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

Average Error: 20.3 → 0.3

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -533213.3045002559 \lor \neg \left(z \leq 8.466378523572911 \cdot 10^{-08}\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + y \cdot 0.0692910599291889\right) - 0.4046220386999212 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)} \cdot \left(0.279195317918525 + z \cdot \left(0.4917317610505968 + z \cdot 0.0692910599291889\right)\right)\\ \end{array}\]

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 (or (<= z -533213.3045002559) (not (<= z 8.466378523572911e-08)))
   (+
    x
    (-
     (+ (* 0.07512208616047561 (/ y z)) (* y 0.0692910599291889))
     (* 0.4046220386999212 (/ y (pow z 2.0)))))
   (+
    x
    (*
     (/ y (+ 3.350343815022304 (* z (+ z 6.012459259764103))))
     (+
      0.279195317918525
      (* z (+ 0.4917317610505968 (* z 0.0692910599291889))))))))

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 <= -533213.3045002559) || !(z <= 8.466378523572911e-08)) {
		tmp = x + (((0.07512208616047561 * (y / z)) + (y * 0.0692910599291889)) - (0.4046220386999212 * (y / pow(z, 2.0))));
	} else {
		tmp = x + ((y / (3.350343815022304 + (z * (z + 6.012459259764103)))) * (0.279195317918525 + (z * (0.4917317610505968 + (z * 0.0692910599291889)))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.3
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if z < -533213.30450025585 or 8.46637852357291077e-8 < z

   1. Initial program 40.4

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

   2. Taylor expanded around inf 0.4

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

   if -533213.30450025585 < z < 8.46637852357291077e-8

   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

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

   3. Simplified 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -533213.3045002559 \lor \neg \left(z \leq 8.466378523572911 \cdot 10^{-08}\right):\\ \;\;\;\;x + \left(\left(0.07512208616047561 \cdot \frac{y}{z} + y \cdot 0.0692910599291889\right) - 0.4046220386999212 \cdot \frac{y}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)} \cdot \left(0.279195317918525 + z \cdot \left(0.4917317610505968 + z \cdot 0.0692910599291889\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))))