math.exp on complex, real part

Average Accuracy: 75.1% → 75.4%

Time: 10.5s

Precision: binary64

Cost: 13252

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 5e-25) (exp re) (* (cos im) (+ re 1.0))))

C

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

↓

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 5e-25) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re + 1.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (exp(re) <= 5d-25) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
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) {
	double tmp;
	if (Math.exp(re) <= 5e-25) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

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

↓

def code(re, im):
	tmp = 0
	if math.exp(re) <= 5e-25:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

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

↓

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 5e-25)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 5e-25)
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 5e-25], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 4.99999999999999962e-25

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if 4.99999999999999962e-25 < (exp.f64 re)

   1. Initial program 67.9%

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

   2. Taylor expanded in re around 0 68.0%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\cos im \cdot \left(re + 1\right)} \]
      Step-by-step derivation

      [Start] 68.0	\[ \cos im \cdot re + \cos im \]
      *-rgt-identity [<=] 68.0	\[ \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]
      distribute-lft-in [<=] 68.0	\[ \color{blue}{\cos im \cdot \left(re + 1\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	74.8%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;re + \cos im\\ \end{array} \]

Alternative 2
Accuracy	75.1%
Cost	12992

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

Alternative 3
Accuracy	74.5%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;re \leq -0.00092:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]

Alternative 4
Accuracy	50.3%
Cost	6464

\[\cos im \]

Alternative 5
Accuracy	28.4%
Cost	192

\[re + 1 \]

Reproduce

herbie shell --seed 2023157 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))