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

Average Error: 19.9 → 0.1

Time: 24.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -40697775657662409605120:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \mathbf{elif}\;z \le 212668326.6286086738109588623046875:\\ \;\;\;\;\left(y \cdot \frac{1}{\sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} \cdot \sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} + x\\ \mathbf{else}:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r20595435 = x;
        double r20595436 = y;
        double r20595437 = z;
        double r20595438 = 0.0692910599291889;
        double r20595439 = r20595437 * r20595438;
        double r20595440 = 0.4917317610505968;
        double r20595441 = r20595439 + r20595440;
        double r20595442 = r20595441 * r20595437;
        double r20595443 = 0.279195317918525;
        double r20595444 = r20595442 + r20595443;
        double r20595445 = r20595436 * r20595444;
        double r20595446 = 6.012459259764103;
        double r20595447 = r20595437 + r20595446;
        double r20595448 = r20595447 * r20595437;
        double r20595449 = 3.350343815022304;
        double r20595450 = r20595448 + r20595449;
        double r20595451 = r20595445 / r20595450;
        double r20595452 = r20595435 + r20595451;
        return r20595452;
}

↓

double f(double x, double y, double z) {
        double r20595453 = z;
        double r20595454 = -4.069777565766241e+22;
        bool r20595455 = r20595453 <= r20595454;
        double r20595456 = 0.0692910599291889;
        double r20595457 = y;
        double r20595458 = r20595456 * r20595457;
        double r20595459 = r20595457 / r20595453;
        double r20595460 = 0.07512208616047561;
        double r20595461 = r20595459 * r20595460;
        double r20595462 = r20595458 + r20595461;
        double r20595463 = x;
        double r20595464 = r20595462 + r20595463;
        double r20595465 = 212668326.62860867;
        bool r20595466 = r20595453 <= r20595465;
        double r20595467 = 1.0;
        double r20595468 = 6.012459259764103;
        double r20595469 = r20595468 + r20595453;
        double r20595470 = r20595453 * r20595469;
        double r20595471 = 3.350343815022304;
        double r20595472 = r20595470 + r20595471;
        double r20595473 = cbrt(r20595472);
        double r20595474 = sqrt(r20595473);
        double r20595475 = r20595473 * r20595473;
        double r20595476 = sqrt(r20595475);
        double r20595477 = r20595474 * r20595476;
        double r20595478 = r20595467 / r20595477;
        double r20595479 = r20595457 * r20595478;
        double r20595480 = r20595453 * r20595456;
        double r20595481 = 0.4917317610505968;
        double r20595482 = r20595480 + r20595481;
        double r20595483 = r20595482 * r20595453;
        double r20595484 = 0.279195317918525;
        double r20595485 = r20595483 + r20595484;
        double r20595486 = sqrt(r20595472);
        double r20595487 = r20595485 / r20595486;
        double r20595488 = r20595479 * r20595487;
        double r20595489 = r20595488 + r20595463;
        double r20595490 = r20595466 ? r20595489 : r20595464;
        double r20595491 = r20595455 ? r20595464 : r20595490;
        return r20595491;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.9
Target	0.2
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 < -4.069777565766241e+22 or 212668326.62860867 < z

   1. Initial program 41.8

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

   2. Taylor expanded around inf 0.0

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

   if -4.069777565766241e+22 < z < 212668326.62860867

   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-sqr-sqrt 0.4

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

   8. Applied *-un-lft-identity 0.4

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

   9. Applied times-frac 0.2

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

   10. Applied associate-*r* 0.2

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

   12. Applied add-cube-cbrt 0.2

       \[\leadsto x + \left(y \cdot \frac{1}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\right) \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

   13. Applied sqrt-prod 0.2

       \[\leadsto x + \left(y \cdot \frac{1}{\color{blue}{\sqrt{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}} \cdot \sqrt{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -40697775657662409605120:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \mathbf{elif}\;z \le 212668326.6286086738109588623046875:\\ \;\;\;\;\left(y \cdot \frac{1}{\sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} \cdot \sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} + x\\ \mathbf{else}:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))