math.log10 on complex, real part

Average Error: 31.5 → 0.6

Time: 12.2s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

(FPCore (re im) :precision binary64 (/ (- (log (hypot re im))) (log 0.1)))

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)) / log(0.1);
}

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.log(0.1);
}

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.hypot(re, im)) / math.log(0.1)

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(hypot(re, im))) / log(0.1))
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)) / log(0.1);
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[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5

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

2. Simplified 0.6

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

   [Start] 31.5	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 0.6	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 0.6

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

4. Simplified 0.6

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

   [Start] 0.6	\[ \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1} \]
   associate-*r/ [=>] 0.6	\[ \color{blue}{\frac{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 1}{\log 0.1}} \]
   *-rgt-identity [=>] 0.6	\[ \frac{\color{blue}{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 0.1} \]

5. Final simplification 0.6

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

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	13380

\[\begin{array}{l} \mathbf{if}\;im \leq 3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\log 0.1}}{\frac{1}{\log im}}\\ \end{array} \]

Alternative 3
Error	35.6
Cost	13252

\[\begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{-108}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \]

Alternative 4
Error	35.6
Cost	13252

\[\begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \]

Alternative 5
Error	35.6
Cost	13252

\[\begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{im}\right)}{\log 0.1}\\ \end{array} \]

Alternative 6
Error	35.6
Cost	13188

\[\begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 7
Error	35.6
Cost	13188

\[\begin{array}{l} \mathbf{if}\;im \leq 2.25 \cdot 10^{-107}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 8
Error	62.2
Cost	12992

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

Alternative 9
Error	46.3
Cost	12992

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

Reproduce

herbie shell --seed 2023038 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))