math.log10 on complex, real part

Average Error: 31.9 → 0.3

Time: 11.2s

Precision: binary64

Cost: 38976

Math

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

↓

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

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) / Float64(-sqrt(log(10.0)))) * Float64(-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 31.9

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

2. Applied egg-rr 0.5

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

3. Simplified 0.8

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

   [Start] 0.5	\[ \frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]
   associate-*l/ [=>] 0.8	\[ \color{blue}{\frac{1 \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}{\sqrt{\log 10}}} \]
   *-lft-identity [=>] 0.8	\[ \frac{\color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \]

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.6
Cost	19520

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

Alternative 2
Error	0.6
Cost	19456

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

Alternative 3
Error	35.5
Cost	13700

\[\begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-68}:\\ \;\;\;\;-\frac{\log \left(im \cdot \left(im \cdot \frac{-0.5}{re}\right) - re\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 4
Error	35.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5
Error	35.3
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6
Error	62.0
Cost	12992

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

Alternative 7
Error	46.4
Cost	12992

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

Reproduce

herbie shell --seed 2023016 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))