Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Average Error: 3.0 → 0.1

Time: 3.7s

Precision: 64

Math

\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

↓

\[x + \frac{\sqrt{1}}{1.12837916709551256 \cdot \frac{e^{z}}{y} - x}\]

C

double code(double x, double y, double z) {
	return ((double) (x + ((double) (y / ((double) (((double) (1.1283791670955126 * ((double) exp(z)))) - ((double) (x * y))))))));
}

↓

double code(double x, double y, double z) {
	return ((double) (x + ((double) (((double) sqrt(1.0)) / ((double) (((double) (1.1283791670955126 * ((double) (((double) exp(z)) / y)))) - x))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.0
Target	0.0
Herbie	0.1

\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

1. Initial program 3.0

   \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
2. Using strategy rm

3. Applied clear-num 3.0

   \[\leadsto x + \color{blue}{\frac{1}{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 3.0

   \[\leadsto x + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}\]

6. Applied associate-/l* 3.0

   \[\leadsto x + \color{blue}{\frac{\sqrt{1}}{\frac{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}{\sqrt{1}}}}\]

7. Simplified 0.1

   \[\leadsto x + \frac{\sqrt{1}}{\color{blue}{1.12837916709551256 \cdot \frac{e^{z}}{y} - x}}\]

8. Final simplification 0.1

   \[\leadsto x + \frac{\sqrt{1}}{1.12837916709551256 \cdot \frac{e^{z}}{y} - x}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))