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

Average Error: 20.3 → 0.5

Time: 14.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -0.6121835230050514109478854152257554233074 \lor \neg \left(z \le 1.477218818365684420300619684695273871322 \cdot 10^{-11}\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r369598 = x;
        double r369599 = y;
        double r369600 = z;
        double r369601 = 0.0692910599291889;
        double r369602 = r369600 * r369601;
        double r369603 = 0.4917317610505968;
        double r369604 = r369602 + r369603;
        double r369605 = r369604 * r369600;
        double r369606 = 0.279195317918525;
        double r369607 = r369605 + r369606;
        double r369608 = r369599 * r369607;
        double r369609 = 6.012459259764103;
        double r369610 = r369600 + r369609;
        double r369611 = r369610 * r369600;
        double r369612 = 3.350343815022304;
        double r369613 = r369611 + r369612;
        double r369614 = r369608 / r369613;
        double r369615 = r369598 + r369614;
        return r369615;
}

↓

double f(double x, double y, double z) {
        double r369616 = z;
        double r369617 = -0.6121835230050514;
        bool r369618 = r369616 <= r369617;
        double r369619 = 1.4772188183656844e-11;
        bool r369620 = r369616 <= r369619;
        double r369621 = !r369620;
        bool r369622 = r369618 || r369621;
        double r369623 = x;
        double r369624 = 0.0692910599291889;
        double r369625 = y;
        double r369626 = r369624 * r369625;
        double r369627 = r369625 / r369616;
        double r369628 = 0.07512208616047561;
        double r369629 = 0.40462203869992125;
        double r369630 = r369629 / r369616;
        double r369631 = r369628 - r369630;
        double r369632 = r369627 * r369631;
        double r369633 = r369626 + r369632;
        double r369634 = r369623 + r369633;
        double r369635 = r369616 * r369624;
        double r369636 = 0.4917317610505968;
        double r369637 = r369635 + r369636;
        double r369638 = r369637 * r369616;
        double r369639 = 0.279195317918525;
        double r369640 = r369638 + r369639;
        double r369641 = 6.012459259764103;
        double r369642 = r369616 + r369641;
        double r369643 = r369642 * r369616;
        double r369644 = 3.350343815022304;
        double r369645 = r369643 + r369644;
        double r369646 = sqrt(r369645);
        double r369647 = r369640 / r369646;
        double r369648 = r369647 / r369646;
        double r369649 = r369625 * r369648;
        double r369650 = r369623 + r369649;
        double r369651 = r369622 ? r369634 : r369650;
        return r369651;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.3
Target	0.1
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if z < -0.6121835230050514 or 1.4772188183656844e-11 < z

   1. Initial program 39.6

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

   2. Taylor expanded around inf 0.8

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

   3. Simplified 0.8

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

   if -0.6121835230050514 < z < 1.4772188183656844e-11

   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-identity 0.1

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

   4. Applied times-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 associate-/r* 0.1

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.6121835230050514109478854152257554233074 \lor \neg \left(z \le 1.477218818365684420300619684695273871322 \cdot 10^{-11}\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))