math.log10 on complex, real part

Average Error: 32.3 → 7.6

Time: 29.5s

Precision: binary64

Cost: 32648

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -8.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{t_0}{\log 10}}{t_0 \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;re \leq -3.15 \cdot 10^{-145}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left({im}^{\left(\frac{\log im}{\log im \cdot \log 10}\right)}\right)\\ \end{array} \]

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 (<= re -8.6e+80)
     (/ (/ t_0 (log 10.0)) (* t_0 (/ 1.0 t_0)))
     (if (<= re -3.15e-145)
       (* (/ (log (+ (* re re) (* im im))) (log 10.0)) 0.5)
       (log (pow im (/ (log im) (* (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(-re);
	double tmp;
	if (re <= -8.6e+80) {
		tmp = (t_0 / log(10.0)) / (t_0 * (1.0 / t_0));
	} else if (re <= -3.15e-145) {
		tmp = (log(((re * re) + (im * im))) / log(10.0)) * 0.5;
	} else {
		tmp = log(pow(im, (log(im) / (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(-re)
    if (re <= (-8.6d+80)) then
        tmp = (t_0 / log(10.0d0)) / (t_0 * (1.0d0 / t_0))
    else if (re <= (-3.15d-145)) then
        tmp = (log(((re * re) + (im * im))) / log(10.0d0)) * 0.5d0
    else
        tmp = log((im ** (log(im) / (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(-re);
	double tmp;
	if (re <= -8.6e+80) {
		tmp = (t_0 / Math.log(10.0)) / (t_0 * (1.0 / t_0));
	} else if (re <= -3.15e-145) {
		tmp = (Math.log(((re * re) + (im * im))) / Math.log(10.0)) * 0.5;
	} else {
		tmp = Math.log(Math.pow(im, (Math.log(im) / (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(-re)
	tmp = 0
	if re <= -8.6e+80:
		tmp = (t_0 / math.log(10.0)) / (t_0 * (1.0 / t_0))
	elif re <= -3.15e-145:
		tmp = (math.log(((re * re) + (im * im))) / math.log(10.0)) * 0.5
	else:
		tmp = math.log(math.pow(im, (math.log(im) / (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 = log(Float64(-re))
	tmp = 0.0
	if (re <= -8.6e+80)
		tmp = Float64(Float64(t_0 / log(10.0)) / Float64(t_0 * Float64(1.0 / t_0)));
	elseif (re <= -3.15e-145)
		tmp = Float64(Float64(log(Float64(Float64(re * re) + Float64(im * im))) / log(10.0)) * 0.5);
	else
		tmp = log((im ^ Float64(log(im) / 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(-re);
	tmp = 0.0;
	if (re <= -8.6e+80)
		tmp = (t_0 / log(10.0)) / (t_0 * (1.0 / t_0));
	elseif (re <= -3.15e-145)
		tmp = (log(((re * re) + (im * im))) / log(10.0)) * 0.5;
	else
		tmp = log((im ^ (log(im) / (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[Log[(-re)], $MachinePrecision]}, If[LessEqual[re, -8.6e+80], 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[re, -3.15e-145], 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[Log[N[Power[im, N[(N[Log[im], $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 < -8.60000000000000008e80

   1. Initial program 48.5

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

   2. Taylor expanded in re around -inf 5.9

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

   3. Simplified 5.9

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

      [Start] 5.9	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational.json-simplify-2 [=>] 5.9	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational.json-simplify-9 [=>] 5.9	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

