math.log10 on complex, real part

Average Error: 32.3 → 8.1

Time: 28.9s

Precision: binary64

Cost: 39368

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

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

↓

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;im \leq 4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{t_0}{\log 10}}{t_0 \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{4}{\frac{\log 10 \cdot 8}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{\log 10}{\log im} \cdot t_1}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (log (- re))) (t_1 (/ (log im) (log 10.0))))
   (if (<= im 4.4e-132)
     (/ (/ t_0 (log 10.0)) (* t_0 (/ 1.0 t_0)))
     (if (<= im 3.2e+73)
       (/ 4.0 (/ (* (log 10.0) 8.0) (log (+ (* re re) (* im im)))))
       (/ t_1 (* (/ (log 10.0) (log im)) t_1))))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

↓

double code(double re, double im) {
	double t_0 = log(-re);
	double t_1 = log(im) / log(10.0);
	double tmp;
	if (im <= 4.4e-132) {
		tmp = (t_0 / log(10.0)) / (t_0 * (1.0 / t_0));
	} else if (im <= 3.2e+73) {
		tmp = 4.0 / ((log(10.0) * 8.0) / log(((re * re) + (im * im))));
	} else {
		tmp = t_1 / ((log(10.0) / log(im)) * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(-re)
    t_1 = log(im) / log(10.0d0)
    if (im <= 4.4d-132) then
        tmp = (t_0 / log(10.0d0)) / (t_0 * (1.0d0 / t_0))
    else if (im <= 3.2d+73) then
        tmp = 4.0d0 / ((log(10.0d0) * 8.0d0) / log(((re * re) + (im * im))))
    else
        tmp = t_1 / ((log(10.0d0) / log(im)) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

↓

public static double code(double re, double im) {
	double t_0 = Math.log(-re);
	double t_1 = Math.log(im) / Math.log(10.0);
	double tmp;
	if (im <= 4.4e-132) {
		tmp = (t_0 / Math.log(10.0)) / (t_0 * (1.0 / t_0));
	} else if (im <= 3.2e+73) {
		tmp = 4.0 / ((Math.log(10.0) * 8.0) / Math.log(((re * re) + (im * im))));
	} else {
		tmp = t_1 / ((Math.log(10.0) / Math.log(im)) * t_1);
	}
	return tmp;
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

↓

def code(re, im):
	t_0 = math.log(-re)
	t_1 = math.log(im) / math.log(10.0)
	tmp = 0
	if im <= 4.4e-132:
		tmp = (t_0 / math.log(10.0)) / (t_0 * (1.0 / t_0))
	elif im <= 3.2e+73:
		tmp = 4.0 / ((math.log(10.0) * 8.0) / math.log(((re * re) + (im * im))))
	else:
		tmp = t_1 / ((math.log(10.0) / math.log(im)) * t_1)
	return tmp

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

↓

function code(re, im)
	t_0 = log(Float64(-re))
	t_1 = Float64(log(im) / log(10.0))
	tmp = 0.0
	if (im <= 4.4e-132)
		tmp = Float64(Float64(t_0 / log(10.0)) / Float64(t_0 * Float64(1.0 / t_0)));
	elseif (im <= 3.2e+73)
		tmp = Float64(4.0 / Float64(Float64(log(10.0) * 8.0) / log(Float64(Float64(re * re) + Float64(im * im)))));
	else
		tmp = Float64(t_1 / Float64(Float64(log(10.0) / log(im)) * t_1));
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

↓

function tmp_2 = code(re, im)
	t_0 = log(-re);
	t_1 = log(im) / log(10.0);
	tmp = 0.0;
	if (im <= 4.4e-132)
		tmp = (t_0 / log(10.0)) / (t_0 * (1.0 / t_0));
	elseif (im <= 3.2e+73)
		tmp = 4.0 / ((log(10.0) * 8.0) / log(((re * re) + (im * im))));
	else
		tmp = t_1 / ((log(10.0) / log(im)) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := Block[{t$95$0 = N[Log[(-re)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.4e-132], N[(N[(t$95$0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.2e+73], N[(4.0 / N[(N[(N[Log[10.0], $MachinePrecision] * 8.0), $MachinePrecision] / N[Log[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.39999999999999981e-132

   1. Initial program 32.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

   2. Taylor expanded in re around -inf 7.3

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10} \]

   3. Simplified 7.3

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]
      Proof

      [Start] 7.3	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational.json-simplify-2 [=>] 7.3	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational.json-simplify-9 [=>] 7.3	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

