Average Error: 19.5 → 0.6
Time: 22.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3562211.514711785:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) \cdot \frac{y}{z}\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3562211.514711785:\\
\;\;\;\;x + \left(0.0692910599291889 \cdot y + \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) \cdot \frac{y}{z}\right)\\

\mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\
\;\;\;\;x + \frac{z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304} \cdot y\\

\mathbf{else}:\\
\;\;\;\;x + \left(0.0692910599291889 \cdot y + \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) \cdot \frac{y}{z}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r21538365 = x;
        double r21538366 = y;
        double r21538367 = z;
        double r21538368 = 0.0692910599291889;
        double r21538369 = r21538367 * r21538368;
        double r21538370 = 0.4917317610505968;
        double r21538371 = r21538369 + r21538370;
        double r21538372 = r21538371 * r21538367;
        double r21538373 = 0.279195317918525;
        double r21538374 = r21538372 + r21538373;
        double r21538375 = r21538366 * r21538374;
        double r21538376 = 6.012459259764103;
        double r21538377 = r21538367 + r21538376;
        double r21538378 = r21538377 * r21538367;
        double r21538379 = 3.350343815022304;
        double r21538380 = r21538378 + r21538379;
        double r21538381 = r21538375 / r21538380;
        double r21538382 = r21538365 + r21538381;
        return r21538382;
}
↓
double f(double x, double y, double z) {
        double r21538383 = z;
        double r21538384 = -3562211.514711785;
        bool r21538385 = r21538383 <= r21538384;
        double r21538386 = x;
        double r21538387 = 0.0692910599291889;
        double r21538388 = y;
        double r21538389 = r21538387 * r21538388;
        double r21538390 = 0.07512208616047561;
        double r21538391 = 0.40462203869992125;
        double r21538392 = r21538391 / r21538383;
        double r21538393 = r21538390 - r21538392;
        double r21538394 = r21538388 / r21538383;
        double r21538395 = r21538393 * r21538394;
        double r21538396 = r21538389 + r21538395;
        double r21538397 = r21538386 + r21538396;
        double r21538398 = 8.728612712058619e-20;
        bool r21538399 = r21538383 <= r21538398;
        double r21538400 = r21538387 * r21538383;
        double r21538401 = 0.4917317610505968;
        double r21538402 = r21538400 + r21538401;
        double r21538403 = r21538383 * r21538402;
        double r21538404 = 0.279195317918525;
        double r21538405 = r21538403 + r21538404;
        double r21538406 = 6.012459259764103;
        double r21538407 = r21538406 + r21538383;
        double r21538408 = r21538383 * r21538407;
        double r21538409 = 3.350343815022304;
        double r21538410 = r21538408 + r21538409;
        double r21538411 = r21538405 / r21538410;
        double r21538412 = r21538411 * r21538388;
        double r21538413 = r21538386 + r21538412;
        double r21538414 = r21538399 ? r21538413 : r21538397;
        double r21538415 = r21538385 ? r21538397 : r21538414;
        return r21538415;
}

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

Original19.5
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3562211.514711785 or 8.728612712058619e-20 < z

    1. Initial program 38.0

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity38.0

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304\right)}}\]

    4. Applied times-frac30.6

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\]

    5. Simplified30.6

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

    6. Taylor expanded around inf 1.1

      \[\leadsto x + \color{blue}{\left(\left(0.0692910599291889 \cdot y + 0.07512208616047561 \cdot \frac{y}{z}\right) - 0.40462203869992125 \cdot \frac{y}{{z}^{2}}\right)}\]

    7. Simplified1.2

      \[\leadsto x + \color{blue}{\left(\frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) + 0.0692910599291889 \cdot y\right)}\]

    if -3562211.514711785 < z < 8.728612712058619e-20

    1. Initial program 0.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.1

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\]

    5. Simplified0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3562211.514711785:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 8.728612712058619 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{z \cdot \left(0.0692910599291889 \cdot z + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(6.012459259764103 + z\right) + 3.350343815022304} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right) \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ 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))))