Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B

Average Error: 19.8 → 0.4

Time: 45.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.569003806210261422708451080373480969642 \cdot 10^{53}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right) \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 540034.0694215781986713409423828125:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(6.012459259764103336465268512256443500519 + z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right) \cdot \frac{y}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r18653501 = x;
        double r18653502 = y;
        double r18653503 = z;
        double r18653504 = 0.0692910599291889;
        double r18653505 = r18653503 * r18653504;
        double r18653506 = 0.4917317610505968;
        double r18653507 = r18653505 + r18653506;
        double r18653508 = r18653507 * r18653503;
        double r18653509 = 0.279195317918525;
        double r18653510 = r18653508 + r18653509;
        double r18653511 = r18653502 * r18653510;
        double r18653512 = 6.012459259764103;
        double r18653513 = r18653503 + r18653512;
        double r18653514 = r18653513 * r18653503;
        double r18653515 = 3.350343815022304;
        double r18653516 = r18653514 + r18653515;
        double r18653517 = r18653511 / r18653516;
        double r18653518 = r18653501 + r18653517;
        return r18653518;
}

↓

double f(double x, double y, double z) {
        double r18653519 = z;
        double r18653520 = -8.569003806210261e+53;
        bool r18653521 = r18653519 <= r18653520;
        double r18653522 = x;
        double r18653523 = 0.0692910599291889;
        double r18653524 = y;
        double r18653525 = r18653523 * r18653524;
        double r18653526 = 0.07512208616047561;
        double r18653527 = 0.40462203869992125;
        double r18653528 = r18653527 / r18653519;
        double r18653529 = r18653526 - r18653528;
        double r18653530 = r18653524 / r18653519;
        double r18653531 = r18653529 * r18653530;
        double r18653532 = r18653525 + r18653531;
        double r18653533 = r18653522 + r18653532;
        double r18653534 = 540034.0694215782;
        bool r18653535 = r18653519 <= r18653534;
        double r18653536 = r18653523 * r18653519;
        double r18653537 = 0.4917317610505968;
        double r18653538 = r18653536 + r18653537;
        double r18653539 = r18653519 * r18653538;
        double r18653540 = 0.279195317918525;
        double r18653541 = r18653539 + r18653540;
        double r18653542 = r18653524 * r18653541;
        double r18653543 = 3.350343815022304;
        double r18653544 = 6.012459259764103;
        double r18653545 = r18653544 + r18653519;
        double r18653546 = r18653519 * r18653545;
        double r18653547 = r18653543 + r18653546;
        double r18653548 = r18653542 / r18653547;
        double r18653549 = r18653522 + r18653548;
        double r18653550 = r18653535 ? r18653549 : r18653533;
        double r18653551 = r18653521 ? r18653533 : r18653550;
        return r18653551;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.8
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if z < -8.569003806210261e+53 or 540034.0694215782 < z

   1. Initial program 43.8

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

   3. Applied *-un-lft-identity 43.8

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

   4. Applied times-frac 35.3

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

   5. Simplified 35.3

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

   7. Applied add-sqr-sqrt 35.3

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

   8. Taylor expanded around inf 0.0

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

   9. Simplified 0.0

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

   if -8.569003806210261e+53 < z < 540034.0694215782

   1. Initial program 0.6

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

   3. Applied *-un-lft-identity 0.6

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   7. Applied associate-*r/ 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.569003806210261422708451080373480969642 \cdot 10^{53}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right) \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 540034.0694215781986713409423828125:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(6.012459259764103336465268512256443500519 + z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right) \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))