math.log10 on complex, real part

Average Error: 32.1 → 7.4

Time: 28.2s

Precision: binary64

Cost: 39428

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ t_1 := \frac{t_0}{\log 10}\\ \mathbf{if}\;im \leq 4.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{t_1}{\frac{\log 10}{t_0} \cdot t_1}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+114}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log 10 \cdot \frac{1}{\log 10}}\\ \end{array} \]

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))) (t_1 (/ t_0 (log 10.0))))
   (if (<= im 4.8e-159)
     (/ t_1 (* (/ (log 10.0) t_0) t_1))
     (if (<= im 8e+114)
       (* (/ (log (+ (* re re) (* im im))) (log 10.0)) 0.5)
       (/ (/ (log im) (log 10.0)) (* (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 t_1 = t_0 / log(10.0);
	double tmp;
	if (im <= 4.8e-159) {
		tmp = t_1 / ((log(10.0) / t_0) * t_1);
	} else if (im <= 8e+114) {
		tmp = (log(((re * re) + (im * im))) / log(10.0)) * 0.5;
	} else {
		tmp = (log(im) / log(10.0)) / (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) :: t_1
    real(8) :: tmp
    t_0 = log(-re)
    t_1 = t_0 / log(10.0d0)
    if (im <= 4.8d-159) then
        tmp = t_1 / ((log(10.0d0) / t_0) * t_1)
    else if (im <= 8d+114) then
        tmp = (log(((re * re) + (im * im))) / log(10.0d0)) * 0.5d0
    else
        tmp = (log(im) / log(10.0d0)) / (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 t_1 = t_0 / Math.log(10.0);
	double tmp;
	if (im <= 4.8e-159) {
		tmp = t_1 / ((Math.log(10.0) / t_0) * t_1);
	} else if (im <= 8e+114) {
		tmp = (Math.log(((re * re) + (im * im))) / Math.log(10.0)) * 0.5;
	} else {
		tmp = (Math.log(im) / Math.log(10.0)) / (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)
	t_1 = t_0 / math.log(10.0)
	tmp = 0
	if im <= 4.8e-159:
		tmp = t_1 / ((math.log(10.0) / t_0) * t_1)
	elif im <= 8e+114:
		tmp = (math.log(((re * re) + (im * im))) / math.log(10.0)) * 0.5
	else:
		tmp = (math.log(im) / math.log(10.0)) / (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))
	t_1 = Float64(t_0 / log(10.0))
	tmp = 0.0
	if (im <= 4.8e-159)
		tmp = Float64(t_1 / Float64(Float64(log(10.0) / t_0) * t_1));
	elseif (im <= 8e+114)
		tmp = Float64(Float64(log(Float64(Float64(re * re) + Float64(im * im))) / log(10.0)) * 0.5);
	else
		tmp = Float64(Float64(log(im) / log(10.0)) / Float64(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);
	t_1 = t_0 / log(10.0);
	tmp = 0.0;
	if (im <= 4.8e-159)
		tmp = t_1 / ((log(10.0) / t_0) * t_1);
	elseif (im <= 8e+114)
		tmp = (log(((re * re) + (im * im))) / log(10.0)) * 0.5;
	else
		tmp = (log(im) / log(10.0)) / (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]}, Block[{t$95$1 = N[(t$95$0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.8e-159], N[(t$95$1 / N[(N[(N[Log[10.0], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8e+114], N[(N[(N[Log[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision] / N[(N[Log[10.0], $MachinePrecision] * N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.79999999999999995e-159

   1. Initial program 33.5

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

   2. Taylor expanded in re around -inf 5.5

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

   3. Simplified 5.5

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

      [Start] 5.5	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational.json-simplify-2 [=>] 5.5	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational.json-simplify-9 [=>] 5.5	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

   4. Applied egg-rr 5.7

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

   5. Applied egg-rr 5.4

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

   if 4.79999999999999995e-159 < im < 8e114

   1. Initial program 12.0

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

   2. Applied egg-rr 12.0

      \[\leadsto \color{blue}{\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5} \]

   if 8e114 < im

   1. Initial program 53.6

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

   2. Taylor expanded in re around 0 4.8

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

   3. Applied egg-rr 5.0

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

   4. Simplified 5.0

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

      [Start] 5.0	\[ \log 10 \cdot \frac{\frac{1}{\log im}}{\frac{\log 10}{\log im} \cdot \frac{\log 10}{\log im}} \]
      rational.json-simplify-47 [=>] 5.0	\[ \log 10 \cdot \color{blue}{\frac{1}{\log im \cdot \left(\frac{\log 10}{\log im} \cdot \frac{\log 10}{\log im}\right)}} \]

   5. Applied egg-rr 4.8

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

   6. Simplified 4.8

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

      [Start] 4.8	\[ \frac{\frac{\log im}{\log 10}}{\log 10 \cdot \frac{\log im}{\log 10 \cdot \log im}} \]
      rational.json-simplify-2 [=>] 4.8	\[ \frac{\frac{\log im}{\log 10}}{\log 10 \cdot \frac{\log im}{\color{blue}{\log im \cdot \log 10}}} \]

   7. Taylor expanded in im around 0 4.7

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{\log \left(-re\right)}{\log 10}}{\frac{\log 10}{\log \left(-re\right)} \cdot \frac{\log \left(-re\right)}{\log 10}}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+114}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log 10 \cdot \frac{1}{\log 10}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	26500

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 5.7 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\frac{t_0}{t_0}}{\frac{1}{t_0}}}{\log 10}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log 10 \cdot \frac{1}{\log 10}}\\ \end{array} \]

Alternative 2
Error	7.6
Cost	26440

\[\begin{array}{l} \mathbf{if}\;im \leq 9.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{2}{\log 10 \cdot \frac{2}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log 10 \cdot \frac{1}{\log 10}}\\ \end{array} \]

Alternative 3
Error	7.5
Cost	26312

\[\begin{array}{l} \mathbf{if}\;im \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\log 10 \cdot \frac{2}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+117}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 10}{\log 10 \cdot \frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 4
Error	7.6
Cost	13768

\[\begin{array}{l} \mathbf{if}\;im \leq 1.35 \cdot 10^{-152}:\\ \;\;\;\;\frac{2}{\log 10 \cdot \frac{2}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 5
Error	10.6
Cost	13516

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{if}\;im \leq 4.7 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	10.8
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;im \leq 1.85 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	30.7
Cost	12992

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

Reproduce

herbie shell --seed 2023066 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))