powComplex, real part

Average Error: 33.9 → 4.7

Time: 19.9s

Precision: binary64

Cost: 40004

Math

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

↓

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_0} \cdot \frac{1}{\frac{1}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{elif}\;y.re \leq 0.00013:\\ \;\;\;\;e^{-t_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-y.re}{0 - \frac{1}{t_1}} - t_0} \cdot 1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp
   (-
    (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
    (* (atan2 x.im x.re) y.im)))
  (cos
   (+
    (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im)
    (* (atan2 x.im x.re) y.re)))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (if (<= y.re -5.8e-6)
     (*
      (exp (- (* t_1 y.re) t_0))
      (/ 1.0 (/ 1.0 (cos (* y.re (atan2 x.im x.re))))))
     (if (<= y.re 0.00013)
       (* (exp (- t_0)) 1.0)
       (* (exp (- (/ (- y.re) (- 0.0 (/ 1.0 t_1))) t_0)) 1.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double tmp;
	if (y_46_re <= -5.8e-6) {
		tmp = exp(((t_1 * y_46_re) - t_0)) * (1.0 / (1.0 / cos((y_46_re * atan2(x_46_im, x_46_re)))));
	} else if (y_46_re <= 0.00013) {
		tmp = exp(-t_0) * 1.0;
	} else {
		tmp = exp(((-y_46_re / (0.0 - (1.0 / t_1))) - t_0)) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * cos(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

↓

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    if (y_46re <= (-5.8d-6)) then
        tmp = exp(((t_1 * y_46re) - t_0)) * (1.0d0 / (1.0d0 / cos((y_46re * atan2(x_46im, x_46re)))))
    else if (y_46re <= 0.00013d0) then
        tmp = exp(-t_0) * 1.0d0
    else
        tmp = exp(((-y_46re / (0.0d0 - (1.0d0 / t_1))) - t_0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double tmp;
	if (y_46_re <= -5.8e-6) {
		tmp = Math.exp(((t_1 * y_46_re) - t_0)) * (1.0 / (1.0 / Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)))));
	} else if (y_46_re <= 0.00013) {
		tmp = Math.exp(-t_0) * 1.0;
	} else {
		tmp = Math.exp(((-y_46_re / (0.0 - (1.0 / t_1))) - t_0)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	tmp = 0
	if y_46_re <= -5.8e-6:
		tmp = math.exp(((t_1 * y_46_re) - t_0)) * (1.0 / (1.0 / math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))))
	elif y_46_re <= 0.00013:
		tmp = math.exp(-t_0) * 1.0
	else:
		tmp = math.exp(((-y_46_re / (0.0 - (1.0 / t_1))) - t_0)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	tmp = 0.0
	if (y_46_re <= -5.8e-6)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_0)) * Float64(1.0 / Float64(1.0 / cos(Float64(y_46_re * atan(x_46_im, x_46_re))))));
	elseif (y_46_re <= 0.00013)
		tmp = Float64(exp(Float64(-t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(Float64(-y_46_re) / Float64(0.0 - Float64(1.0 / t_1))) - t_0)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = 0.0;
	if (y_46_re <= -5.8e-6)
		tmp = exp(((t_1 * y_46_re) - t_0)) * (1.0 / (1.0 / cos((y_46_re * atan2(x_46_im, x_46_re)))));
	elseif (y_46_re <= 0.00013)
		tmp = exp(-t_0) * 1.0;
	else
		tmp = exp(((-y_46_re / (0.0 - (1.0 / t_1))) - t_0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e-6], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 / N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.00013], N[(N[Exp[(-t$95$0)], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[((-y$46$re) / N[(0.0 - N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.8000000000000004e-6

   1. Initial program 37.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 1.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Applied egg-rr 1.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\frac{1}{\frac{1}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}} \]

   if -5.8000000000000004e-6 < y.re < 1.29999999999999989e-4

   1. Initial program 34.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 30.7

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 30.8

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   4. Taylor expanded in y.re around 0 6.9

      \[\leadsto \color{blue}{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

   if 1.29999999999999989e-4 < y.re

   1. Initial program 20.3

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 9.9

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 10.2

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   4. Applied egg-rr 1.4

      \[\leadsto e^{\color{blue}{\frac{-y.re}{0 - \frac{1}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \frac{1}{\frac{1}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{elif}\;y.re \leq 0.00013:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-y.re}{0 - \frac{1}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array} \]

Alternatives

Alternative 1
Error	4.7
Cost	39748

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 0.00024:\\ \;\;\;\;e^{-t_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-y.re}{0 - \frac{1}{t_1}} - t_0} \cdot 1\\ \end{array} \]

Alternative 2
Error	4.7
Cost	27208

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_0} \cdot 1\\ \mathbf{elif}\;y.re \leq 0.000112:\\ \;\;\;\;e^{-t_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-y.re}{0 - \frac{1}{t_1}} - t_0} \cdot 1\\ \end{array} \]

Alternative 3
Error	6.2
Cost	26888

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_0} \cdot 1\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;e^{-t_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	7.1
Cost	26244

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;{\left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+58}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \]

Alternative 5
Error	9.7
Cost	20236

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -7.8 \cdot 10^{-256}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - t_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 3.8 \cdot 10^{-209}:\\ \;\;\;\;e^{-t_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 1.15 \cdot 10^{-200}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_0} \cdot 1\\ \end{array} \]

Alternative 6
Error	9.8
Cost	20100

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(x.im \cdot -1\right) \cdot y.re - t_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_0} \cdot 1\\ \end{array} \]

Alternative 7
Error	13.1
Cost	13512

\[\begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{+58}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.7
Cost	7176

\[\begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.7
Cost	6920

\[\begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	38.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))