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

Average Error: 20.2 → 0.2

Time: 8.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.001875260796471 \cdot 10^{+24} \lor \neg \left(z \leq 495134.48440773255\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{z \cdot \left(z \cdot \left(z \cdot 0.004801250986110447\right) + -0.24180012482592123\right)}{z \cdot 0.0692910599291889 - 0.4917317610505968} + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \end{array}\]

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 -2.001875260796471e+24) (not (<= z 495134.48440773255)))
   (+
    x
    (+
     (* y 0.0692910599291889)
     (* (/ y z) (- 0.07512208616047561 (/ 0.4046220386999212 z)))))
   (+
    x
    (/
     (*
      y
      (+
       (/
        (* z (+ (* z (* z 0.004801250986110447)) -0.24180012482592123))
        (- (* z 0.0692910599291889) 0.4917317610505968))
       0.279195317918525))
     (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))))

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 <= -2.001875260796471e+24) || !(z <= 495134.48440773255)) {
		tmp = x + ((y * 0.0692910599291889) + ((y / z) * (0.07512208616047561 - (0.4046220386999212 / z))));
	} else {
		tmp = x + ((y * (((z * ((z * (z * 0.004801250986110447)) + -0.24180012482592123)) / ((z * 0.0692910599291889) - 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.2
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if z < -2.0018752607964711e24 or 495134.484407732554 < z

   1. Initial program 42.3

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

   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

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

   if -2.0018752607964711e24 < z < 495134.484407732554

   1. Initial program 0.3

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

   3. Applied flip-+_binary64_10964 0.3

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

   4. Applied associate-*l/_binary64_10933 0.3

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

   5. Simplified 0.3

      \[\leadsto x + \frac{y \cdot \left(\frac{\color{blue}{z \cdot \left(z \cdot \left(0.004801250986110447 \cdot z\right) + -0.24180012482592123\right)}}{z \cdot 0.0692910599291889 - 0.4917317610505968} + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.001875260796471 \cdot 10^{+24} \lor \neg \left(z \leq 495134.48440773255\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{z \cdot \left(z \cdot \left(z \cdot 0.004801250986110447\right) + -0.24180012482592123\right)}{z \cdot 0.0692910599291889 - 0.4917317610505968} + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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))))