   4. Applied egg-rr 7.4

      \[\leadsto \color{blue}{\left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right) \cdot \frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}} \]

   5. Simplified 7.3

      \[\leadsto \color{blue}{\frac{2}{\log \left(-re\right)} \cdot \frac{\log \left(-re\right)}{2 \cdot \frac{\log 10}{\log \left(-re\right)}}} \]
      Proof

      [Start] 7.4	\[ \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right) \cdot \frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \]
      rational.json-simplify-2 [=>] 7.4	\[ \color{blue}{\frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \]
      rational.json-simplify-43 [<=] 7.4	\[ \color{blue}{\frac{2}{\log \left(-re\right)} \cdot \left(\frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \cdot \log 10\right)} \]
      rational.json-simplify-7 [<=] 7.4	\[ \frac{2}{\log \left(-re\right)} \cdot \left(\frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \cdot \color{blue}{\frac{\log 10}{1}}\right) \]
      rational.json-simplify-55 [=>] 7.4	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\frac{\log 10}{1}}{\frac{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}{1}}} \]
      rational.json-simplify-7 [=>] 7.4	\[ \frac{2}{\log \left(-re\right)} \cdot \frac{\color{blue}{\log 10}}{\frac{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}{1}} \]
      rational.json-simplify-7 [=>] 7.4	\[ \frac{2}{\log \left(-re\right)} \cdot \frac{\log 10}{\color{blue}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}} \]
      rational.json-simplify-46 [=>] 7.3	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\frac{\log 10}{\frac{\log 10}{\log \left(-re\right)}}}{\log 10 \cdot \frac{2}{\log \left(-re\right)}}} \]
      rational.json-simplify-46 [=>] 7.3	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\frac{\frac{\log 10}{\frac{\log 10}{\log \left(-re\right)}}}{\log 10}}{\frac{2}{\log \left(-re\right)}}} \]
      rational.json-simplify-61 [=>] 7.4	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\log \left(-re\right)}{\frac{2}{\frac{\frac{\log 10}{\frac{\log 10}{\log \left(-re\right)}}}{\log 10}}}} \]

   6. Applied egg-rr 7.3

      \[\leadsto \color{blue}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\log \left(-re\right) \cdot \frac{1}{\log \left(-re\right)}}} \]

   if 4.39999999999999981e-132 < im < 3.19999999999999982e73

   1. Initial program 11.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

   2. Applied egg-rr 25.1

      \[\leadsto \color{blue}{\frac{1}{\log 10 \cdot 4} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{2}\right)} \]

   3. Applied egg-rr 11.5

      \[\leadsto \color{blue}{\frac{4}{\frac{\log 10 \cdot 8}{\log \left(re \cdot re + im \cdot im\right)}}} \]

   if 3.19999999999999982e73 < im

   1. Initial program 47.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

   2. Taylor expanded in re around 0 6.8

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10} \]

   3. Applied egg-rr 6.8

      \[\leadsto \color{blue}{\frac{\log 10}{\log im} \cdot \frac{2}{\frac{\log 10}{\log im} \cdot \left(2 \cdot \frac{\log 10}{\log im}\right)}} \]

   4. Applied egg-rr 6.7

      \[\leadsto \color{blue}{\frac{\frac{\log im}{\log 10}}{\frac{\log 10}{\log im} \cdot \frac{\log im}{\log 10}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\log \left(-re\right)}{\log 10}}{\log \left(-re\right) \cdot \frac{1}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{4}{\frac{\log 10 \cdot 8}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\frac{\log 10}{\log im} \cdot \frac{\log im}{\log 10}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.1
Cost	26500

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{t_0}{\log 10}}{t_0 \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{4}{\frac{\log 10 \cdot 8}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 10}{\frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

Alternative 2
Error	8.1
Cost	26312

\[\begin{array}{l} \mathbf{if}\;im \leq 2.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{4}{\frac{\log 10 \cdot 8}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 10}{\frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

Alternative 3
Error	8.1
Cost	13896

\[\begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{-131}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{4}{\frac{\log 10 \cdot 8}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 4
Error	8.1
Cost	13768

\[\begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+74}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5
Error	10.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 6
Error	10.9
Cost	13188

\[\begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 7
Error	30.8
Cost	12992

\[\frac{\log im}{\log 10} \]

Reproduce

herbie shell --seed 2023074 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))