math.log10 on complex, real part

Average Error: 31.5 → 7.4

Time: 35.9s

Precision: binary64

Cost: 39428

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 1.7 \cdot 10^{-175}:\\ \;\;\;\;\log 10 \cdot \frac{t_0}{\frac{\log 10}{\frac{t_0}{\log 10 \cdot t_0}}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\log 10}{\frac{\log im}{\log 10}}}}{\frac{1}{\log 10}}\\ \end{array} \]

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))))
   (if (<= im 1.7e-175)
     (* (log 10.0) (/ t_0 (/ (log 10.0) (/ t_0 (* (log 10.0) t_0)))))
     (if (<= im 5.8e+85)
       (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0))
       (/
        (/ 1.0 (/ (log 10.0) (/ (log im) (log 10.0))))
        (/ 1.0 (log 10.0)))))))

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 tmp;
	if (im <= 1.7e-175) {
		tmp = log(10.0) * (t_0 / (log(10.0) / (t_0 / (log(10.0) * t_0))));
	} else if (im <= 5.8e+85) {
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	} else {
		tmp = (1.0 / (log(10.0) / (log(im) / log(10.0)))) / (1.0 / log(10.0));
	}
	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) :: tmp
    t_0 = log(-re)
    if (im <= 1.7d-175) then
        tmp = log(10.0d0) * (t_0 / (log(10.0d0) / (t_0 / (log(10.0d0) * t_0))))
    else if (im <= 5.8d+85) then
        tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
    else
        tmp = (1.0d0 / (log(10.0d0) / (log(im) / log(10.0d0)))) / (1.0d0 / log(10.0d0))
    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 tmp;
	if (im <= 1.7e-175) {
		tmp = Math.log(10.0) * (t_0 / (Math.log(10.0) / (t_0 / (Math.log(10.0) * t_0))));
	} else if (im <= 5.8e+85) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
	} else {
		tmp = (1.0 / (Math.log(10.0) / (Math.log(im) / Math.log(10.0)))) / (1.0 / Math.log(10.0));
	}
	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)
	tmp = 0
	if im <= 1.7e-175:
		tmp = math.log(10.0) * (t_0 / (math.log(10.0) / (t_0 / (math.log(10.0) * t_0))))
	elif im <= 5.8e+85:
		tmp = math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
	else:
		tmp = (1.0 / (math.log(10.0) / (math.log(im) / math.log(10.0)))) / (1.0 / math.log(10.0))
	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))
	tmp = 0.0
	if (im <= 1.7e-175)
		tmp = Float64(log(10.0) * Float64(t_0 / Float64(log(10.0) / Float64(t_0 / Float64(log(10.0) * t_0)))));
	elseif (im <= 5.8e+85)
		tmp = Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0));
	else
		tmp = Float64(Float64(1.0 / Float64(log(10.0) / Float64(log(im) / log(10.0)))) / Float64(1.0 / log(10.0)));
	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);
	tmp = 0.0;
	if (im <= 1.7e-175)
		tmp = log(10.0) * (t_0 / (log(10.0) / (t_0 / (log(10.0) * t_0))));
	elseif (im <= 5.8e+85)
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	else
		tmp = (1.0 / (log(10.0) / (log(im) / log(10.0)))) / (1.0 / log(10.0));
	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]}, If[LessEqual[im, 1.7e-175], N[(N[Log[10.0], $MachinePrecision] * N[(t$95$0 / N[(N[Log[10.0], $MachinePrecision] / N[(t$95$0 / N[(N[Log[10.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e+85], N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.7e-175

   1. Initial program 32.0

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

   2. Taylor expanded in re around -inf 4.5

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

   3. Simplified 4.5

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

      [Start] 4.5	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational.json-simplify-2 [=>] 4.5	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational.json-simplify-9 [=>] 4.5	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

