math.log/2 on complex, real part

Average Error: 32.4 → 0.4

Time: 6.4s

Precision: binary64

Math

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

↓

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

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 (pow (sqrt (hypot re im)) 2.0)) (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(pow(sqrt(hypot(re, im)), 2.0)) / 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.pow(Math.sqrt(Math.hypot(re, im)), 2.0)) / 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.pow(math.sqrt(math.hypot(re, im)), 2.0)) / 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((sqrt(hypot(re, im)) ^ 2.0)) / 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((sqrt(hypot(re, im)) ^ 2.0)) / 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[Power[N[Sqrt[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 32.4

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

2. Simplified 0.4

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

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022150 
(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))))