r345186 = 6.012459259764103;
        double r345187 = r345177 + r345186;
        double r345188 = r345187 * r345177;
        double r345189 = 3.350343815022304;
        double r345190 = r345188 + r345189;
        double r345191 = r345185 / r345190;
        double r345192 = r345175 + r345191;
        return r345192;
}
↓
double f(double x, double y, double z) {
        double r345193 = z;
        double r345194 = -6.267617289862141e+53;
        bool r345195 = r345193 <= r345194;
        double r345196 = 4.469388652929308e-07;
        bool r345197 = r345193 <= r345196;
        double r345198 = !r345197;
        bool r345199 = r345195 || r345198;
        double r345200 = x;
        double r345201 = 0.07512208616047561;
        double r345202 = y;
        double r345203 = r345202 / r345193;
        double r345204 = r345201 * r345203;
        double r345205 = 0.0692910599291889;
        double r345206 = r345205 * r345202;
        double r345207 = r345204 + r345206;
        double r345208 = 0.40462203869992125;
        double r345209 = 2.0;
        double r345210 = pow(r345193, r345209);
        double r345211 = r345202 / r345210;
        double r345212 = r345208 * r345211;
        double r345213 = r345207 - r345212;
        double r345214 = r345200 + r345213;
        double r345215 = r345193 * r345205;
        double r345216 = 0.4917317610505968;
        double r345217 = r345215 + r345216;
        double r345218 = r345217 * r345193;
        double r345219 = 0.279195317918525;
        double r345220 = r345218 + r345219;
        double r345221 = 6.012459259764103;
        double r345222 = r345193 + r345221;
        double r345223 = r345222 * r345193;
        double r345224 = 3.350343815022304;
        double r345225 = r345223 + r345224;
        double r345226 = r345220 / r345225;
        double r345227 = r345202 * r345226;
        double r345228 = r345200 + r345227;
        double r345229 = r345199 ? r345214 : r345228;
        return r345229;
}

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. Taylor expanded around inf 0.6

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

    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

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

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))))