math.log/2 on complex, real part

Average Error: 32.0 → 0.4

Time: 13.3s

Precision: binary64

Cost: 25984

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

↓

\[2 \cdot \frac{\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)}{\log base} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

↓

(FPCore (re im base)
 :precision binary64
 (* 2.0 (/ (log (sqrt (hypot re im))) (log base))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

↓

double code(double re, double im, double base) {
	return 2.0 * (log(sqrt(hypot(re, im))) / log(base));
}

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

↓

public static double code(double re, double im, double base) {
	return 2.0 * (Math.log(Math.sqrt(Math.hypot(re, im))) / Math.log(base));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

↓

def code(re, im, base):
	return 2.0 * (math.log(math.sqrt(math.hypot(re, im))) / math.log(base))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

↓

function code(re, im, base)
	return Float64(2.0 * Float64(log(sqrt(hypot(re, im))) / log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

↓

function tmp = code(re, im, base)
	tmp = 2.0 * (log(sqrt(hypot(re, im))) / log(base));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_, base_] := N[(2.0 * N[(N[Log[N[Sqrt[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.0

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

2. Simplified 0.4

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

   [Start] 32.0	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   mul0-rgt [=>] 32.0	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
   +-rgt-identity [=>] 32.0	\[ \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]
   metadata-eval [=>] 32.0	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]
   +-rgt-identity [=>] 32.0	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]
   times-frac [=>] 31.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \frac{\log base}{\log base}} \]
   *-inverses [=>] 31.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \color{blue}{1} \]
   *-rgt-identity [=>] 31.9	\[ \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]
   hypot-def [=>] 0.4	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log base} \]

3. Applied egg-rr 0.4

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

4. Applied egg-rr 0.4

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

5. Simplified 0.4

   \[\leadsto \color{blue}{2 \cdot \frac{\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)}{\log base}} \]
   Proof

   [Start] 0.4	\[ \frac{1}{\log base} \cdot \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) + \frac{1}{\log base} \cdot \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) \]
   count-2 [=>] 0.4	\[ \color{blue}{2 \cdot \left(\frac{1}{\log base} \cdot \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)\right)} \]
   *-commutative [=>] 0.4	\[ 2 \cdot \color{blue}{\left(\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) \cdot \frac{1}{\log base}\right)} \]
   associate-*r/ [=>] 0.4	\[ 2 \cdot \color{blue}{\frac{\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) \cdot 1}{\log base}} \]
   associate-/l* [=>] 0.4	\[ 2 \cdot \color{blue}{\frac{\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)}{\frac{\log base}{1}}} \]
   /-rgt-identity [=>] 0.4	\[ 2 \cdot \frac{\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)}{\color{blue}{\log base}} \]

6. Final simplification 0.4

   \[\leadsto 2 \cdot \frac{\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)}{\log base} \]

Alternatives

Alternative 1
Error	0.4
Cost	19456

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

Alternative 2
Error	32.5
Cost	13896

\[\begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \frac{0.25 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 3
Error	36.4
Cost	13453

\[\begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{-29} \lor \neg \left(im \leq 1.2 \cdot 10^{+15}\right) \land im \leq 6.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 4
Error	36.3
Cost	13453

\[\begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+14} \lor \neg \left(im \leq 3.4 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \end{array} \]

Alternative 5
Error	46.5
Cost	12992

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

Reproduce

herbie shell --seed 2023045 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))