math.log/1 on complex, real part

Average Error: 31.9 → 0

Time: 2.3s

Precision: binary64

Cost: 12928

Math

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

↓

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

FPCore

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

↓

(FPCore (re im) :precision binary64 (log (hypot re im)))

C

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

↓

double code(double re, double im) {
	return log(hypot(re, im));
}

Java

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

↓

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im));
}

Python

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

↓

def code(re, im):
	return math.log(math.hypot(re, im))

Julia

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

↓

function code(re, im)
	return log(hypot(re, im))
end

MATLAB

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

↓

function tmp = code(re, im)
	tmp = log(hypot(re, im));
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

↓

code[re_, im_] := N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 31.9

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

2. Simplified 0

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

3. Final simplification 0

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

Alternatives

Alternative 1
Error	35.9
Cost	7108

\[\begin{array}{l} \mathbf{if}\;im \leq 1.6189431069546294 \cdot 10^{-158}:\\ \;\;\;\;\log \left(im \cdot \frac{im \cdot -0.5}{re} - re\right)\\ \mathbf{elif}\;im \leq 6.720785491000028 \cdot 10^{-141}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;im \leq 5.4594895982152474 \cdot 10^{-83}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Error	35.7
Cost	6924

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 1.6189431069546294 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.720785491000028 \cdot 10^{-141}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;im \leq 5.4594895982152474 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 3
Error	47.0
Cost	6464

\[\log im \]

Reproduce

herbie shell --seed 2022228 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))