math.log10 on complex, real part

Average Accuracy: 50.3% → 99.7%

Time: 12.8s

Precision: binary64

Cost: 38656

Math

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

↓

\[\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{-2}}\right)}\right) \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (sqrt (pow (log 10.0) -2.0)))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

↓

double code(double re, double im) {
	return log(pow(hypot(re, im), sqrt(pow(log(10.0), -2.0))));
}

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) {
	return Math.log(Math.pow(Math.hypot(re, im), Math.sqrt(Math.pow(Math.log(10.0), -2.0))));
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

↓

def code(re, im):
	return math.log(math.pow(math.hypot(re, im), math.sqrt(math.pow(math.log(10.0), -2.0))))

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

↓

function code(re, im)
	return log((hypot(re, im) ^ sqrt((log(10.0) ^ -2.0))))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

↓

function tmp = code(re, im)
	tmp = log((hypot(re, im) ^ sqrt((log(10.0) ^ -2.0))));
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_] := N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Sqrt[N[Power[N[Log[10.0], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

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

2. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
   Proof

   [Start] 50.3	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 99.1	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}\right)} \]
   Proof

   [Start] 99.1	\[ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]
   add-log-exp [=>] 99.1	\[ \color{blue}{\log \left(e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}\right)} \]
   div-inv [=>] 98.5	\[ \log \left(e^{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log 10}}}\right) \]
   exp-to-pow [=>] 98.5	\[ \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}\right)} \]

4. Applied egg-rr 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\sqrt{{\log 10}^{-2}}\right)}}\right) \]
   Proof

   [Start] 98.5	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}\right) \]
   add-sqr-sqrt [=>] 99.7	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \sqrt{\frac{1}{\log 10}}\right)}}\right) \]
   sqrt-unprod [=>] 98.5	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\sqrt{\frac{1}{\log 10} \cdot \frac{1}{\log 10}}\right)}}\right) \]
   inv-pow [=>] 98.5	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\color{blue}{{\log 10}^{-1}} \cdot \frac{1}{\log 10}}\right)}\right) \]
   metadata-eval [<=] 98.5	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{\color{blue}{\left(-1\right)}} \cdot \frac{1}{\log 10}}\right)}\right) \]
   inv-pow [=>] 98.5	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{\left(-1\right)} \cdot \color{blue}{{\log 10}^{-1}}}\right)}\right) \]
   metadata-eval [<=] 98.5	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{\left(-1\right)} \cdot {\log 10}^{\color{blue}{\left(-1\right)}}}\right)}\right) \]
   pow-sqr [=>] 99.7	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\color{blue}{{\log 10}^{\left(2 \cdot \left(-1\right)\right)}}}\right)}\right) \]
   metadata-eval [=>] 99.7	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{\left(2 \cdot \color{blue}{-1}\right)}}\right)}\right) \]
   metadata-eval [=>] 99.7	\[ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{\color{blue}{-2}}}\right)}\right) \]

5. Final simplification 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{{\log 10}^{-2}}\right)}\right) \]

Alternatives

Alternative 1
Accuracy	42.5%
Cost	19852

\[\begin{array}{l} t_0 := \frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{if}\;re \leq -2.6 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log \left({im}^{\left(\frac{-1}{\log 0.1}\right)}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	19456

\[\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]

Alternative 3
Accuracy	42.5%
Cost	13580

\[\begin{array}{l} t_0 := \frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{if}\;re \leq -3.1 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \end{array} \]

Alternative 4
Accuracy	42.5%
Cost	13516

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{if}\;re \leq -2.85 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \end{array} \]

Alternative 5
Accuracy	3.1%
Cost	12992

\[\frac{\log im}{\log 0.1} \]

Alternative 6
Accuracy	27.3%
Cost	12992

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

Reproduce

herbie shell --seed 2023131 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))