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

Average Error: 20.1 → 0.1

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1811801313.489081 \lor \neg \left(z \leq 736883.8876437305\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \left(\frac{0.07512208616047561}{z} - \frac{0.40462203869992125}{z \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\\ \end{array}\]

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 (or (<= z -1811801313.489081) (not (<= z 736883.8876437305)))
   (+
    x
    (*
     y
     (+
      0.0692910599291889
      (- (/ 0.07512208616047561 z) (/ 0.40462203869992125 (* z z))))))
   (+
    x
    (*
     (/ y (sqrt (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))
     (/
      (+
       (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
       0.279195317918525)
      (sqrt (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))))))

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((z <= -1811801313.489081) || !(z <= 736883.8876437305))) {
		VAR = ((double) (x + ((double) (y * ((double) (0.0692910599291889 + ((double) ((0.07512208616047561 / z) - (0.40462203869992125 / ((double) (z * z)))))))))));
	} else {
		VAR = ((double) (x + ((double) ((y / ((double) sqrt(((double) (((double) (z * ((double) (z + 6.012459259764103)))) + 3.350343815022304))))) * (((double) (((double) (z * ((double) (((double) (z * 0.0692910599291889)) + 0.4917317610505968)))) + 0.279195317918525)) / ((double) sqrt(((double) (((double) (z * ((double) (z + 6.012459259764103)))) + 3.350343815022304)))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.1
Target	0.1
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 2 regimes

2. if z < -1811801313.4890809 or 736883.8876437305 < z

   1. Initial program 40.7

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

   2. Simplified 32.9

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

   3. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{1}{z} + 0.0692910599291889\right) - 0.40462203869992125 \cdot \frac{1}{{z}^{2}}\right)}\]

   4. Simplified 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + \left(\frac{0.07512208616047561}{z} - \frac{0.40462203869992125}{z \cdot z}\right)\right)}\]

   if -1811801313.4890809 < z < 736883.8876437305

   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. Simplified 0.1

      \[\leadsto \color{blue}{x + y \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.5

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

   5. Applied *-un-lft-identity 0.5

      \[\leadsto x + y \cdot \frac{\color{blue}{1 \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \cdot \sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\]

   6. Applied times-frac 0.2

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\right)}\]

   7. Applied associate-*r* 0.2

      \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\right) \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}}\]

   8. Simplified 0.2

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1811801313.489081 \lor \neg \left(z \leq 736883.8876437305\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \left(\frac{0.07512208616047561}{z} - \frac{0.40462203869992125}{z \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}} \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\sqrt{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))))