Average Error: 3.0 → 0.1
Time: 3.3s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{1}{\left(\sqrt{e^{z}} \cdot 1.12837916709551256\right) \cdot \frac{\sqrt{e^{z}}}{y} - \frac{x}{1}}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
x + \frac{1}{\left(\sqrt{e^{z}} \cdot 1.12837916709551256\right) \cdot \frac{\sqrt{e^{z}}}{y} - \frac{x}{1}}
double code(double x, double y, double z) {
	return (x + (y / ((1.1283791670955126 * exp(z)) - (x * y))));
}
↓
double code(double x, double y, double z) {
	return (x + (1.0 / (((sqrt(exp(z)) * 1.1283791670955126) * (sqrt(exp(z)) / y)) - (x / 1.0))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.0
Target0.0
Herbie0.1
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 3.0

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

  3. Applied clear-num3.0

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

  5. Applied div-sub3.0

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

  6. Simplified3.0

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

  7. Simplified0.1

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

  9. Applied *-un-lft-identity0.1

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

  10. Applied add-sqr-sqrt0.1

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

  11. Applied times-frac0.1

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

  12. Applied associate-*r*0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020049 +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)))))