Average Error: 20.1 → 0.5
Time: 25.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -45463414175.9846343994140625:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{elif}\;z \le 3.147343088148751899527381405232459143234 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \end{array}\]
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 -45463414175.9846343994140625:\\
\;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\

\mathbf{elif}\;z \le 3.147343088148751899527381405232459143234 \cdot 10^{-12}:\\
\;\;\;\;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}\\

\mathbf{else}:\\
\;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r82139433 = x;
        double r82139434 = y;
        double r82139435 = z;
        double r82139436 = 0.0692910599291889;
        double r82139437 = r82139435 * r82139436;
        double r82139438 = 0.4917317610505968;
        double r82139439 = r82139437 + r82139438;
        double r82139440 = r82139439 * r82139435;
        double r82139441 = 0.279195317918525;
        double r82139442 = r82139440 + r82139441;
        double r82139443 = r82139434 * r82139442;
        double r82139444 = 6.012459259764103;
        double r82139445 = r82139435 + r82139444;
        double r82139446 = r82139445 * r82139435;
        double r82139447 = 3.350343815022304;
        double r82139448 = r82139446 + r82139447;
        double r82139449 = r82139443 / r82139448;
        double r82139450 = r82139433 + r82139449;
        return r82139450;
}
↓
double f(double x, double y, double z) {
        double r82139451 = z;
        double r82139452 = -45463414175.984634;
        bool r82139453 = r82139451 <= r82139452;
        double r82139454 = x;
        double r82139455 = 0.0692910599291889;
        double r82139456 = y;
        double r82139457 = r82139455 * r82139456;
        double r82139458 = r82139456 / r82139451;
        double r82139459 = 0.07512208616047561;
        double r82139460 = 0.40462203869992125;
        double r82139461 = r82139460 / r82139451;
        double r82139462 = r82139459 - r82139461;
        double r82139463 = r82139458 * r82139462;
        double r82139464 = r82139457 + r82139463;
        double r82139465 = r82139454 + r82139464;
        double r82139466 = 3.147343088148752e-12;
        bool r82139467 = r82139451 <= r82139466;
        double r82139468 = r82139451 * r82139455;
        double r82139469 = 0.4917317610505968;
        double r82139470 = r82139468 + r82139469;
        double r82139471 = r82139470 * r82139451;
        double r82139472 = 0.279195317918525;
        double r82139473 = r82139471 + r82139472;
        double r82139474 = r82139456 * r82139473;
        double r82139475 = 6.012459259764103;
        double r82139476 = r82139451 + r82139475;
        double r82139477 = r82139476 * r82139451;
        double r82139478 = 3.350343815022304;
        double r82139479 = r82139477 + r82139478;
        double r82139480 = r82139474 / r82139479;
        double r82139481 = r82139454 + r82139480;
        double r82139482 = r82139467 ? r82139481 : r82139465;
        double r82139483 = r82139453 ? r82139465 : r82139482;
        return r82139483;
}

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

Original20.1
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -45463414175.984634 or 3.147343088148752e-12 < z

    1. Initial program 40.0

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

    3. Applied *-un-lft-identity40.0

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

    4. Applied times-frac31.9

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

    5. Simplified31.9

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

    6. Taylor expanded around inf 0.8

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

    7. Simplified0.8

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

    if -45463414175.984634 < z < 3.147343088148752e-12

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

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

    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 associate-*r/0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -45463414175.9846343994140625:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{elif}\;z \le 3.147343088148751899527381405232459143234 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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)) (/ 1.0 (+ (* (+ 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))))