math.log/2 on complex, real part

Average Error: 32.0 → 7.0

Time: 1.0min

Precision: binary64

Cost: 13768

\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.22 \cdot 10^{+148}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.95 \cdot 10^{-136}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log base} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

↓

(FPCore (re im base)
 :precision binary64
 (if (<= re -1.22e+148)
   (/ (log (- re)) (log base))
   (if (<= re -1.95e-136)
     (* (/ (log (+ (* re re) (* im im))) (log base)) 0.5)
     (- (/ (log (/ 1.0 im)) (log base))))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

↓

double code(double re, double im, double base) {
	double tmp;
	if (re <= -1.22e+148) {
		tmp = log(-re) / log(base);
	} else if (re <= -1.95e-136) {
		tmp = (log(((re * re) + (im * im))) / log(base)) * 0.5;
	} else {
		tmp = -(log((1.0 / im)) / log(base));
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

↓

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (re <= (-1.22d+148)) then
        tmp = log(-re) / log(base)
    else if (re <= (-1.95d-136)) then
        tmp = (log(((re * re) + (im * im))) / log(base)) * 0.5d0
    else
        tmp = -(log((1.0d0 / im)) / log(base))
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

↓

public static double code(double re, double im, double base) {
	double tmp;
	if (re <= -1.22e+148) {
		tmp = Math.log(-re) / Math.log(base);
	} else if (re <= -1.95e-136) {
		tmp = (Math.log(((re * re) + (im * im))) / Math.log(base)) * 0.5;
	} else {
		tmp = -(Math.log((1.0 / im)) / Math.log(base));
	}
	return tmp;
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

↓

def code(re, im, base):
	tmp = 0
	if re <= -1.22e+148:
		tmp = math.log(-re) / math.log(base)
	elif re <= -1.95e-136:
		tmp = (math.log(((re * re) + (im * im))) / math.log(base)) * 0.5
	else:
		tmp = -(math.log((1.0 / im)) / math.log(base))
	return tmp

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

↓

function code(re, im, base)
	tmp = 0.0
	if (re <= -1.22e+148)
		tmp = Float64(log(Float64(-re)) / log(base));
	elseif (re <= -1.95e-136)
		tmp = Float64(Float64(log(Float64(Float64(re * re) + Float64(im * im))) / log(base)) * 0.5);
	else
		tmp = Float64(-Float64(log(Float64(1.0 / im)) / log(base)));
	end
	return tmp
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

↓

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (re <= -1.22e+148)
		tmp = log(-re) / log(base);
	elseif (re <= -1.95e-136)
		tmp = (log(((re * re) + (im * im))) / log(base)) * 0.5;
	else
		tmp = -(log((1.0 / im)) / log(base));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_, base_] := If[LessEqual[re, -1.22e+148], N[(N[Log[(-re)], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.95e-136], N[(N[(N[Log[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-N[(N[Log[N[(1.0 / im), $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if re < -1.22000000000000007e148

   1. Initial program 61.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   2. Simplified 61.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]
      Proof

      [Start] 61.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-65 [=>] 61.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]
      rational_best-simplify-17 [=>] 61.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} - \tan^{-1}_* \frac{im}{re}}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-5 [=>] 61.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-64 [=>] 61.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + 0}{\log base \cdot \log base + 0 \cdot 0}} \]
      trig-simplify-13 [=>] 61.9	\[ \frac{\color{blue}{0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-6 [=>] 61.9	\[ \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 61.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]
      rational_best-simplify-3 [=>] 61.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{0 + \log base \cdot \log base}} \]
      rational_best-simplify-6 [=>] 61.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]
      rational_best-simplify-90 [=>] 61.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} + \frac{0}{\log base}} \]
      rational_best-simplify-64 [=>] 61.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) + 0}{\log base}} \]
      rational_best-simplify-3 [=>] 61.9	\[ \frac{\color{blue}{0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base} \]

   3. Taylor expanded in re around -inf 3.7

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log base} \]

   4. Simplified 3.7

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log base} \]
      Proof

      [Start] 3.7	\[ \frac{\log \left(-1 \cdot re\right)}{\log base} \]
      rational_best-simplify-1 [=>] 3.7	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log base} \]
      rational_best-simplify-11 [<=] 3.7	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log base} \]

   if -1.22000000000000007e148 < re < -1.94999999999999988e-136

   1. Initial program 10.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   2. Simplified 10.8

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]
      Proof

      [Start] 10.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-65 [=>] 10.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]
      rational_best-simplify-17 [=>] 10.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} - \tan^{-1}_* \frac{im}{re}}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-5 [=>] 10.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-64 [=>] 10.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + 0}{\log base \cdot \log base + 0 \cdot 0}} \]
      trig-simplify-13 [=>] 10.9	\[ \frac{\color{blue}{0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-6 [=>] 10.9	\[ \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 10.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]
      rational_best-simplify-3 [=>] 10.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{0 + \log base \cdot \log base}} \]
      rational_best-simplify-6 [=>] 10.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]
      rational_best-simplify-90 [=>] 10.8	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} + \frac{0}{\log base}} \]
      rational_best-simplify-64 [=>] 10.8	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) + 0}{\log base}} \]
      rational_best-simplify-3 [=>] 10.8	\[ \frac{\color{blue}{0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base} \]

   3. Applied egg-rr 10.8

      \[\leadsto \color{blue}{\frac{\log \left(re \cdot re + im \cdot im\right)}{\log base} \cdot 0.5} \]

   if -1.94999999999999988e-136 < re

   1. Initial program 30.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   2. Simplified 30.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]
      Proof

      [Start] 30.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-65 [=>] 30.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]
      rational_best-simplify-17 [=>] 30.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} - \tan^{-1}_* \frac{im}{re}}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-5 [=>] 30.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + 0 \cdot 0} + \frac{\color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-64 [=>] 30.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + 0}{\log base \cdot \log base + 0 \cdot 0}} \]
      trig-simplify-13 [=>] 30.9	\[ \frac{\color{blue}{0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
      rational_best-simplify-6 [=>] 30.9	\[ \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 30.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]
      rational_best-simplify-3 [=>] 30.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{0 + \log base \cdot \log base}} \]
      rational_best-simplify-6 [=>] 30.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]
      rational_best-simplify-90 [=>] 30.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} + \frac{0}{\log base}} \]
      rational_best-simplify-64 [=>] 30.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) + 0}{\log base}} \]
      rational_best-simplify-3 [=>] 30.9	\[ \frac{\color{blue}{0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base} \]

   3. Taylor expanded in im around inf 5.9

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log base}} \]

   4. Simplified 5.9

      \[\leadsto \color{blue}{-\frac{\log \left(\frac{1}{im}\right)}{\log base}} \]
      Proof

      [Start] 5.9	\[ -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log base} \]
      rational_best-simplify-1 [=>] 5.9	\[ \color{blue}{\frac{\log \left(\frac{1}{im}\right)}{\log base} \cdot -1} \]
      rational_best-simplify-10 [=>] 5.9	\[ \color{blue}{-\frac{\log \left(\frac{1}{im}\right)}{\log base}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.22 \cdot 10^{+148}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.95 \cdot 10^{-136}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log base} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.9
Cost	13768

\[\begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{0.5}{\log base} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	13316

\[\begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 4
Error	31.2
Cost	12992

\[\frac{\log im}{\log base} \]

Reproduce

herbie shell --seed 2023099 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))