Average Error: 19.9 → 0.1
Time: 1.0m
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -583385.136662959121167659759521484375 \lor \neg \left(z \le 117711952.00199253857135772705078125\right):\\ \;\;\;\;x + \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}
↓
\begin{array}{l}
\mathbf{if}\;z \le -583385.136662959121167659759521484375 \lor \neg \left(z \le 117711952.00199253857135772705078125\right):\\
\;\;\;\;x + \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\

\end{array}
double f(double x, double y, double z) {
        double r311221 = x;
        double r311222 = y;
        double r311223 = z;
        double r311224 = 0.0692910599291889;
        double r311225 = r311223 * r311224;
        double r311226 = 0.4917317610505968;
        double r311227 = r311225 + r311226;
        double r311228 = r311227 * r311223;
        double r311229 = 0.279195317918525;
        double r311230 = r311228 + r311229;
        double r311231 = r311222 * r311230;
        double r311232 = 6.012459259764103;
        double r311233 = r311223 + r311232;
        double r311234 = r311233 * r311223;
        double r311235 = 3.350343815022304;
        double r311236 = r311234 + r311235;
        double r311237 = r311231 / r311236;
        double r311238 = r311221 + r311237;
        return r311238;
}
↓
double f(double x, double y, double z) {
        double r311239 = z;
        double r311240 = -583385.1366629591;
        bool r311241 = r311239 <= r311240;
        double r311242 = 117711952.00199254;
        bool r311243 = r311239 <= r311242;
        double r311244 = !r311243;
        bool r311245 = r311241 || r311244;
        double r311246 = x;
        double r311247 = 0.07512208616047561;
        double r311248 = y;
        double r311249 = r311248 / r311239;
        double r311250 = r311247 * r311249;
        double r311251 = 0.0692910599291889;
        double r311252 = r311251 * r311248;
        double r311253 = r311250 + r311252;
        double r311254 = 0.40462203869992125;
        double r311255 = 2.0;
        double r311256 = pow(r311239, r311255);
        double r311257 = r311248 / r311256;
        double r311258 = r311254 * r311257;
        double r311259 = r311253 - r311258;
        double r311260 = r311246 + r311259;
        double r311261 = r311239 * r311251;
        double r311262 = 0.4917317610505968;
        double r311263 = r311261 + r311262;
        double r311264 = r311263 * r311239;
        double r311265 = 0.279195317918525;
        double r311266 = r311264 + r311265;
        double r311267 = 6.012459259764103;
        double r311268 = r311239 + r311267;
        double r311269 = r311268 * r311239;
        double r311270 = 3.350343815022304;
        double r311271 = r311269 + r311270;
        double r311272 = cbrt(r311271);
        double r311273 = r311272 * r311272;
        double r311274 = r311266 / r311273;
        double r311275 = r311274 / r311272;
        double r311276 = r311248 * r311275;
        double r311277 = r311246 + r311276;
        double r311278 = r311245 ? r311260 : r311277;
        return r311278;
}

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

Original19.9
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -583385.1366629591 or 117711952.00199254 < z

    1. Initial program 41.0

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)}\]

    if -583385.1366629591 < z < 117711952.00199254

    1. Initial program 0.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
    2. Using strategy rm

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

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

    5. Simplified0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.4

      \[\leadsto x + y \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\color{blue}{\left(\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\right) \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}\]

    8. Applied associate-/r*0.2

      \[\leadsto x + y \cdot \color{blue}{\frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -583385.136662959121167659759521484375 \lor \neg \left(z \le 117711952.00199253857135772705078125\right):\\ \;\;\;\;x + \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.6524566747) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (/ 1 (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394))))