Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 10.2s

Precision: binary64

Math

\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

↓

\[\left(x + \left(y \cdot \log \left(\frac{1}{y}\right) + \left(y + \log \left(\frac{1}{y}\right) \cdot 0.5\right)\right)\right) - z\]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

↓

(FPCore (x y z)
 :precision binary64
 (- (+ x (+ (* y (log (/ 1.0 y))) (+ y (* (log (/ 1.0 y)) 0.5)))) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}

↓

double code(double x, double y, double z) {
	return (x + ((y * log(1.0 / y)) + (y + (log(1.0 / y) * 0.5)))) - z;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

1. Initial program 0.1

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right) - z}\]

3. Taylor expanded around inf 0.1

   \[\leadsto \left(x - \color{blue}{\left(-\left(y \cdot \log \left(\frac{1}{y}\right) + \left(y + 0.5 \cdot \log \left(\frac{1}{y}\right)\right)\right)\right)}\right) - z\]

4. Final simplification 0.1

   \[\leadsto \left(x + \left(y \cdot \log \left(\frac{1}{y}\right) + \left(y + \log \left(\frac{1}{y}\right) \cdot 0.5\right)\right)\right) - z\]

Reproduce

herbie shell --seed 2021176 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))