math.log10 on complex, real part

Average Error: 49.5% → 99.5%

Time: 20.7s

Precision: binary64

Cost: 32448.00

Math

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

↓

\[{\left(\frac{1}{\sqrt{\log 10}}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (pow (/ 1.0 (sqrt (log 10.0))) 2.0) (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((1.0 / sqrt(log(10.0))), 2.0) * 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((1.0 / Math.sqrt(Math.log(10.0))), 2.0) * 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((1.0 / math.sqrt(math.log(10.0))), 2.0) * 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(1.0 / sqrt(log(10.0))) ^ 2.0) * 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 = ((1.0 / sqrt(log(10.0))) ^ 2.0) * 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[Power[N[(1.0 / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5

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

2. Simplified 99.1

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

   [Start] 49.5	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 99.1	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 98.0

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

4. Applied egg-rr 97.8

   \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}}}\right)}^{3} \]

5. Simplified 97.8

   \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1}}}\right)}^{3} \]
   Proof

   [Start] 97.8	\[ {\left(\sqrt[3]{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}}\right)}^{3} \]
   associate-*r/ [=>] 97.8	\[ {\left(\sqrt[3]{\color{blue}{\frac{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 1}{\log 0.1}}}\right)}^{3} \]
   *-rgt-identity [=>] 97.8	\[ {\left(\sqrt[3]{\frac{\color{blue}{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 0.1}}\right)}^{3} \]

6. Applied egg-rr 99.0

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

7. Applied egg-rr 99.4

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

8. Simplified 99.5

   \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{\log 10}}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   Proof

   [Start] 99.4	\[ \frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]
   associate-*r* [=>] 99.5	\[ \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   unpow2 [<=] 99.5	\[ \color{blue}{{\left(\frac{1}{\sqrt{\log 10}}\right)}^{2}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

9. Final simplification 99.5

   \[\leadsto {\left(\frac{1}{\sqrt{\log 10}}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

Alternatives

Alternative 1
Error	99.1%
Cost	19456.00

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

Alternative 2
Error	42.8%
Cost	13772.00

\[\begin{array}{l} t_0 := \log \left(\frac{-1}{re}\right)\\ \mathbf{if}\;im \leq 6 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_0}{\log 0.1}\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;im \leq 0.062:\\ \;\;\;\;\frac{\frac{-1}{\log 0.1}}{\frac{-1}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 3
Error	42.8%
Cost	13516.00

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{if}\;im \leq 6 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;im \leq 0.02:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 4
Error	27.0%
Cost	13120.00

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

Alternative 5
Error	3.0%
Cost	12992.00

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

Alternative 6
Error	27.0%
Cost	12992.00

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

Reproduce

herbie shell --seed 2023093 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))