Average Error: 0.0 → 0.1
Time: 6.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{{\left(\frac{1}{\sqrt{e^{im}}}\right)}^{3} \cdot {\left(\frac{0.5}{\sqrt{e^{im}}}\right)}^{3}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{{\left(\frac{1}{\sqrt{e^{im}}}\right)}^{3} \cdot {\left(\frac{0.5}{\sqrt{e^{im}}}\right)}^{3}}\right) \cdot \cos re
double code(double re, double im) {
	return ((0.5 * cos(re)) * (exp(-im) + exp(im)));
}
↓
double code(double re, double im) {
	return (fma(0.5, exp(im), cbrt((pow((1.0 / sqrt(exp(im))), 3.0) * pow((0.5 / sqrt(exp(im))), 3.0)))) * cos(re));
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{\color{blue}{\sqrt[3]{\left(e^{im} \cdot e^{im}\right) \cdot e^{im}}}}\right) \cdot \cos re\]

  5. Applied add-cbrt-cube0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{\sqrt[3]{\left(0.5 \cdot 0.5\right) \cdot 0.5}}}{\sqrt[3]{\left(e^{im} \cdot e^{im}\right) \cdot e^{im}}}\right) \cdot \cos re\]

  6. Applied cbrt-undiv0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\sqrt[3]{\frac{\left(0.5 \cdot 0.5\right) \cdot 0.5}{\left(e^{im} \cdot e^{im}\right) \cdot e^{im}}}}\right) \cdot \cos re\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{\color{blue}{{\left(\frac{0.5}{e^{im}}\right)}^{3}}}\right) \cdot \cos re\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{{\left(\frac{0.5}{\color{blue}{\sqrt{e^{im}} \cdot \sqrt{e^{im}}}}\right)}^{3}}\right) \cdot \cos re\]

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

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{{\left(\frac{\color{blue}{1 \cdot 0.5}}{\sqrt{e^{im}} \cdot \sqrt{e^{im}}}\right)}^{3}}\right) \cdot \cos re\]

  11. Applied times-frac0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{{\color{blue}{\left(\frac{1}{\sqrt{e^{im}}} \cdot \frac{0.5}{\sqrt{e^{im}}}\right)}}^{3}}\right) \cdot \cos re\]

  12. Applied unpow-prod-down0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{\color{blue}{{\left(\frac{1}{\sqrt{e^{im}}}\right)}^{3} \cdot {\left(\frac{0.5}{\sqrt{e^{im}}}\right)}^{3}}}\right) \cdot \cos re\]

  13. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \sqrt[3]{{\left(\frac{1}{\sqrt{e^{im}}}\right)}^{3} \cdot {\left(\frac{0.5}{\sqrt{e^{im}}}\right)}^{3}}\right) \cdot \cos re\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))