Average Error: 0.0 → 0.1
Time: 5.4s
Precision: binary64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
↓
\[\begin{array}{l} t_0 := e^{-im}\\ \left(0.5 \cdot \cos re\right) \cdot \frac{{t_0}^{3} + {\left(e^{im}\right)}^{3}}{{t_0}^{2} + \mathsf{fma}\left(e^{im}, e^{im}, -e^{im - im}\right)} \end{array} \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (exp (- im))))
   (*
    (* 0.5 (cos re))
    (/
     (+ (pow t_0 3.0) (pow (exp im) 3.0))
     (+ (pow t_0 2.0) (fma (exp im) (exp im) (- (exp (- im im)))))))))
double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}
↓
double code(double re, double im) {
	double t_0 = exp(-im);
	return (0.5 * cos(re)) * ((pow(t_0, 3.0) + pow(exp(im), 3.0)) / (pow(t_0, 2.0) + fma(exp(im), exp(im), -exp((im - im)))));
}

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end
↓
function code(re, im)
	t_0 = exp(Float64(-im))
	return Float64(Float64(0.5 * cos(re)) * Float64(Float64((t_0 ^ 3.0) + (exp(im) ^ 3.0)) / Float64((t_0 ^ 2.0) + fma(exp(im), exp(im), Float64(-exp(Float64(im - im)))))))
end

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_] := Block[{t$95$0 = N[Exp[(-im)], $MachinePrecision]}, N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[N[Exp[im], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Exp[im], $MachinePrecision] * N[Exp[im], $MachinePrecision] + (-N[Exp[N[(im - im), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\begin{array}{l}
t_0 := e^{-im}\\
\left(0.5 \cdot \cos re\right) \cdot \frac{{t_0}^{3} + {\left(e^{im}\right)}^{3}}{{t_0}^{2} + \mathsf{fma}\left(e^{im}, e^{im}, -e^{im - im}\right)}
\end{array}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.1

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\frac{{\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{-im} \cdot e^{-im} + \left(e^{im} \cdot e^{im} - e^{-im} \cdot e^{im}\right)}} \]

  3. Applied egg-rr0.1

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

  4. Taylor expanded in im around inf 0.1

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \frac{{\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{\color{blue}{{\left(e^{-im}\right)}^{2}} + \mathsf{fma}\left(e^{im}, e^{im}, -e^{\left(-im\right) + im}\right)} \]

  5. Final simplification0.1

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \frac{{\left(e^{-im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{{\left(e^{-im}\right)}^{2} + \mathsf{fma}\left(e^{im}, e^{im}, -e^{im - im}\right)} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))