math.log10 on complex, real part

Average Error: 32.1 → 0.3

Time: 7.2s

Precision: binary64

Cost: 32320

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 32.1

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

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)): 127 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.9

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

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	36.2
Cost	13516

\[\begin{array}{l} t_0 := \frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{if}\;im \leq 1.04 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 4
Error	36.2
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ \mathbf{if}\;im \leq 3.1 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5
Error	35.8
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -7.1 \cdot 10^{-152}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6
Error	62.1
Cost	12992

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

Alternative 7
Error	46.9
Cost	12992

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

Reproduce

herbie shell --seed 2022329 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))