math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%
Time: 4.4s
Alternatives: 1
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(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]

\begin{array}{l}

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

Sampling outcomes in binary64 precision:

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.

Accuracy vs Speed?

Herbie found 1 alternatives:

AlternativeAccuracySpeedup
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.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(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]

\begin{array}{l}

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

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \end{array} \]
(FPCore (re im)
 :precision binary64
 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)))
double code(double re, double im) {
	return (exp(im) + exp(-im)) * (cos(re) * 0.5);
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (exp(im) + exp(-im)) * (cos(re) * 0.5d0)
end function

public static double code(double re, double im) {
	return (Math.exp(im) + Math.exp(-im)) * (Math.cos(re) * 0.5);
}

def code(re, im):
	return (math.exp(im) + math.exp(-im)) * (math.cos(re) * 0.5)

function code(re, im)
	return Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
end

function tmp = code(re, im)
	tmp = (exp(im) + exp(-im)) * (cos(re) * 0.5);
end

code[re_, im_] := N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)
\end{array}
Derivation

  1. Initial program 100.0%

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

  3. Final simplification100.0%

    \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \]
  4. Add Preprocessing

Reproduce

?
herbie shell --seed 2024341 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))