math.log10 on complex, real part

Average Error: 32.1 → 7.2

Time: 16.5s

Precision: binary64

Cost: 26952

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1.5}{\log 10}\\ \mathbf{if}\;im \leq 3.95 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{-1.5}{\log 10} + \left(-\frac{-1}{\log 10}\right)\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+114}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(t_0 - \frac{1}{\log 10}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ 1.5 (log 10.0))))
   (if (<= im 3.95e-159)
     (*
      (+ (/ -1.5 (log 10.0)) (- (/ -1.0 (log 10.0))))
      (/ -1.0 (/ 0.5 (log (- re)))))
     (if (<= im 8e+114)
       (*
        (* 2.0 (log (sqrt (+ (* re re) (* im im)))))
        (- t_0 (/ 1.0 (log 10.0))))
       (* (- (log im)) (- (/ 0.5 (log 10.0)) t_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 = 1.5 / log(10.0);
	double tmp;
	if (im <= 3.95e-159) {
		tmp = ((-1.5 / log(10.0)) + -(-1.0 / log(10.0))) * (-1.0 / (0.5 / log(-re)));
	} else if (im <= 8e+114) {
		tmp = (2.0 * log(sqrt(((re * re) + (im * im))))) * (t_0 - (1.0 / log(10.0)));
	} else {
		tmp = -log(im) * ((0.5 / log(10.0)) - t_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 = 1.5d0 / log(10.0d0)
    if (im <= 3.95d-159) then
        tmp = (((-1.5d0) / log(10.0d0)) + -((-1.0d0) / log(10.0d0))) * ((-1.0d0) / (0.5d0 / log(-re)))
    else if (im <= 8d+114) then
        tmp = (2.0d0 * log(sqrt(((re * re) + (im * im))))) * (t_0 - (1.0d0 / log(10.0d0)))
    else
        tmp = -log(im) * ((0.5d0 / log(10.0d0)) - t_0)
    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 = 1.5 / Math.log(10.0);
	double tmp;
	if (im <= 3.95e-159) {
		tmp = ((-1.5 / Math.log(10.0)) + -(-1.0 / Math.log(10.0))) * (-1.0 / (0.5 / Math.log(-re)));
	} else if (im <= 8e+114) {
		tmp = (2.0 * Math.log(Math.sqrt(((re * re) + (im * im))))) * (t_0 - (1.0 / Math.log(10.0)));
	} else {
		tmp = -Math.log(im) * ((0.5 / Math.log(10.0)) - t_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 = 1.5 / math.log(10.0)
	tmp = 0
	if im <= 3.95e-159:
		tmp = ((-1.5 / math.log(10.0)) + -(-1.0 / math.log(10.0))) * (-1.0 / (0.5 / math.log(-re)))
	elif im <= 8e+114:
		tmp = (2.0 * math.log(math.sqrt(((re * re) + (im * im))))) * (t_0 - (1.0 / math.log(10.0)))
	else:
		tmp = -math.log(im) * ((0.5 / math.log(10.0)) - t_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(1.5 / log(10.0))
	tmp = 0.0
	if (im <= 3.95e-159)
		tmp = Float64(Float64(Float64(-1.5 / log(10.0)) + Float64(-Float64(-1.0 / log(10.0)))) * Float64(-1.0 / Float64(0.5 / log(Float64(-re)))));
	elseif (im <= 8e+114)
		tmp = Float64(Float64(2.0 * log(sqrt(Float64(Float64(re * re) + Float64(im * im))))) * Float64(t_0 - Float64(1.0 / log(10.0))));
	else
		tmp = Float64(Float64(-log(im)) * Float64(Float64(0.5 / log(10.0)) - t_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 = 1.5 / log(10.0);
	tmp = 0.0;
	if (im <= 3.95e-159)
		tmp = ((-1.5 / log(10.0)) + -(-1.0 / log(10.0))) * (-1.0 / (0.5 / log(-re)));
	elseif (im <= 8e+114)
		tmp = (2.0 * log(sqrt(((re * re) + (im * im))))) * (t_0 - (1.0 / log(10.0)));
	else
		tmp = -log(im) * ((0.5 / log(10.0)) - t_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[(1.5 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.95e-159], N[(N[(N[(-1.5 / N[Log[10.0], $MachinePrecision]), $MachinePrecision] + (-N[(-1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(-1.0 / N[(0.5 / N[Log[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8e+114], N[(N[(2.0 * N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Log[im], $MachinePrecision]) * N[(N[(0.5 / N[Log[10.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.94999999999999986e-159

   1. Initial program 33.5

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

   2. Applied egg-rr 33.6

      \[\leadsto \color{blue}{\frac{-0.5}{\log 10} \cdot \frac{-1}{\frac{0.5}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   3. Taylor expanded in re around -inf 5.8

      \[\leadsto \frac{-0.5}{\log 10} \cdot \frac{-1}{\frac{0.5}{\log \color{blue}{\left(-1 \cdot re\right)}}} \]

   4. Simplified 5.8

      \[\leadsto \frac{-0.5}{\log 10} \cdot \frac{-1}{\frac{0.5}{\log \color{blue}{\left(-re\right)}}} \]
      Proof

