Average Error: 0.0 → 0.1
Time: 2.6s
Precision: 64
\[e^{\left(x \cdot y\right) \cdot y}\]
↓
\[\frac{{\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}\right)}^{3} - 1}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}, e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} + 1, 1\right)}\]
e^{\left(x \cdot y\right) \cdot y}
↓
\frac{{\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}\right)}^{3} - 1}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}, e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} + 1, 1\right)}
double code(double x, double y) {
	return ((double) exp(((double) (((double) (x * y)) * y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) pow(((double) exp(((double) log1p(((double) exp(((double) (((double) (x * y)) * y)))))))), 3.0)) - 1.0)) / ((double) fma(((double) exp(((double) log1p(((double) exp(((double) (((double) (x * y)) * y)))))))), ((double) (((double) exp(((double) log1p(((double) exp(((double) (((double) (x * y)) * y)))))))) + 1.0)), 1.0))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[e^{\left(x \cdot y\right) \cdot y}\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)\right)}\]
  4. Using strategy rm

  5. Applied expm1-udef0.1

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} - 1}\]
  6. Using strategy rm

  7. Applied flip3--0.1

    \[\leadsto \color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} \cdot e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} \cdot 1\right)}}\]

  8. Simplified0.1

    \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}\right)}^{3} - 1}}{e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} \cdot e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} \cdot 1\right)}\]

  9. Simplified0.1

    \[\leadsto \frac{{\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}, e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} + 1, 1\right)}}\]

  10. Final simplification0.1

    \[\leadsto \frac{{\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}\right)}^{3} - 1}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)}, e^{\mathsf{log1p}\left(e^{\left(x \cdot y\right) \cdot y}\right)} + 1, 1\right)}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))