math.log10 on complex, real part

Average Error: 32.3 → 7.8

Time: 48.0s

Precision: binary64

Cost: 59848

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{if}\;im \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\log \left(-re\right)}{\log 10} + -1\right) + 1\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0))))
   (if (<= im 5e-159)
     (+ (+ (/ (log (- re)) (log 10.0)) -1.0) 1.0)
     (if (<= im 4.3e+98)
       (* (/ 1.0 t_0) (* t_0 t_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(sqrt(((re * re) + (im * im)))) / log(10.0);
	double tmp;
	if (im <= 5e-159) {
		tmp = ((log(-re) / log(10.0)) + -1.0) + 1.0;
	} else if (im <= 4.3e+98) {
		tmp = (1.0 / t_0) * (t_0 * t_0);
	} else {
		tmp = 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(sqrt(((re * re) + (im * im)))) / log(10.0d0)
    if (im <= 5d-159) then
        tmp = ((log(-re) / log(10.0d0)) + (-1.0d0)) + 1.0d0
    else if (im <= 4.3d+98) then
        tmp = (1.0d0 / t_0) * (t_0 * t_0)
    else
        tmp = 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(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
	double tmp;
	if (im <= 5e-159) {
		tmp = ((Math.log(-re) / Math.log(10.0)) + -1.0) + 1.0;
	} else if (im <= 4.3e+98) {
		tmp = (1.0 / t_0) * (t_0 * t_0);
	} else {
		tmp = 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(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
	tmp = 0
	if im <= 5e-159:
		tmp = ((math.log(-re) / math.log(10.0)) + -1.0) + 1.0
	elif im <= 4.3e+98:
		tmp = (1.0 / t_0) * (t_0 * t_0)
	else:
		tmp = 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(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
	tmp = 0.0
	if (im <= 5e-159)
		tmp = Float64(Float64(Float64(log(Float64(-re)) / log(10.0)) + -1.0) + 1.0);
	elseif (im <= 4.3e+98)
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_0 * t_0));
	else
		tmp = 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(sqrt(((re * re) + (im * im)))) / log(10.0);
	tmp = 0.0;
	if (im <= 5e-159)
		tmp = ((log(-re) / log(10.0)) + -1.0) + 1.0;
	elseif (im <= 4.3e+98)
		tmp = (1.0 / t_0) * (t_0 * t_0);
	else
		tmp = 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[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5e-159], N[(N[(N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[im, 4.3e+98], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.00000000000000032e-159

   1. Initial program 33.1

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

   2. Applied egg-rr 33.1

      \[\leadsto \color{blue}{\left(0 - \left(-1 - \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)\right) - 1} \]

   3. Applied egg-rr 33.1

      \[\leadsto \left(0 - \color{blue}{\left(\left(-2 - \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right) + 1\right)}\right) - 1 \]

   4. Taylor expanded in re around -inf 5.8

      \[\leadsto \left(0 - \left(\left(-2 - \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\right) + 1\right)\right) - 1 \]

   5. Simplified 5.8

      \[\leadsto \left(0 - \left(\left(-2 - \frac{\log \color{blue}{\left(-re\right)}}{\log 10}\right) + 1\right)\right) - 1 \]
      Proof

      [Start] 5.8	\[ \left(0 - \left(\left(-2 - \frac{\log \left(-1 \cdot re\right)}{\log 10}\right) + 1\right)\right) - 1 \]
      rational.json-simplify-2 [=>] 5.8	\[ \left(0 - \left(\left(-2 - \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10}\right) + 1\right)\right) - 1 \]
      rational.json-simplify-9 [=>] 5.8	\[ \left(0 - \left(\left(-2 - \frac{\log \color{blue}{\left(-re\right)}}{\log 10}\right) + 1\right)\right) - 1 \]

   6. Applied egg-rr 5.8

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

   if 5.00000000000000032e-159 < im < 4.3000000000000001e98

   1. Initial program 12.5

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

   2. Applied egg-rr 12.5

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

   if 4.3000000000000001e98 < im

   1. Initial program 50.0

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

   2. Taylor expanded in re around 0 5.9

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\log \left(-re\right)}{\log 10} + -1\right) + 1\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \cdot \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	20424

\[\begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\left(\frac{\log \left(-re\right)}{\log 10} + -1\right) + 1\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 2
Error	7.8
Cost	20040

\[\begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;\left(\frac{\log \left(-re\right)}{\log 10} + -1\right) + 1\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 3
Error	10.5
Cost	13444

\[\begin{array}{l} \mathbf{if}\;im \leq 1.8 \cdot 10^{-95}:\\ \;\;\;\;\left(\frac{\log \left(-re\right)}{\log 10} + -1\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log im}{\log 10} + 1\right) + -1\\ \end{array} \]

Alternative 4
Error	10.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log im}{\log 10} + 1\right) + -1\\ \end{array} \]

Alternative 5
Error	10.5
Cost	13188

\[\begin{array}{l} \mathbf{if}\;im \leq 6.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6
Error	31.1
Cost	12992

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

Reproduce

herbie shell --seed 2023077 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))