   4. Applied egg-rr 4.6

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

   5. Simplified 4.5

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

      [Start] 4.6	\[ \log 10 \cdot \frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\log \left(-re\right)} \cdot \frac{\log 10}{\log \left(-re\right)}} \]
      rational.json-simplify-46 [=>] 4.6	\[ \log 10 \cdot \color{blue}{\frac{\frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\log \left(-re\right)}}}{\frac{\log 10}{\log \left(-re\right)}}} \]
      rational.json-simplify-61 [=>] 4.5	\[ \log 10 \cdot \color{blue}{\frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\log \left(-re\right)}}}}} \]
      rational.json-simplify-7 [<=] 4.5	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\color{blue}{\frac{\log \left(-re\right)}{1}}}}}} \]
      rational.json-simplify-61 [=>] 4.6	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{\log \left(-re\right)}}{\color{blue}{\frac{1}{\frac{\log \left(-re\right)}{\log 10}}}}}} \]
      rational.json-simplify-61 [<=] 4.6	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\color{blue}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\frac{1}{\frac{1}{\log \left(-re\right)}}}}}} \]
      rational.json-simplify-61 [=>] 4.5	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\color{blue}{\frac{\log \left(-re\right)}{\frac{1}{1}}}}}} \]
      metadata-eval [=>] 4.5	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\frac{\log \left(-re\right)}{\color{blue}{1}}}}} \]
      rational.json-simplify-7 [=>] 4.5	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\color{blue}{\log \left(-re\right)}}}} \]
      rational.json-simplify-47 [=>] 4.5	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\color{blue}{\frac{\log \left(-re\right)}{\log 10 \cdot \log \left(-re\right)}}}} \]

   if 1.7e-175 < im < 5.79999999999999995e85

   1. Initial program 13.1

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

   if 5.79999999999999995e85 < im

   1. Initial program 48.0

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

   2. Taylor expanded in re around 0 5.2

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

   3. Applied egg-rr 5.4

      \[\leadsto \color{blue}{\log 10 \cdot \frac{\frac{1}{\log im}}{\frac{\log 10}{\log im} \cdot \frac{\log 10}{\log im}}} \]

   4. Simplified 5.4

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

      [Start] 5.4	\[ \log 10 \cdot \frac{\frac{1}{\log im}}{\frac{\log 10}{\log im} \cdot \frac{\log 10}{\log im}} \]
      rational.json-simplify-47 [=>] 5.4	\[ \log 10 \cdot \color{blue}{\frac{1}{\log im \cdot \left(\frac{\log 10}{\log im} \cdot \frac{\log 10}{\log im}\right)}} \]

   5. Applied egg-rr 5.2

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

   6. Applied egg-rr 5.2

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\log 10}{\frac{\log im}{\log 10}}}}}{\frac{1}{\log 10}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{-175}:\\ \;\;\;\;\log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\log \left(-re\right)}{\log 10 \cdot \log \left(-re\right)}}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\log 10}{\frac{\log im}{\log 10}}}}{\frac{1}{\log 10}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	26628

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 1.55 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5}{t_0} \cdot \left(t_0 \cdot \frac{t_0}{\frac{\log 10}{2}}\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\log 10}{\frac{\log im}{\log 10}}}}{\frac{1}{\log 10}}\\ \end{array} \]

Alternative 2
Error	7.3
Cost	26568

\[\begin{array}{l} t_0 := \frac{1}{\log 10}\\ \mathbf{if}\;im \leq 4.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{\log \left(-re\right)}{\log 10 \cdot \log 10}}{t_0}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\log 10}{\frac{\log im}{\log 10}}}}{t_0}\\ \end{array} \]

Alternative 3
Error	7.4
Cost	26376

\[\begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 10^{+87}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{{\log 10}^{2}}}{\frac{1}{\log 10}}\\ \end{array} \]

Alternative 4
Error	7.3
Cost	26376

\[\begin{array}{l} t_0 := \frac{1}{\log 10}\\ \mathbf{if}\;im \leq 4.4 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{\log \left(-re\right)}{\log 10 \cdot \log 10}}{t_0}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{{\log 10}^{2}}}{t_0}\\ \end{array} \]

Alternative 5
Error	7.4
Cost	26248

\[\begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\log im}^{2}}{\log 10}}{\log im}\\ \end{array} \]

Alternative 6
Error	7.5
Cost	20040

\[\begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{-179}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+87}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 7
Error	12.2
Cost	13644

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{4}{\frac{4}{\frac{\log im}{\log 10}}}\\ \mathbf{if}\;re \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.1
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;re \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -7 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -5.6 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	12.1
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ \mathbf{if}\;re \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.35 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;re \leq -2 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 10
Error	30.0
Cost	12992

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

Reproduce

herbie shell --seed 2023065 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))