math.log10 on complex, real part

Average Accuracy: 49.1% → 99.1%

Time: 11.1s

Precision: binary64

Cost: 38912

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\log 10}\\ \frac{1}{t_0} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{t_0} \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (log 10.0)))) (* (/ 1.0 t_0) (/ (log (hypot re im)) t_0))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

↓

double code(double re, double im) {
	double t_0 = sqrt(log(10.0));
	return (1.0 / t_0) * (log(hypot(re, im)) / t_0);
}

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) {
	double t_0 = Math.sqrt(Math.log(10.0));
	return (1.0 / t_0) * (Math.log(Math.hypot(re, im)) / t_0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

↓

def code(re, im):
	t_0 = math.sqrt(math.log(10.0))
	return (1.0 / t_0) * (math.log(math.hypot(re, im)) / t_0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

↓

function code(re, im)
	t_0 = sqrt(log(10.0))
	return Float64(Float64(1.0 / t_0) * Float64(log(hypot(re, im)) / t_0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

↓

function tmp = code(re, im)
	t_0 = sqrt(log(10.0));
	tmp = (1.0 / t_0) * (log(hypot(re, im)) / t_0);
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_] := Block[{t$95$0 = N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.8%

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

2. Simplified 99.1%

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

   [Start] 51.8	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 99.1	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 99.2%

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

   [Start] 99.1	\[ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]
   *-un-lft-identity [=>] 99.1	\[ \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
   add-sqr-sqrt [=>] 99.1	\[ \frac{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}} \]
   times-frac [=>] 99.2	\[ \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]

4. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	19456

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

Alternative 2
Accuracy	43.8%
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{-70}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 3
Accuracy	43.8%
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 4
Accuracy	34.6%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt[3]{1000000}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5
Accuracy	5.0%
Cost	6464

\[\sqrt[3]{-6} \]

Alternative 6
Accuracy	11.2%
Cost	6464

\[\sqrt[3]{1.6666666666666667} \]

Alternative 7
Accuracy	11.5%
Cost	6464

\[\sqrt[3]{18} \]

Alternative 8
Accuracy	12.2%
Cost	6464

\[\sqrt[3]{729} \]

Alternative 9
Accuracy	12.3%
Cost	6464

\[\sqrt[3]{1001} \]

Alternative 10
Accuracy	12.8%
Cost	6464

\[\sqrt[3]{10000} \]

Alternative 11
Accuracy	13.1%
Cost	6464

\[\sqrt[3]{19683} \]

Alternative 12
Accuracy	14.7%
Cost	6464

\[\sqrt[3]{1000000} \]

Reproduce

herbie shell --seed 2023157 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))