   4. Applied egg-rr 6.0

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

   5. Simplified 6.0

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

      [Start] 6.0	\[ \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right) \cdot \frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \]
      rational.json-simplify-2 [=>] 6.0	\[ \color{blue}{\frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \]
      rational.json-simplify-43 [<=] 6.1	\[ \color{blue}{\frac{2}{\log \left(-re\right)} \cdot \left(\frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \cdot \log 10\right)} \]
      rational.json-simplify-7 [<=] 6.1	\[ \frac{2}{\log \left(-re\right)} \cdot \left(\frac{1}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)} \cdot \color{blue}{\frac{\log 10}{1}}\right) \]
      rational.json-simplify-55 [=>] 6.0	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\frac{\log 10}{1}}{\frac{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}{1}}} \]
      rational.json-simplify-7 [=>] 6.0	\[ \frac{2}{\log \left(-re\right)} \cdot \frac{\color{blue}{\log 10}}{\frac{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}{1}} \]
      rational.json-simplify-7 [=>] 6.0	\[ \frac{2}{\log \left(-re\right)} \cdot \frac{\log 10}{\color{blue}{\frac{\log 10}{\log \left(-re\right)} \cdot \left(\log 10 \cdot \frac{2}{\log \left(-re\right)}\right)}} \]
      rational.json-simplify-46 [=>] 6.0	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\frac{\log 10}{\frac{\log 10}{\log \left(-re\right)}}}{\log 10 \cdot \frac{2}{\log \left(-re\right)}}} \]
      rational.json-simplify-46 [=>] 6.0	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\frac{\frac{\log 10}{\frac{\log 10}{\log \left(-re\right)}}}{\log 10}}{\frac{2}{\log \left(-re\right)}}} \]
      rational.json-simplify-61 [=>] 6.0	\[ \frac{2}{\log \left(-re\right)} \cdot \color{blue}{\frac{\log \left(-re\right)}{\frac{2}{\frac{\frac{\log 10}{\frac{\log 10}{\log \left(-re\right)}}}{\log 10}}}} \]

   6. Applied egg-rr 5.9

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

   if -8.60000000000000008e80 < re < -3.15e-145

   1. Initial program 11.5

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

   2. Applied egg-rr 11.5

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

   if -3.15e-145 < re

   1. Initial program 32.4

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

   2. Taylor expanded in re around 0 6.5

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

   3. Applied egg-rr 6.6

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

   4. Simplified 6.5

      \[\leadsto \color{blue}{\frac{2}{\log im} \cdot \frac{\frac{\log im}{\frac{\log 10}{\log im}}}{2}} \]
      Proof

      [Start] 6.6	\[ \frac{2}{\log im} \cdot \frac{\log 10}{\frac{\log 10}{\log im} \cdot \left(2 \cdot \frac{\log 10}{\log im}\right)} \]
      rational.json-simplify-46 [=>] 6.6	\[ \frac{2}{\log im} \cdot \color{blue}{\frac{\frac{\log 10}{\frac{\log 10}{\log im}}}{2 \cdot \frac{\log 10}{\log im}}} \]
      rational.json-simplify-2 [=>] 6.6	\[ \frac{2}{\log im} \cdot \frac{\frac{\log 10}{\frac{\log 10}{\log im}}}{\color{blue}{\frac{\log 10}{\log im} \cdot 2}} \]
      rational.json-simplify-46 [=>] 6.6	\[ \frac{2}{\log im} \cdot \color{blue}{\frac{\frac{\frac{\log 10}{\frac{\log 10}{\log im}}}{\frac{\log 10}{\log im}}}{2}} \]
      rational.json-simplify-61 [=>] 6.5	\[ \frac{2}{\log im} \cdot \frac{\frac{\color{blue}{\frac{\log im}{\frac{\log 10}{\log 10}}}}{\frac{\log 10}{\log im}}}{2} \]
      rational.json-simplify-60 [=>] 6.5	\[ \frac{2}{\log im} \cdot \frac{\frac{\color{blue}{\log im}}{\frac{\log 10}{\log im}}}{2} \]

