math.exp on complex, real part

Average Error: 0.01% → 0.02%

Time: 8.4s

Precision: binary64

Cost: 13056

Math

\[e^{re} \cdot \cos im \]

↓

\[\frac{\cos im}{e^{-re}} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

↓

(FPCore (re im) :precision binary64 (/ (cos im) (exp (- re))))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

↓

double code(double re, double im) {
	return cos(im) / exp(-re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(im) / exp(-re)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

↓

public static double code(double re, double im) {
	return Math.cos(im) / Math.exp(-re);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

↓

def code(re, im):
	return math.cos(im) / math.exp(-re)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

↓

function code(re, im)
	return Float64(cos(im) / exp(Float64(-re)))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

↓

function tmp = code(re, im)
	tmp = cos(im) / exp(-re);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[Cos[im], $MachinePrecision] / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.01

   \[e^{re} \cdot \cos im \]

2. Applied egg-rr 0.53

   \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \cos im\right) - 1} \]

3. Applied egg-rr 0.15

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{e^{re} \cdot \cos im}}} \]

4. Applied egg-rr 25.36

   \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{e^{-re}}{\cos im}\right)} - 1}} \]

5. Simplified 0.15

   \[\leadsto \frac{1}{\color{blue}{\frac{e^{-re}}{\cos im}}} \]
   Proof

   [Start] 25.36	\[ \frac{1}{e^{\mathsf{log1p}\left(\frac{e^{-re}}{\cos im}\right)} - 1} \]
   expm1-def [=>] 25.16	\[ \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-re}}{\cos im}\right)\right)}} \]
   expm1-log1p [=>] 0.15	\[ \frac{1}{\color{blue}{\frac{e^{-re}}{\cos im}}} \]

6. Taylor expanded in re around inf 0.02

   \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

7. Final simplification 0.02

   \[\leadsto \frac{\cos im}{e^{-re}} \]

Alternatives

Alternative 1
Error	0.64%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \]

Alternative 2
Error	0.75%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3
Error	0.98%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 4
Error	1.39%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im + re\\ \end{array} \]

Alternative 5
Error	1.76%
Cost	12996

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]

Alternative 6
Error	0.01%
Cost	12992

\[\cos im \cdot e^{re} \]

Alternative 7
Error	33.36%
Cost	6464

\[\cos im \]

Alternative 8
Error	62.64%
Cost	320

\[\left(re + 2\right) + -1 \]

Alternative 9
Error	62.63%
Cost	192

\[re + 1 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))