math.log10 on complex, real part

Average Error: 32.4 → 0.5

Time: 11.5s

Precision: binary64

Cost: 38848

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (/ (pow (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((log(10.0) ^ -0.5) / Float64(sqrt(log(10.0)) / 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 = (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[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision] / N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.4

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

2. Simplified 0.6

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
   Proof
   (/.f64 (log.f64 (hypot.f64 re im)) (log.f64 10)): 0 points increase in error, 0 points decrease in error
   (/.f64 (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))))) (log.f64 10)): 126 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.5

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

5. Final simplification 0.5

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

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	36.0
Cost	13316

\[\begin{array}{l} \mathbf{if}\;re \leq -1.8802694785784264 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log im}}\\ \end{array} \]

Alternative 4
Error	36.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;re \leq -1.8802694785784264 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log im}}\\ \end{array} \]

Alternative 5
Error	46.3
Cost	13120

\[\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log im}} \]

Alternative 6
Error	46.3
Cost	13056

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

Alternative 7
Error	46.3
Cost	12992

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

Reproduce

herbie shell --seed 2022318 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))