math.exp on complex, real part

?

Percentage Accurate: 100.0% → 100.0%
Time: 9.5s
Precision: binary64
Cost: 12992

?

\[e^{re} \cdot \cos im \]
↓
\[e^{re} \cdot \cos im \]
(FPCore (re im) :precision binary64 (* (exp re) (cos im)))
↓
(FPCore (re im) :precision binary64 (* (exp re) (cos im)))
double code(double re, double im) {
	return exp(re) * cos(im);
}
↓
double code(double re, double im) {
	return exp(re) * cos(im);
}

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 = exp(re) * cos(im)
end function

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.exp(re) * Math.cos(im);
}

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

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

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

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

e^{re} \cdot \cos im
↓
e^{re} \cdot \cos im

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 12 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 100.0%

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

  2. Final simplification100.0%

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

Alternatives

Alternative 1
Accuracy100.0%
Cost12992
\[e^{re} \cdot \cos im \]
Alternative 2
Accuracy92.8%
Cost19528
\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000000002:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
Alternative 3
Accuracy97.1%
Cost7880
\[\begin{array}{l} t_0 := re \cdot 0.16666666666666666 + 0.5\\ \mathbf{if}\;re \leq -0.025:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(\left(1 + re \cdot \left(re \cdot t_0\right)\right) + -1\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + t_0 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
Alternative 4
Accuracy97.1%
Cost7757
\[\begin{array}{l} \mathbf{if}\;re \leq -0.028 \lor \neg \left(re \leq 2.4 \cdot 10^{-11}\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot 0.16666666666666666 + 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
Alternative 5
Accuracy96.1%
Cost7368
\[\begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -2.05:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + t_0\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot t_0\\ \end{array} \]
Alternative 6
Accuracy96.1%
Cost7244
\[\begin{array}{l} \mathbf{if}\;re \leq -0.00019:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 7
Accuracy93.3%
Cost6984
\[\begin{array}{l} \mathbf{if}\;re \leq -0.00155:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
Alternative 8
Accuracy62.6%
Cost6596
\[\begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \end{array} \]
Alternative 9
Accuracy41.1%
Cost832
\[1 + \left(re + re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\right) \]
Alternative 10
Accuracy41.0%
Cost704
\[1 + \left(re + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \]
Alternative 11
Accuracy38.2%
Cost576
\[1 + \left(re + re \cdot \left(re \cdot 0.5\right)\right) \]
Alternative 12
Accuracy29.4%
Cost192
\[re + 1 \]

Reproduce?

herbie shell --seed 2023272 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))