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

Average Error: 20.1 → 0.1

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1061330829675691048335048704 \lor \neg \left(z \le 458842.024295310140587389469146728515625\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204} \cdot \sqrt[3]{z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204}\right) \cdot \left(\sqrt[3]{z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204} \cdot z\right) + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r450411 = x;
        double r450412 = y;
        double r450413 = z;
        double r450414 = 0.0692910599291889;
        double r450415 = r450413 * r450414;
        double r450416 = 0.4917317610505968;
        double r450417 = r450415 + r450416;
        double r450418 = r450417 * r450413;
        double r450419 = 0.279195317918525;
        double r450420 = r450418 + r450419;
        double r450421 = r450412 * r450420;
        double r450422 = 6.012459259764103;
        double r450423 = r450413 + r450422;
        double r450424 = r450423 * r450413;
        double r450425 = 3.350343815022304;
        double r450426 = r450424 + r450425;
        double r450427 = r450421 / r450426;
        double r450428 = r450411 + r450427;
        return r450428;
}

↓

double f(double x, double y, double z) {
        double r450429 = z;
        double r450430 = -1.061330829675691e+27;
        bool r450431 = r450429 <= r450430;
        double r450432 = 458842.02429531014;
        bool r450433 = r450429 <= r450432;
        double r450434 = !r450433;
        bool r450435 = r450431 || r450434;
        double r450436 = x;
        double r450437 = y;
        double r450438 = 0.07512208616047561;
        double r450439 = 1.0;
        double r450440 = r450439 / r450429;
        double r450441 = r450438 * r450440;
        double r450442 = 0.0692910599291889;
        double r450443 = r450441 + r450442;
        double r450444 = 0.40462203869992125;
        double r450445 = 2.0;
        double r450446 = pow(r450429, r450445);
        double r450447 = r450439 / r450446;
        double r450448 = r450444 * r450447;
        double r450449 = r450443 - r450448;
        double r450450 = r450437 * r450449;
        double r450451 = r450436 + r450450;
        double r450452 = r450429 * r450442;
        double r450453 = 0.4917317610505968;
        double r450454 = r450452 + r450453;
        double r450455 = cbrt(r450454);
        double r450456 = r450455 * r450455;
        double r450457 = r450455 * r450429;
        double r450458 = r450456 * r450457;
        double r450459 = 0.279195317918525;
        double r450460 = r450458 + r450459;
        double r450461 = 6.012459259764103;
        double r450462 = r450429 + r450461;
        double r450463 = r450462 * r450429;
        double r450464 = 3.350343815022304;
        double r450465 = r450463 + r450464;
        double r450466 = r450460 / r450465;
        double r450467 = r450437 * r450466;
        double r450468 = r450436 + r450467;
        double r450469 = r450435 ? r450451 : r450468;
        return r450469;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.1
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if z < -1.061330829675691e+27 or 458842.02429531014 < z

   1. Initial program 41.6

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

   3. Applied *-un-lft-identity 41.6

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

   4. Applied times-frac 33.0

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

   5. Simplified 33.0

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

   6. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)}\]

   if -1.061330829675691e+27 < z < 458842.02429531014

   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-identity 0.2

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

   4. Applied times-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 add-cube-cbrt 0.2

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

   8. Applied associate-*l* 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1061330829675691048335048704 \lor \neg \left(z \le 458842.024295310140587389469146728515625\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\sqrt[3]{z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204} \cdot \sqrt[3]{z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204}\right) \cdot \left(\sqrt[3]{z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204} \cdot z\right) + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))