math.log10 on complex, real part

Average Error: 32.2 → 8.5

Time: 37.6s

Precision: binary64

Cost: 39428

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\log 10 \cdot \frac{t_0}{\frac{\log 10}{\frac{t_0}{\log 10 \cdot t_0}}}\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16200000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log im \cdot \frac{1}{\log im}}\\ \end{array} \]

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 4.1e-209)
     (* (log 10.0) (/ t_0 (/ (log 10.0) (/ t_0 (* (log 10.0) t_0)))))
     (if (<= im 8.6e-176)
       (/ 1.0 (/ (log 10.0) (log im)))
       (if (<= im 16200000.0)
         (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0))
         (/ (/ (log im) (log 10.0)) (* (log im) (/ 1.0 (log im)))))))))

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 <= 4.1e-209) {
		tmp = log(10.0) * (t_0 / (log(10.0) / (t_0 / (log(10.0) * t_0))));
	} else if (im <= 8.6e-176) {
		tmp = 1.0 / (log(10.0) / log(im));
	} else if (im <= 16200000.0) {
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	} else {
		tmp = (log(im) / log(10.0)) / (log(im) * (1.0 / log(im)));
	}
	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 <= 4.1d-209) then
        tmp = log(10.0d0) * (t_0 / (log(10.0d0) / (t_0 / (log(10.0d0) * t_0))))
    else if (im <= 8.6d-176) then
        tmp = 1.0d0 / (log(10.0d0) / log(im))
    else if (im <= 16200000.0d0) then
        tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
    else
        tmp = (log(im) / log(10.0d0)) / (log(im) * (1.0d0 / log(im)))
    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 <= 4.1e-209) {
		tmp = Math.log(10.0) * (t_0 / (Math.log(10.0) / (t_0 / (Math.log(10.0) * t_0))));
	} else if (im <= 8.6e-176) {
		tmp = 1.0 / (Math.log(10.0) / Math.log(im));
	} else if (im <= 16200000.0) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
	} else {
		tmp = (Math.log(im) / Math.log(10.0)) / (Math.log(im) * (1.0 / Math.log(im)));
	}
	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 <= 4.1e-209:
		tmp = math.log(10.0) * (t_0 / (math.log(10.0) / (t_0 / (math.log(10.0) * t_0))))
	elif im <= 8.6e-176:
		tmp = 1.0 / (math.log(10.0) / math.log(im))
	elif im <= 16200000.0:
		tmp = math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
	else:
		tmp = (math.log(im) / math.log(10.0)) / (math.log(im) * (1.0 / math.log(im)))
	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 <= 4.1e-209)
		tmp = Float64(log(10.0) * Float64(t_0 / Float64(log(10.0) / Float64(t_0 / Float64(log(10.0) * t_0)))));
	elseif (im <= 8.6e-176)
		tmp = Float64(1.0 / Float64(log(10.0) / log(im)));
	elseif (im <= 16200000.0)
		tmp = Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0));
	else
		tmp = Float64(Float64(log(im) / log(10.0)) / Float64(log(im) * Float64(1.0 / log(im))));
	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 <= 4.1e-209)
		tmp = log(10.0) * (t_0 / (log(10.0) / (t_0 / (log(10.0) * t_0))));
	elseif (im <= 8.6e-176)
		tmp = 1.0 / (log(10.0) / log(im));
	elseif (im <= 16200000.0)
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	else
		tmp = (log(im) / log(10.0)) / (log(im) * (1.0 / log(im)));
	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, 4.1e-209], 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, 8.6e-176], N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 16200000.0], 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[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision] / N[(N[Log[im], $MachinePrecision] * N[(1.0 / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 4.09999999999999977e-209

   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 2.6

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

   3. Simplified 2.6

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

      [Start] 2.6	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational.json-simplify-2 [=>] 2.6	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational.json-simplify-9 [=>] 2.6	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

