Average Error: 2.8 → 0.1
Time: 4.8s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{1}{1 \cdot \left(\frac{1.12837916709551256 \cdot \left(\sqrt[3]{e^{z}} \cdot \sqrt[3]{e^{z}}\right)}{\frac{y}{\sqrt[3]{e^{z}}}} - x\right)}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
x + \frac{1}{1 \cdot \left(\frac{1.12837916709551256 \cdot \left(\sqrt[3]{e^{z}} \cdot \sqrt[3]{e^{z}}\right)}{\frac{y}{\sqrt[3]{e^{z}}}} - x\right)}
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 / (1.0 * (((1.1283791670955126 * (cbrt(exp(z)) * cbrt(exp(z)))) / (y / cbrt(exp(z)))) - x))));
}

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

Original2.8
Target0.1
Herbie0.1
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 2.8

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

  3. Applied clear-num2.8

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

  5. Applied *-un-lft-identity2.8

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

  6. Applied *-un-lft-identity2.8

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

  7. Applied times-frac2.8

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

  8. Simplified2.8

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

  9. Simplified0.1

    \[\leadsto x + \frac{1}{1 \cdot \color{blue}{\left(1.12837916709551256 \cdot \frac{e^{z}}{y} - x\right)}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt0.1

    \[\leadsto x + \frac{1}{1 \cdot \left(1.12837916709551256 \cdot \frac{\color{blue}{\left(\sqrt[3]{e^{z}} \cdot \sqrt[3]{e^{z}}\right) \cdot \sqrt[3]{e^{z}}}}{y} - x\right)}\]

  12. Applied associate-/l*0.1

    \[\leadsto x + \frac{1}{1 \cdot \left(1.12837916709551256 \cdot \color{blue}{\frac{\sqrt[3]{e^{z}} \cdot \sqrt[3]{e^{z}}}{\frac{y}{\sqrt[3]{e^{z}}}}} - x\right)}\]

  13. Applied associate-*r/0.1

    \[\leadsto x + \frac{1}{1 \cdot \left(\color{blue}{\frac{1.12837916709551256 \cdot \left(\sqrt[3]{e^{z}} \cdot \sqrt[3]{e^{z}}\right)}{\frac{y}{\sqrt[3]{e^{z}}}}} - x\right)}\]

  14. Final simplification0.1

    \[\leadsto x + \frac{1}{1 \cdot \left(\frac{1.12837916709551256 \cdot \left(\sqrt[3]{e^{z}} \cdot \sqrt[3]{e^{z}}\right)}{\frac{y}{\sqrt[3]{e^{z}}}} - x\right)}\]

Reproduce

herbie shell --seed 2020066 +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)))))