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

Average Error: 20.0 → 0.2

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -954854560823597498647977604008640512 \lor \neg \left(z \le 0.6231595884632936677149928073049522936344\right):\\ \;\;\;\;y \cdot \frac{1}{\left(14.43187621926893804413793986896052956581 + \frac{\frac{101.237333520038163214849191717803478241}{z}}{z}\right) - \frac{15.64635683029203505611803848296403884888}{z}} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.2791953179185249767080279070796677842736 + \left(z \cdot 0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot \left(z \cdot z\right)\right)\right) \cdot \frac{y}{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r245634 = x;
        double r245635 = y;
        double r245636 = z;
        double r245637 = 0.0692910599291889;
        double r245638 = r245636 * r245637;
        double r245639 = 0.4917317610505968;
        double r245640 = r245638 + r245639;
        double r245641 = r245640 * r245636;
        double r245642 = 0.279195317918525;
        double r245643 = r245641 + r245642;
        double r245644 = r245635 * r245643;
        double r245645 = 6.012459259764103;
        double r245646 = r245636 + r245645;
        double r245647 = r245646 * r245636;
        double r245648 = 3.350343815022304;
        double r245649 = r245647 + r245648;
        double r245650 = r245644 / r245649;
        double r245651 = r245634 + r245650;
        return r245651;
}

↓

double f(double x, double y, double z) {
        double r245652 = z;
        double r245653 = -9.548545608235975e+35;
        bool r245654 = r245652 <= r245653;
        double r245655 = 0.6231595884632937;
        bool r245656 = r245652 <= r245655;
        double r245657 = !r245656;
        bool r245658 = r245654 || r245657;
        double r245659 = y;
        double r245660 = 1.0;
        double r245661 = 14.431876219268938;
        double r245662 = 101.23733352003816;
        double r245663 = r245662 / r245652;
        double r245664 = r245663 / r245652;
        double r245665 = r245661 + r245664;
        double r245666 = 15.646356830292035;
        double r245667 = r245666 / r245652;
        double r245668 = r245665 - r245667;
        double r245669 = r245660 / r245668;
        double r245670 = r245659 * r245669;
        double r245671 = x;
        double r245672 = r245670 + r245671;
        double r245673 = 0.279195317918525;
        double r245674 = 0.4917317610505968;
        double r245675 = r245652 * r245674;
        double r245676 = 0.0692910599291889;
        double r245677 = r245652 * r245652;
        double r245678 = r245676 * r245677;
        double r245679 = r245675 + r245678;
        double r245680 = r245673 + r245679;
        double r245681 = 6.012459259764103;
        double r245682 = r245681 + r245652;
        double r245683 = r245652 * r245682;
        double r245684 = 3.350343815022304;
        double r245685 = r245683 + r245684;
        double r245686 = r245659 / r245685;
        double r245687 = r245680 * r245686;
        double r245688 = r245671 + r245687;
        double r245689 = r245658 ? r245672 : r245688;
        return r245689;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.0
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if z < -9.548545608235975e+35 or 0.6231595884632937 < z

   1. Initial program 42.6

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

   2. Simplified 34.2

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

   3. Taylor expanded around 0 34.2

      \[\leadsto \frac{\color{blue}{\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right)} + 0.2791953179185249767080279070796677842736}{z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084} \cdot y + x\]

   4. Simplified 34.2

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

   6. Applied clear-num 34.2

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

   7. Simplified 34.2

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

   8. Taylor expanded around inf 0.2

      \[\leadsto \frac{1}{\color{blue}{\left(101.237333520038163214849191717803478241 \cdot \frac{1}{{z}^{2}} + 14.43187621926893804413793986896052956581\right) - 15.64635683029203505611803848296403884888 \cdot \frac{1}{z}}} \cdot y + x\]

   9. Simplified 0.2

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{\frac{101.237333520038163214849191717803478241}{z}}{z} + 14.43187621926893804413793986896052956581\right) - \frac{15.64635683029203505611803848296403884888}{z}}} \cdot y + x\]

   if -9.548545608235975e+35 < z < 0.6231595884632937

   1. Initial program 0.4

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

   2. Simplified 0.1

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

   3. Taylor expanded around 0 0.1

      \[\leadsto \frac{\color{blue}{\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right)} + 0.2791953179185249767080279070796677842736}{z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084} \cdot y + x\]

   4. Simplified 0.1

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

   6. Applied associate-*l* 0.1

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

   8. Applied div-inv 0.4

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

   9. Applied associate-*l* 0.2

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

   10. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -954854560823597498647977604008640512 \lor \neg \left(z \le 0.6231595884632936677149928073049522936344\right):\\ \;\;\;\;y \cdot \frac{1}{\left(14.43187621926893804413793986896052956581 + \frac{\frac{101.237333520038163214849191717803478241}{z}}{z}\right) - \frac{15.64635683029203505611803848296403884888}{z}} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.2791953179185249767080279070796677842736 + \left(z \cdot 0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot \left(z \cdot z\right)\right)\right) \cdot \frac{y}{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))))