math.log10 on complex, real part

Average Accuracy: 50.4% → 99.6%

Time: 12.5s

Precision: binary64

Cost: 32320

Math

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

↓

\[\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\left(\mathsf{log1p}\left(9\right)\right)}^{-3}} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (log (hypot re im)) (cbrt (pow (log1p 9.0) -3.0))))

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(hypot(re, im)) * cbrt(pow(log1p(9.0), -3.0));
}

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.hypot(re, im)) * Math.cbrt(Math.pow(Math.log1p(9.0), -3.0));
}

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 Float64(log(hypot(re, im)) * cbrt((log1p(9.0) ^ -3.0)))
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[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Log[1 + 9.0], $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

2. Simplified 99.1%

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

   [Start] 50.4	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 99.1	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}{{\log 10}^{3}}}} \]
   Proof

   [Start] 99.1	\[ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]
   add-cbrt-cube [=>] 98.9	\[ \frac{\color{blue}{\sqrt[3]{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}}{\log 10} \]
   add-cbrt-cube [=>] 97.9	\[ \frac{\sqrt[3]{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}} \]
   cbrt-undiv [=>] 98.9	\[ \color{blue}{\sqrt[3]{\frac{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}} \]
   pow3 [=>] 98.9	\[ \sqrt[3]{\frac{\color{blue}{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}} \]
   pow3 [=>] 98.9	\[ \sqrt[3]{\frac{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}{\color{blue}{{\log 10}^{3}}}} \]

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\left(\mathsf{log1p}\left(9\right)\right)}^{-3}}} \]
   Proof

   [Start] 98.9	\[ \sqrt[3]{\frac{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}{{\log 10}^{3}}} \]
   div-inv [=>] 98.9	\[ \sqrt[3]{\color{blue}{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3} \cdot \frac{1}{{\log 10}^{3}}}} \]
   cbrt-prod [=>] 98.7	\[ \color{blue}{\sqrt[3]{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}} \cdot \sqrt[3]{\frac{1}{{\log 10}^{3}}}} \]
   rem-cbrt-cube [=>] 98.6	\[ \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \sqrt[3]{\frac{1}{{\log 10}^{3}}} \]
   pow-flip [=>] 99.6	\[ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{\color{blue}{{\log 10}^{\left(-3\right)}}} \]
   metadata-eval [<=] 99.6	\[ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log \color{blue}{\left(1 + 9\right)}}^{\left(-3\right)}} \]
   metadata-eval [<=] 99.6	\[ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log \left(1 + \color{blue}{3 \cdot 3}\right)}^{\left(-3\right)}} \]
   log1p-udef [<=] 99.6	\[ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{log1p}\left(3 \cdot 3\right)\right)}}^{\left(-3\right)}} \]
   metadata-eval [=>] 99.6	\[ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\left(\mathsf{log1p}\left(\color{blue}{9}\right)\right)}^{\left(-3\right)}} \]
   metadata-eval [=>] 99.6	\[ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\left(\mathsf{log1p}\left(9\right)\right)}^{\color{blue}{-3}}} \]

5. Final simplification 99.6%

   \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\left(\mathsf{log1p}\left(9\right)\right)}^{-3}} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	19584

\[\frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

Alternative 2
Accuracy	99.0%
Cost	19520

\[\frac{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1} \]

Alternative 3
Accuracy	99.1%
Cost	19456

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

Alternative 4
Accuracy	43.2%
Cost	13517

\[\begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{-80} \lor \neg \left(im \leq 1.4 \cdot 10^{-68}\right) \land im \leq 1.16 \cdot 10^{-33}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 5
Accuracy	27.3%
Cost	13120

\[\frac{1}{\frac{\log 10}{\log im}} \]

Alternative 6
Accuracy	27.3%
Cost	12992

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

Alternative 7
Accuracy	2.5%
Cost	7232

\[\frac{1}{\frac{re}{im} \cdot \left(\frac{re}{im} \cdot \frac{\mathsf{log1p}\left(9\right)}{0.5}\right)} \]

Alternative 8
Accuracy	2.5%
Cost	7104

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

Alternative 9
Accuracy	2.5%
Cost	7104

\[\frac{im}{re} \cdot \left(\frac{im}{re} \cdot \frac{0.5}{\mathsf{log1p}\left(9\right)}\right) \]

Reproduce

herbie shell --seed 2023153 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))