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

Average Error: 19.8 → 0.1

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1900238327.22550368 \lor \neg \left(z \le 520855.793016935873\right):\\ \;\;\;\;x + \left(0.0692910599291888946 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.404622038699921249}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{z \cdot \left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) + 0.279195317918524977}{\sqrt{z \cdot \left(z + 6.0124592597641033\right) + 3.35034381502230394}}}{\sqrt{z \cdot \left(z + 6.0124592597641033\right) + 3.35034381502230394}}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) (x + ((double) (((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 <= -1900238327.2255037) || !(z <= 520855.7930169359))) {
		VAR = ((double) (x + ((double) (((double) (0.0692910599291889 * y)) + ((double) (((double) (y / z)) * ((double) (0.07512208616047561 - ((double) (0.40462203869992125 / z))))))))));
	} else {
		VAR = ((double) (x + ((double) (y * ((double) (((double) (((double) (((double) (z * ((double) (((double) (z * 0.0692910599291889)) + 0.4917317610505968)))) + 0.279195317918525)) / ((double) sqrt(((double) (((double) (z * ((double) (z + 6.012459259764103)))) + 3.350343815022304)))))) / ((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	19.8
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if z < -1900238327.22550368 or 520855.793016935873 < z

   1. Initial program 40.3

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

   2. Simplified 32.6

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

   3. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047561 \cdot \frac{y}{z} + 0.0692910599291888946 \cdot y\right) - 0.404622038699921249 \cdot \frac{y}{{z}^{2}}\right)}\]

   4. Simplified 0.0

      \[\leadsto x + \color{blue}{\left(0.0692910599291888946 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.404622038699921249}{z}\right)\right)}\]

   if -1900238327.22550368 < z < 520855.793016935873

   1. Initial program 0.2

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x + y \cdot \frac{z \cdot \left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) + 0.279195317918524977}{z \cdot \left(z + 6.0124592597641033\right) + 3.35034381502230394}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.5

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

   5. Applied associate-/r* 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1900238327.22550368 \lor \neg \left(z \le 520855.793016935873\right):\\ \;\;\;\;x + \left(0.0692910599291888946 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.404622038699921249}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{z \cdot \left(z \cdot 0.0692910599291888946 + 0.49173176105059679\right) + 0.279195317918524977}{\sqrt{z \cdot \left(z + 6.0124592597641033\right) + 3.35034381502230394}}}{\sqrt{z \cdot \left(z + 6.0124592597641033\right) + 3.35034381502230394}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))