Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 6.2s

Precision: binary64

Math

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

↓

\[\left(y + \left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\left(y + 0.5\right) \cdot 0.3333333333333333\right) \cdot \log y\right)\right)\right) - z\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (-
  (+
   y
   (-
    x
    (+
     (* (+ y 0.5) (* 2.0 (log (cbrt y))))
     (* (* (+ y 0.5) 0.3333333333333333) (log y)))))
  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 (y + (x - (((y + 0.5) * (2.0 * log(cbrt(y)))) + (((y + 0.5) * 0.3333333333333333) * log(y))))) - 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. Using strategy rm

3. Applied add-cube-cbrt_binary64_11230 0.1

   \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right) + y\right) - z\]

4. Applied log-prod_binary64_11281 0.2

   \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z\]

5. Applied distribute-rgt-in_binary64_11150 0.2

   \[\leadsto \left(\left(x - \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(y + 0.5\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)}\right) + y\right) - z\]

6. Simplified 0.2

   \[\leadsto \left(\left(x - \left(\color{blue}{\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

7. Simplified 0.2

   \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)}\right)\right) + y\right) - z\]
8. Using strategy rm

9. Applied pow1/3_binary64_11277 0.2

   \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(y + 0.5\right) \cdot \log \color{blue}{\left({y}^{0.3333333333333333}\right)}\right)\right) + y\right) - z\]

10. Applied log-pow_binary64_11284 0.2

    \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(y + 0.5\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \log y\right)}\right)\right) + y\right) - z\]

11. Applied associate-*r*_binary64_11140 0.1

    \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\left(\left(y + 0.5\right) \cdot 0.3333333333333333\right) \cdot \log y}\right)\right) + y\right) - z\]

12. Final simplification 0.1

    \[\leadsto \left(y + \left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\left(y + 0.5\right) \cdot 0.3333333333333333\right) \cdot \log y\right)\right)\right) - z\]

Reproduce

herbie shell --seed 2020280 
(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))