   5. Applied egg-rr 6.6

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

   6. Simplified 6.6

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

      [Start] 6.6	\[ \frac{\frac{2}{\log im}}{\frac{\log 10}{\log im \cdot \log \left(\sqrt{im}\right)}} \]
      rational.json-simplify-44 [=>] 6.6	\[ \color{blue}{\frac{\frac{2}{\frac{\log 10}{\log im \cdot \log \left(\sqrt{im}\right)}}}{\log im}} \]
      rational.json-simplify-35 [=>] 6.6	\[ \frac{\frac{2}{\color{blue}{\frac{\log 10 + \log 10}{\log im \cdot \log \left(\sqrt{im}\right) + \log im \cdot \log \left(\sqrt{im}\right)}}}}{\log im} \]
      rational.json-simplify-61 [=>] 6.6	\[ \frac{\color{blue}{\frac{\log im \cdot \log \left(\sqrt{im}\right) + \log im \cdot \log \left(\sqrt{im}\right)}{\frac{\log 10 + \log 10}{2}}}}{\log im} \]
      rational.json-simplify-2 [=>] 6.6	\[ \frac{\frac{\log im \cdot \log \left(\sqrt{im}\right) + \color{blue}{\log \left(\sqrt{im}\right) \cdot \log im}}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      rational.json-simplify-51 [=>] 6.6	\[ \frac{\frac{\color{blue}{\log im \cdot \left(\log \left(\sqrt{im}\right) + \log \left(\sqrt{im}\right)\right)}}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      exponential.json-simplify-12 [=>] 6.6	\[ \frac{\frac{\log im \cdot \left(\color{blue}{0.5 \cdot \log im} + \log \left(\sqrt{im}\right)\right)}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      rational.json-simplify-2 [=>] 6.6	\[ \frac{\frac{\log im \cdot \left(\color{blue}{\log im \cdot 0.5} + \log \left(\sqrt{im}\right)\right)}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      exponential.json-simplify-12 [=>] 6.6	\[ \frac{\frac{\log im \cdot \left(\log im \cdot 0.5 + \color{blue}{0.5 \cdot \log im}\right)}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      rational.json-simplify-51 [=>] 6.6	\[ \frac{\frac{\log im \cdot \color{blue}{\left(\log im \cdot \left(0.5 + 0.5\right)\right)}}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      metadata-eval [=>] 6.6	\[ \frac{\frac{\log im \cdot \left(\log im \cdot \color{blue}{1}\right)}{\frac{\log 10 + \log 10}{2}}}{\log im} \]
      metadata-eval [<=] 6.6	\[ \frac{\frac{\log im \cdot \left(\log im \cdot 1\right)}{\frac{\log 10 + \log 10}{\color{blue}{1 + 1}}}}{\log im} \]
      rational.json-simplify-35 [<=] 6.6	\[ \frac{\frac{\log im \cdot \left(\log im \cdot 1\right)}{\color{blue}{\frac{\log 10}{1}}}}{\log im} \]
      rational.json-simplify-7 [=>] 6.6	\[ \frac{\frac{\log im \cdot \left(\log im \cdot 1\right)}{\color{blue}{\log 10}}}{\log im} \]

   7. Applied egg-rr 62.1

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

   8. Simplified 6.5

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

      [Start] 62.1	\[ \log \left({\left({im}^{\log im}\right)}^{\left(\frac{\frac{1}{\log 10}}{\log im}\right)}\right) \]
      exponential.json-simplify-32 [=>] 6.8	\[ \log \color{blue}{\left({im}^{\left(\log im \cdot \frac{\frac{1}{\log 10}}{\log im}\right)}\right)} \]
      rational.json-simplify-2 [=>] 6.8	\[ \log \left({im}^{\color{blue}{\left(\frac{\frac{1}{\log 10}}{\log im} \cdot \log im\right)}}\right) \]
      rational.json-simplify-47 [=>] 6.5	\[ \log \left({im}^{\left(\color{blue}{\frac{1}{\log 10 \cdot \log im}} \cdot \log im\right)}\right) \]
      rational.json-simplify-2 [<=] 6.5	\[ \log \left({im}^{\left(\frac{1}{\color{blue}{\log im \cdot \log 10}} \cdot \log im\right)}\right) \]
      rational.json-simplify-7 [<=] 6.5	\[ \log \left({im}^{\left(\frac{1}{\log im \cdot \log 10} \cdot \color{blue}{\frac{\log im}{1}}\right)}\right) \]
      rational.json-simplify-55 [=>] 6.5	\[ \log \left({im}^{\color{blue}{\left(\frac{\frac{\log im}{1}}{\frac{\log im \cdot \log 10}{1}}\right)}}\right) \]
      rational.json-simplify-7 [=>] 6.5	\[ \log \left({im}^{\left(\frac{\color{blue}{\log im}}{\frac{\log im \cdot \log 10}{1}}\right)}\right) \]
      rational.json-simplify-7 [=>] 6.5	\[ \log \left({im}^{\left(\frac{\log im}{\color{blue}{\log im \cdot \log 10}}\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\log \left(-re\right)}{\log 10}}{\log \left(-re\right) \cdot \frac{1}{\log \left(-re\right)}}\\ \mathbf{elif}\;re \leq -3.15 \cdot 10^{-145}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left({im}^{\left(\frac{\log im}{\log im \cdot \log 10}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	26500

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -1.02 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{t_0}{\log 10}}{t_0 \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;re \leq -3.95 \cdot 10^{-143}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\log im \cdot \frac{\log im}{\log im \cdot \log 100}\right)\\ \end{array} \]

Alternative 2
Error	7.5
Cost	26440

\[\begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-154}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\log im \cdot \frac{\log im}{\log im \cdot \log 100}\right)\\ \end{array} \]

Alternative 3
Error	7.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 4
Error	10.6
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5
Error	31.2
Cost	12992

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

Reproduce

herbie shell --seed 2023073 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))