math.log10 on complex, real part

Average Error: 31.8 → 0.3

Time: 12.1s

Precision: binary64

Cost: 19712

Math

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

↓

\[\frac{-0.3333333333333333}{\frac{\log 0.1}{3}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (/ -0.3333333333333333 (/ (log 0.1) 3.0)) (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 (-0.3333333333333333 / (log(0.1) / 3.0)) * 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 (-0.3333333333333333 / (Math.log(0.1) / 3.0)) * 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 (-0.3333333333333333 / (math.log(0.1) / 3.0)) * 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(-0.3333333333333333 / Float64(log(0.1) / 3.0)) * 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 = (-0.3333333333333333 / (log(0.1) / 3.0)) * 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[(-0.3333333333333333 / N[(N[Log[0.1], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8

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

2. Simplified 0.6

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

   [Start] 31.8	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 0.6	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 0.6

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

4. Applied egg-rr 0.6

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

5. Simplified 0.3

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

   [Start] 0.6	\[ \frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
   associate-/r/ [=>] 0.3	\[ \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

6. Applied egg-rr 0.9

   \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log \left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right) + \frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)} \]

7. Simplified 0.6

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

   [Start] 0.9	\[ \frac{1}{\log 10} \cdot \log \left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right) + \frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \]
   *-commutative [<=] 0.9	\[ \color{blue}{\log \left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right) \cdot \frac{1}{\log 10}} + \frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \]
   log-pow [=>] 0.9	\[ \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)} \cdot \frac{1}{\log 10} + \frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \]
   associate-*l* [=>] 0.9	\[ \color{blue}{2 \cdot \left(\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \frac{1}{\log 10}\right)} + \frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \]
   *-commutative [<=] 0.9	\[ 2 \cdot \color{blue}{\left(\frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)} + \frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \]
   distribute-lft1-in [=>] 0.9	\[ \color{blue}{\left(2 + 1\right) \cdot \left(\frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)} \]
   metadata-eval [=>] 0.9	\[ \color{blue}{3} \cdot \left(\frac{1}{\log 10} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right) \]
   associate-*l/ [=>] 0.6	\[ 3 \cdot \color{blue}{\frac{1 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}{\log 10}} \]
   *-lft-identity [=>] 0.6	\[ 3 \cdot \frac{\color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

8. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\frac{\log 0.1}{3}}} \]

9. Simplified 0.3

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{\log 0.1}{3}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   Proof

   [Start] 0.4	\[ \frac{-0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\frac{\log 0.1}{3}} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{-0.3333333333333333}{\frac{\frac{\log 0.1}{3}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
   associate-/r/ [=>] 0.3	\[ \color{blue}{\frac{-0.3333333333333333}{\frac{\log 0.1}{3}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

10. Final simplification 0.3

    \[\leadsto \frac{-0.3333333333333333}{\frac{\log 0.1}{3}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	19456

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

Alternative 2
Error	35.6
Cost	13781

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{if}\;re \leq -2.25 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \mathbf{elif}\;re \leq -2.8 \cdot 10^{-98} \lor \neg \left(re \leq -1.7 \cdot 10^{-127}\right) \land re \leq -3.1 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 3
Error	46.5
Cost	12992

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

Alternative 4
Error	62.8
Cost	7104

\[-0.5 \cdot \frac{re \cdot re}{\log 0.1 \cdot \left(im \cdot im\right)} \]

Alternative 5
Error	62.7
Cost	7104

\[re \cdot \left(0.5 \cdot \frac{re}{\log 0.1 \cdot \left(im \cdot im\right)}\right) \]

Alternative 6
Error	62.4
Cost	7104

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

Alternative 7
Error	62.4
Cost	7104

\[\frac{\frac{re}{\log 10}}{im} \cdot \frac{re \cdot 0.5}{im} \]

Reproduce

herbie shell --seed 2023045 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))