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

Average Error: 19.6 → 0.3

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -5.380316622776481 \lor \neg \left(z \leq 3.3483315204785487\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533453 + -0.0027777777775172103\right)\right)\\ \end{array}\]

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 -5.380316622776481) (not (<= z 3.3483315204785487)))
   (+
    x
    (+
     (* y 0.0692910599291889)
     (* (/ y z) (- 0.07512208616047561 (/ 0.4046220386999212 z)))))
   (+
    x
    (*
     y
     (+
      0.08333333333333323
      (* z (+ (* z 0.0007936505811533453) -0.0027777777775172103)))))))

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 <= -5.380316622776481) || !(z <= 3.3483315204785487)) {
		tmp = x + ((y * 0.0692910599291889) + ((y / z) * (0.07512208616047561 - (0.4046220386999212 / z))));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533453) + -0.0027777777775172103))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.6
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 < -5.38031662277648071 or 3.34833152047854865 < z

   1. Initial program 39.3

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.3

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

   if -5.38031662277648071 < z < 3.34833152047854865

   1. Initial program 0.1

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

   3. Applied *-un-lft-identity_binary64 0.1

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

   4. Applied times-frac_binary64 0.1

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

   7. Taylor expanded around 0 0.3

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.0007936505811533453 \cdot {z}^{2} + 0.08333333333333323\right) - 0.0027777777775172103 \cdot z\right)}\]

   8. Simplified 0.3

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533453 + -0.0027777777775172103\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.380316622776481 \lor \neg \left(z \leq 3.3483315204785487\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533453 + -0.0027777777775172103\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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))))