   4. Applied egg-rr 2.8

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

   5. Simplified 2.6

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

      [Start] 2.8	\[ \log 10 \cdot \frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\log \left(-re\right)} \cdot \frac{\log 10}{\log \left(-re\right)}} \]
      rational.json-simplify-46 [=>] 2.8	\[ \log 10 \cdot \color{blue}{\frac{\frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\log \left(-re\right)}}}{\frac{\log 10}{\log \left(-re\right)}}} \]
      rational.json-simplify-61 [=>] 2.7	\[ \log 10 \cdot \color{blue}{\frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\log \left(-re\right)}}}}} \]
      rational.json-simplify-7 [<=] 2.7	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{\log \left(-re\right)}}{\frac{\log 10}{\color{blue}{\frac{\log \left(-re\right)}{1}}}}}} \]
      rational.json-simplify-61 [=>] 2.8	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{1}{\log \left(-re\right)}}{\color{blue}{\frac{1}{\frac{\log \left(-re\right)}{\log 10}}}}}} \]
      rational.json-simplify-61 [<=] 2.7	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\color{blue}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\frac{1}{\frac{1}{\log \left(-re\right)}}}}}} \]
      rational.json-simplify-61 [=>] 2.7	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\color{blue}{\frac{\log \left(-re\right)}{\frac{1}{1}}}}}} \]
      metadata-eval [=>] 2.7	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\frac{\log \left(-re\right)}{\color{blue}{1}}}}} \]
      rational.json-simplify-7 [=>] 2.7	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\frac{\log \left(-re\right)}{\log 10}}{\color{blue}{\log \left(-re\right)}}}} \]
      rational.json-simplify-47 [=>] 2.6	\[ \log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\color{blue}{\frac{\log \left(-re\right)}{\log 10 \cdot \log \left(-re\right)}}}} \]

   if 4.09999999999999977e-209 < im < 8.60000000000000025e-176

   1. Initial program 36.0

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

   2. Taylor expanded in re around 0 41.0

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

   3. Applied egg-rr 41.1

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

   4. Applied egg-rr 41.0

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

   if 8.60000000000000025e-176 < im < 1.62e7

   1. Initial program 15.0

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

   if 1.62e7 < im

   1. Initial program 40.1

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

   2. Taylor expanded in re around 0 7.8

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

   3. Applied egg-rr 8.1

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

   4. Applied egg-rr 7.8

      \[\leadsto \color{blue}{\frac{\frac{\log im}{\log 10}}{\log im \cdot \frac{1}{\log im}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\log 10 \cdot \frac{\log \left(-re\right)}{\frac{\log 10}{\frac{\log \left(-re\right)}{\log 10 \cdot \log \left(-re\right)}}}\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16200000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log im \cdot \frac{1}{\log im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	26572

\[\begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16200000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log im} \cdot \frac{\log im \cdot \log im}{\log 10}\\ \end{array} \]

Alternative 2
Error	8.5
Cost	26572

\[\begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16000000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log im \cdot \frac{1}{\log im}}\\ \end{array} \]

Alternative 3
Error	8.5
Cost	26572

\[\begin{array}{l} \mathbf{if}\;im \leq 1.45 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{{\log \left(\frac{-1}{re}\right)}^{2}}{\log 10}}{\log \left(-re\right)}\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16200000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log im \cdot \frac{1}{\log im}}\\ \end{array} \]

Alternative 4
Error	8.4
Cost	26572

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{t_0}{t_0 \cdot \log 10}}{\frac{1}{t_0}}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16000000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log im}{\log 10}}{\log im \cdot \frac{1}{\log im}}\\ \end{array} \]

Alternative 5
Error	8.4
Cost	26380

\[\begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 6 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16200000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\log im}^{2}}{\log 10}}{\log im}\\ \end{array} \]

Alternative 6
Error	8.5
Cost	20172

\[\begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{elif}\;im \leq 16200000:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) + -1\\ \end{array} \]

Alternative 7
Error	11.3
Cost	14036

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) + -1\\ \end{array} \]

Alternative 8
Error	11.3
Cost	13844

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{im}\right)}{-\log 10}\\ \end{array} \]

Alternative 9
Error	11.3
Cost	13716

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;im \leq 3.2 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	11.3
Cost	13716

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{if}\;im \leq 4.1 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.22 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 11
Error	31.0
Cost	12992

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

Reproduce

herbie shell --seed 2023064 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))