math.log/2 on complex, real part

Average Error: 31.6 → 0.4

Time: 29.6s

Precision: binary64

Cost: 19456

Math

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

↓

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

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 (/ (log (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 log(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 Math.log(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 math.log(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(log(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 = log(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[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6

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

2. Simplified 0.5

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

3. Applied egg-rr 0.4

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

4. Taylor expanded in base around 0 31.5

   \[\leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}} \]

5. Simplified 0.4

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

Alternatives

Alternative 1
Error	25.5
Cost	13972

\[\begin{array}{l} t_0 := -\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ t_1 := \log \left(\frac{-1}{im}\right)\\ \mathbf{if}\;re \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{-307}:\\ \;\;\;\;-\frac{t_1}{\log base}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-117}:\\ \;\;\;\;-\frac{1}{\log base} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{re}\right)}{\log base}\\ \end{array} \]

Alternative 2
Error	25.4
Cost	13844

\[\begin{array}{l} t_0 := -\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ t_1 := -\frac{\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{if}\;re \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \leq -8.4 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{re}\right)}{\log base}\\ \end{array} \]

Alternative 3
Error	25.8
Cost	13712

\[\begin{array}{l} t_0 := -\frac{\log \left(\frac{-1}{re}\right)}{\log base}\\ t_1 := -\frac{\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.2 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\log \left(\frac{1}{im}\right)}{\log base}\\ \end{array} \]

Alternative 4
Error	36.0
Cost	13580

\[\begin{array}{l} t_0 := -\frac{\log \left(\frac{-1}{re}\right)}{\log base}\\ t_1 := -\frac{\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -9.8 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.9 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	46.3
Cost	13184

\[-\frac{\log \left(\frac{-1}{im}\right)}{\log base} \]

Reproduce

herbie shell --seed 2023010 
(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))))