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

Average Error: 19.7 → 0.1

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -29316570.78157575 \lor \neg \left(z \leq 256.5068760213334\right):\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \sqrt[3]{{\left(\frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\right)}^{3}}\\ \end{array}\]

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 <= -29316570.78157575) || !(z <= 256.5068760213334))) {
		VAR = ((double) (x + ((double) (((double) (0.0692910599291889 * y)) + ((double) ((y / z) * ((double) (0.07512208616047561 - (0.40462203869992125 / z)))))))));
	} else {
		VAR = ((double) (x + ((double) (y * ((double) cbrt(((double) pow((((double) (((double) (z * ((double) (((double) (z * 0.0692910599291889)) + 0.4917317610505968)))) + 0.279195317918525)) / ((double) (((double) (z * ((double) (z + 6.012459259764103)))) + 3.350343815022304))), 3.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.7
Target	0.2
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 < -29316570.7815757506 or 256.506876021333426 < z

   1. Initial program Error: 40.0 bits

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

   2. Simplified Error: 32.3 bits

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

   4. Applied div-inv Error: 32.3 bits

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

   5. Applied associate-*r* Error: 40.0 bits

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

   6. Simplified Error: 40.0 bits

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

   7. Taylor expanded around inf Error: 0.0 bits

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

   8. Simplified Error: 0.0 bits

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

   if -29316570.7815757506 < z < 256.506876021333426

   1. Initial program Error: 0.2 bits

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

   2. Simplified Error: 0.1 bits

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

   4. Applied add-cbrt-cube Error: 0.1 bits

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

   5. Applied add-cbrt-cube Error: 0.2 bits

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

   6. Applied cbrt-undiv Error: 0.1 bits

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

   7. Simplified Error: 0.1 bits

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

4. Final simplification Error: 0.1 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29316570.78157575 \lor \neg \left(z \leq 256.5068760213334\right):\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.40462203869992125}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \sqrt[3]{{\left(\frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))