powComplex, real part

Average Error: 32.9 → 6.2

Time: 26.8s

Precision: binary64

Cost: 13384

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.0686730943674026 \cdot 10^{-14}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2600000000000:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \]

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
 (if (<= y.re -1.0686730943674026e-14)
   (pow (hypot x.im x.re) y.re)
   (if (<= y.re 2600000000000.0)
     (exp (* (atan2 x.im x.re) (- y.im)))
     (pow x.re y.re))))

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 tmp;
	if (y_46_re <= -1.0686730943674026e-14) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 2600000000000.0) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = pow(x_46_re, y_46_re);
	}
	return tmp;
}

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 tmp;
	if (y_46_re <= -1.0686730943674026e-14) {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 2600000000000.0) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.pow(x_46_re, y_46_re);
	}
	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):
	tmp = 0
	if y_46_re <= -1.0686730943674026e-14:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 2600000000000.0:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.pow(x_46_re, y_46_re)
	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)
	tmp = 0.0
	if (y_46_re <= -1.0686730943674026e-14)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	elseif (y_46_re <= 2600000000000.0)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = x_46_re ^ y_46_re;
	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)
	tmp = 0.0;
	if (y_46_re <= -1.0686730943674026e-14)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	elseif (y_46_re <= 2600000000000.0)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = x_46_re ^ y_46_re;
	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_] := If[LessEqual[y$46$re, -1.0686730943674026e-14], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 2600000000000.0], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Power[x$46$re, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.0686730943674026e-14

   1. Initial program 36.6

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

   2. Taylor expanded in y.im around 0 1.6

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

   3. Taylor expanded in y.re around 0 1.7

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

   4. Taylor expanded in y.im around 0 4.0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot 1 \]

   5. Simplified 3.9

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \cdot 1 \]
      Proof
      (pow.f64 (hypot.f64 x.im x.re) y.re): 0 points increase in error, 0 points decrease in error
      (pow.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x.im x.im) (*.f64 x.re x.re)))) y.re): 65 points increase in error, 13 points decrease in error
      (pow.f64 (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.im 2)) (*.f64 x.re x.re))) y.re): 0 points increase in error, 0 points decrease in error
      (pow.f64 (sqrt.f64 (+.f64 (pow.f64 x.im 2) (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)))) y.re): 0 points increase in error, 0 points decrease in error

   if -1.0686730943674026e-14 < y.re < 2.6e12

   1. Initial program 33.8

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

   2. Taylor expanded in y.im around 0 29.7

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

   3. Taylor expanded in y.re around 0 29.8

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

   4. Taylor expanded in y.re around 0 6.7

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

   if 2.6e12 < y.re

   1. Initial program 18.2

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

   2. Taylor expanded in y.im around 0 8.9

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

   3. Taylor expanded in y.re around 0 8.8

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

   4. Taylor expanded in y.im around 0 16.0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot 1 \]

   5. Simplified 16.0

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \cdot 1 \]
      Proof
      (pow.f64 (hypot.f64 x.im x.re) y.re): 0 points increase in error, 0 points decrease in error
      (pow.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x.im x.im) (*.f64 x.re x.re)))) y.re): 65 points increase in error, 13 points decrease in error
      (pow.f64 (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.im 2)) (*.f64 x.re x.re))) y.re): 0 points increase in error, 0 points decrease in error
      (pow.f64 (sqrt.f64 (+.f64 (pow.f64 x.im 2) (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)))) y.re): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in x.im around 0 9.7

      \[\leadsto {\color{blue}{x.re}}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.0686730943674026 \cdot 10^{-14}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2600000000000:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.4
Cost	12992

\[{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

Alternative 2
Error	22.4
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.7 \cdot 10^{-98}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \]

Alternative 3
Error	21.8
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.6 \cdot 10^{-291}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \]

Alternative 4
Error	28.3
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x.re \leq 7.2 \cdot 10^{-194}:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \]

Alternative 5
Error	34.3
Cost	6528

\[{x.re}^{y.re} \]

Reproduce

herbie shell --seed 2022316 
(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)))))