math.log10 on complex, real part

Average Accuracy: 49.6% → 99.5%

Time: 12.6s

Precision: binary64

Cost: 38848

Math

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

↓

\[\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (/ (pow (log 10.0) -0.5) (sqrt (log 10.0))) (log (hypot re im))))

C

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

↓

double code(double re, double im) {
	return (pow(log(10.0), -0.5) / sqrt(log(10.0))) * log(hypot(re, im));
}

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.pow(Math.log(10.0), -0.5) / Math.sqrt(Math.log(10.0))) * Math.log(Math.hypot(re, im));
}

Python

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

↓

def code(re, im):
	return (math.pow(math.log(10.0), -0.5) / math.sqrt(math.log(10.0))) * math.log(math.hypot(re, im))

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 Float64(Float64((log(10.0) ^ -0.5) / sqrt(log(10.0))) * log(hypot(re, im)))
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(10.0) ^ -0.5) / sqrt(log(10.0))) * log(hypot(re, im));
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[(N[(N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.6%

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

2. Simplified 99.1%

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

   [Start] 49.6	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 99.1	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 99.1%

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

   [Start] 99.1	\[ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]
   *-un-lft-identity [=>] 99.1	\[ \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
   add-sqr-sqrt [=>] 99.1	\[ \frac{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}} \]
   times-frac [=>] 99.1	\[ \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
   Proof

   [Start] 99.1	\[ \frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]
   clear-num [=>] 99.1	\[ \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
   un-div-inv [=>] 99.1	\[ \color{blue}{\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
   pow1/2 [=>] 99.1	\[ \frac{\frac{1}{\color{blue}{{\log 10}^{0.5}}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
   pow-flip [=>] 99.1	\[ \frac{\color{blue}{{\log 10}^{\left(-0.5\right)}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
   metadata-eval [=>] 99.1	\[ \frac{{\log 10}^{\color{blue}{-0.5}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   Proof

   [Start] 99.1	\[ \frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
   associate-/r/ [=>] 99.5	\[ \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

6. Final simplification 99.5%

   \[\leadsto \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	19456

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

Alternative 2
Accuracy	44.7%
Cost	13444

\[\begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;1 + \left(-1 - \frac{\log \left(-re\right)}{\log 0.1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 3
Accuracy	44.7%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 4
Accuracy	44.7%
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5
Accuracy	35.1%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-270}:\\ \;\;\;\;101\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6
Accuracy	5.1%
Cost	64

\[-3 \]

Alternative 7
Accuracy	11.4%
Cost	64

\[2 \]

Alternative 8
Accuracy	11.6%
Cost	64

\[3 \]

Alternative 9
Accuracy	11.7%
Cost	64

\[4 \]

Alternative 10
Accuracy	11.8%
Cost	64

\[5 \]

Alternative 11
Accuracy	12.1%
Cost	64

\[8 \]

Alternative 12
Accuracy	12.3%
Cost	64

\[10 \]

Alternative 13
Accuracy	12.3%
Cost	64

\[11 \]

Alternative 14
Accuracy	12.4%
Cost	64

\[12 \]

Alternative 15
Accuracy	12.8%
Cost	64

\[20 \]

Alternative 16
Accuracy	13.1%
Cost	64

\[28 \]

Alternative 17
Accuracy	14.4%
Cost	64

\[81 \]

Alternative 18
Accuracy	14.6%
Cost	64

\[100 \]

Alternative 19
Accuracy	14.7%
Cost	64

\[101 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))