math.log10 on complex, real part

Average Error: 32.7 → 7.9

Time: 34.9s

Precision: binary64

Cost: 39556

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\log 10}{\log \left(-re\right)}\\ \mathbf{if}\;re \leq -7 \cdot 10^{+75}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_0 \cdot t_0}\\ \mathbf{elif}\;re \leq -2.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log 10} \cdot \frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (log 10.0) (log (- re)))))
   (if (<= re -7e+75)
     (* t_0 (/ 1.0 (* t_0 t_0)))
     (if (<= re -2.5e-129)
       (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0))
       (/
        1.0
        (* (/ 1.0 (log 10.0)) (/ (log 10.0) (/ (log im) (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(10.0) / log(-re);
	double tmp;
	if (re <= -7e+75) {
		tmp = t_0 * (1.0 / (t_0 * t_0));
	} else if (re <= -2.5e-129) {
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	} else {
		tmp = 1.0 / ((1.0 / log(10.0)) * (log(10.0) / (log(im) / 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(10.0d0) / log(-re)
    if (re <= (-7d+75)) then
        tmp = t_0 * (1.0d0 / (t_0 * t_0))
    else if (re <= (-2.5d-129)) then
        tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
    else
        tmp = 1.0d0 / ((1.0d0 / log(10.0d0)) * (log(10.0d0) / (log(im) / 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(10.0) / Math.log(-re);
	double tmp;
	if (re <= -7e+75) {
		tmp = t_0 * (1.0 / (t_0 * t_0));
	} else if (re <= -2.5e-129) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
	} else {
		tmp = 1.0 / ((1.0 / Math.log(10.0)) * (Math.log(10.0) / (Math.log(im) / 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(10.0) / math.log(-re)
	tmp = 0
	if re <= -7e+75:
		tmp = t_0 * (1.0 / (t_0 * t_0))
	elif re <= -2.5e-129:
		tmp = math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
	else:
		tmp = 1.0 / ((1.0 / math.log(10.0)) * (math.log(10.0) / (math.log(im) / 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 = Float64(log(10.0) / log(Float64(-re)))
	tmp = 0.0
	if (re <= -7e+75)
		tmp = Float64(t_0 * Float64(1.0 / Float64(t_0 * t_0)));
	elseif (re <= -2.5e-129)
		tmp = Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / log(10.0)) * Float64(log(10.0) / Float64(log(im) / 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(10.0) / log(-re);
	tmp = 0.0;
	if (re <= -7e+75)
		tmp = t_0 * (1.0 / (t_0 * t_0));
	elseif (re <= -2.5e-129)
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	else
		tmp = 1.0 / ((1.0 / log(10.0)) * (log(10.0) / (log(im) / 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[(N[Log[10.0], $MachinePrecision] / N[Log[(-re)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7e+75], N[(t$95$0 * N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -2.5e-129], N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision] * N[(N[Log[10.0], $MachinePrecision] / N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.9999999999999997e75

   1. Initial program 46.7

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

   2. Taylor expanded in re around -inf 6.2

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

   3. Simplified 6.2

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

      [Start] 6.2	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational.json-simplify-2 [=>] 6.2	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational.json-simplify-9 [=>] 6.2	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

   4. Applied egg-rr 6.3

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

   if -6.9999999999999997e75 < re < -2.50000000000000014e-129

   1. Initial program 12.4

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

   if -2.50000000000000014e-129 < re

   1. Initial program 32.8

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

   2. Taylor expanded in re around 0 6.7

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

   3. Applied egg-rr 7.0

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

   4. Applied egg-rr 6.7

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

   5. Applied egg-rr 6.7

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log 10}{\log \left(-re\right)} \cdot \frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \frac{\log 10}{\log \left(-re\right)}}\\ \mathbf{elif}\;re \leq -2.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log 10} \cdot \frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	39428

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -8.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\log 10}{t_0} \cdot \left(t_0 \cdot \frac{\frac{t_0}{\log 10}}{\log 10}\right)\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log 10} \cdot \frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

Alternative 2
Error	7.9
Cost	39428

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -4 \cdot 10^{+75}:\\ \;\;\;\;\frac{t_0}{\log 10 \cdot \left(\frac{\log 10}{t_0} \cdot \frac{t_0}{\log 10}\right)}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log 10} \cdot \frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

Alternative 3
Error	7.9
Cost	26568

\[\begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -2.55 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log 10} \cdot \frac{\log 10}{\frac{\log im}{\log 10}}}\\ \end{array} \]

Alternative 4
Error	8.1
Cost	26504

\[\begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -4.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\log im \cdot \left(\frac{-\log im}{\log 10} \cdot \frac{-1}{\log im}\right)\\ \end{array} \]

Alternative 5
Error	7.9
Cost	26440

\[\begin{array}{l} t_0 := -\log im\\ \mathbf{if}\;re \leq -7.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\log im \cdot \frac{t_0}{t_0 \cdot \log 10}\\ \end{array} \]

Alternative 6
Error	8.0
Cost	20040

\[\begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -1.15 \cdot 10^{-125}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 7
Error	10.5
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 8
Error	30.7
Cost	12992

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

Reproduce

herbie shell --seed 2023063 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))