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

Average Error: 19.4 → 0.1

Time: 6.5s

Precision: binary64

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -12402545.026031014:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{elif}\;z \leq 87664462.12876986:\\ \;\;\;\;x + \frac{y \cdot \left(0.0692910599291889 \cdot {z}^{2} + \left(z \cdot 0.4917317610505968 + 0.279195317918525\right)\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot 0.07512208616047561\right)\\ \end{array}\]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -12402545.026031014)
   (+
    x
    (+
     (* 0.0692910599291889 y)
     (* (/ y z) (- 0.07512208616047561 (/ 0.4046220386999212 z)))))
   (if (<= z 87664462.12876986)
     (+
      x
      (/
       (*
        y
        (+
         (* 0.0692910599291889 (pow z 2.0))
         (+ (* z 0.4917317610505968) 0.279195317918525)))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))
     (+ x (+ (* 0.0692910599291889 y) (* (/ y z) 0.07512208616047561))))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (z <= -12402545.026031014) {
		tmp = x + ((0.0692910599291889 * y) + ((y / z) * (0.07512208616047561 - (0.4046220386999212 / z))));
	} else if (z <= 87664462.12876986) {
		tmp = x + ((y * ((0.0692910599291889 * pow(z, 2.0)) + ((z * 0.4917317610505968) + 0.279195317918525))) / ((z * (z + 6.012459259764103)) + 3.350343815022304));
	} else {
		tmp = x + ((0.0692910599291889 * y) + ((y / z) * 0.07512208616047561));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.4
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -12402545.026031014

   1. Initial program 40.5

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291889 \cdot y\right) - 0.4046220386999212 \cdot \frac{y}{{z}^{2}}\right)}\]

   3. Simplified 0.0

      \[\leadsto x + \color{blue}{\left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)}\]

   if -12402545.026031014 < z < 87664462.1287698597

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

   2. Taylor expanded around 0 0.2

      \[\leadsto x + \frac{\color{blue}{\left(0.0692910599291889 \cdot {z}^{2} + \left(0.4917317610505968 \cdot z + 0.279195317918525\right)\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

   if 87664462.1287698597 < z

   1. Initial program 40.8

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291889 \cdot y\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -12402545.026031014:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{elif}\;z \leq 87664462.12876986:\\ \;\;\;\;x + \frac{y \cdot \left(0.0692910599291889 \cdot {z}^{2} + \left(z \cdot 0.4917317610505968 + 0.279195317918525\right)\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot 0.07512208616047561\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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 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))))