      [Start] 5.8	\[ \frac{-0.5}{\log 10} \cdot \frac{-1}{\frac{0.5}{\log \left(-1 \cdot re\right)}} \]
      rational.json-simplify-39 [=>] 5.8	\[ \frac{-0.5}{\log 10} \cdot \frac{-1}{\frac{0.5}{\log \color{blue}{\left(re \cdot -1\right)}}} \]
      rational.json-simplify-72 [=>] 5.8	\[ \frac{-0.5}{\log 10} \cdot \frac{-1}{\frac{0.5}{\log \color{blue}{\left(-re\right)}}} \]

   5. Applied egg-rr 5.3

      \[\leadsto \color{blue}{\left(\frac{-1}{\log 10} \cdot -1 + \left(\frac{-0.5}{\log 10} + \frac{-1}{\log 10}\right)\right)} \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]

   6. Simplified 5.3

      \[\leadsto \color{blue}{\left(\frac{-1.5}{\log 10} + \left(-\frac{-1}{\log 10}\right)\right)} \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]
      Proof

      [Start] 5.3	\[ \left(\frac{-1}{\log 10} \cdot -1 + \left(\frac{-0.5}{\log 10} + \frac{-1}{\log 10}\right)\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]
      rational.json-simplify-41 [=>] 5.3	\[ \color{blue}{\left(\left(\frac{-0.5}{\log 10} + \frac{-1}{\log 10}\right) + \frac{-1}{\log 10} \cdot -1\right)} \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]
      rational.json-simplify-38 [=>] 5.3	\[ \left(\color{blue}{\frac{-0.5 + -1}{\log 10}} + \frac{-1}{\log 10} \cdot -1\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]
      metadata-eval [=>] 5.3	\[ \left(\frac{\color{blue}{-1.5}}{\log 10} + \frac{-1}{\log 10} \cdot -1\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]
      rational.json-simplify-72 [=>] 5.3	\[ \left(\frac{-1.5}{\log 10} + \color{blue}{\left(-\frac{-1}{\log 10}\right)}\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}} \]

   if 3.94999999999999986e-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 12.2

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot 3 - \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\frac{\log 10}{2}}} \]

   3. Simplified 12.4

      \[\leadsto \color{blue}{3 \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{2}{\log 10}} \]
      Proof

      [Start] 12.2	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot 3 - \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\frac{\log 10}{2}} \]
      rational.json-simplify-39 [=>] 12.2	\[ \color{blue}{3 \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} - \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\frac{\log 10}{2}} \]
      rational.json-simplify-17 [=>] 12.4	\[ 3 \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} - \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{2}{\log 10}} \]

   4. Applied egg-rr 11.7

      \[\leadsto \color{blue}{\left(2 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\frac{1.5}{\log 10} - \frac{1}{\log 10}\right)} \]

   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 4.6

      \[\leadsto \color{blue}{\left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \left(\frac{0.5}{\log 10} + \frac{1}{\log 10}\right)\right)} \]

   4. Applied egg-rr 4.6

      \[\leadsto \left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \color{blue}{\left(\frac{0.5}{\log 10} - \frac{-1}{\log 10}\right)}\right) \]

   5. Simplified 4.6

      \[\leadsto \left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \color{blue}{\frac{1.5}{\log 10}}\right) \]
      Proof

      [Start] 4.6	\[ \left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \left(\frac{0.5}{\log 10} - \frac{-1}{\log 10}\right)\right) \]
      rational.json-simplify-32 [=>] 4.6	\[ \left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \color{blue}{\frac{0.5 - -1}{\log 10}}\right) \]
      metadata-eval [=>] 4.6	\[ \left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \frac{\color{blue}{1.5}}{\log 10}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.95 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{-1.5}{\log 10} + \left(-\frac{-1}{\log 10}\right)\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+114}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\frac{1.5}{\log 10} - \frac{1}{\log 10}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \frac{1.5}{\log 10}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	20292

\[\begin{array}{l} \mathbf{if}\;im \leq 5.7 \cdot 10^{-154}:\\ \;\;\;\;\left(\frac{-1.5}{\log 10} + \left(-\frac{-1}{\log 10}\right)\right) \cdot \frac{-1}{\frac{0.5}{\log \left(-re\right)}}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \frac{1.5}{\log 10}\right)\\ \end{array} \]

Alternative 2
Error	7.5
Cost	20168

\[\begin{array}{l} \mathbf{if}\;im \leq 9.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{\log \left(\frac{-1}{re}\right)}}}{\log 10}\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \frac{1.5}{\log 10}\right)\\ \end{array} \]

Alternative 3
Error	7.4
Cost	20104

\[\begin{array}{l} \mathbf{if}\;im \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{\log \left(\frac{-1}{re}\right)}}}{\log 10}\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+115}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\left(-\log im\right) \cdot \left(\frac{0.5}{\log 10} - \frac{1.5}{\log 10}\right)\\ \end{array} \]

Alternative 4
Error	7.6
Cost	20040

\[\begin{array}{l} \mathbf{if}\;im \leq 4.4 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{\log \left(\frac{-1}{re}\right)}}}{\log 10}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 5
Error	10.7
Cost	13772

\[\begin{array}{l} t_0 := \frac{\frac{1}{\frac{-1}{\log \left(\frac{-1}{re}\right)}}}{\log 10}\\ \mathbf{if}\;im \leq 4.3 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\log im}}}{\log 10}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 6
Error	10.8
Cost	13516

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{1}{\frac{\log 10}{\log im}}\\ \mathbf{if}\;im \leq 3 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	10.8
Cost	13516

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ \mathbf{if}\;im \leq 3.4 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\log im}}}{\log 10}\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 8
Error	10.6
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;im \leq 2.7 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
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)))