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

Average Error: 20.3 → 0.2

Time: 4.1s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.2730222216892933 \cdot 10^{26} \lor \neg \left(z \le 72489.2439122225187\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r189981 = x;
        double r189982 = y;
        double r189983 = z;
        double r189984 = 0.0692910599291889;
        double r189985 = r189983 * r189984;
        double r189986 = 0.4917317610505968;
        double r189987 = r189985 + r189986;
        double r189988 = r189987 * r189983;
        double r189989 = 0.279195317918525;
        double r189990 = r189988 + r189989;
        double r189991 = r189982 * r189990;
        double r189992 = 6.012459259764103;
        double r189993 = r189983 + r189992;
        double r189994 = r189993 * r189983;
        double r189995 = 3.350343815022304;
        double r189996 = r189994 + r189995;
        double r189997 = r189991 / r189996;
        double r189998 = r189981 + r189997;
        return r189998;
}

↓

double f(double x, double y, double z) {
        double r189999 = z;
        double r190000 = -1.2730222216892933e+26;
        bool r190001 = r189999 <= r190000;
        double r190002 = 72489.24391222252;
        bool r190003 = r189999 <= r190002;
        double r190004 = !r190003;
        bool r190005 = r190001 || r190004;
        double r190006 = x;
        double r190007 = 0.07512208616047561;
        double r190008 = y;
        double r190009 = r190008 / r189999;
        double r190010 = r190007 * r190009;
        double r190011 = 0.0692910599291889;
        double r190012 = r190011 * r190008;
        double r190013 = r190010 + r190012;
        double r190014 = r190006 + r190013;
        double r190015 = r189999 * r190011;
        double r190016 = 0.4917317610505968;
        double r190017 = r190015 + r190016;
        double r190018 = r190017 * r189999;
        double r190019 = 0.279195317918525;
        double r190020 = r190018 + r190019;
        double r190021 = r190008 * r190020;
        double r190022 = 6.012459259764103;
        double r190023 = r189999 + r190022;
        double r190024 = r190023 * r189999;
        double r190025 = 3.350343815022304;
        double r190026 = r190024 + r190025;
        double r190027 = r190021 / r190026;
        double r190028 = r190006 + r190027;
        double r190029 = r190005 ? r190014 : r190028;
        return r190029;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.3
Target	0.1
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737680000:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)\right) \cdot \frac{1}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291888946\right) \cdot y - \left(\frac{0.404622038699921249 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.2730222216892933e+26 or 72489.24391222252 < z

   1. Initial program 42.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)}\]

   if -1.2730222216892933e+26 < z < 72489.24391222252

   1. Initial program 0.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2730222216892933 \cdot 10^{26} \lor \neg \left(z \le 72489.2439122225187\right):\\ \;\;\;\;x + \left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) \cdot z + 0.279195317918524977\right)}{\left(z + 6.0124592597641033\right) \cdot z + 3.35034381502230394}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))