math.log/1 on complex, real part

Average Error: 32.4 → 0

Time: 2.4s

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 32.4

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

2. Simplified 0

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   Proof
   (log.f64 (hypot.f64 re im)): 0 points increase in error, 0 points decrease in error
   (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))))): 144 points increase in error, 0 points decrease in error

3. Final simplification 0

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

Alternatives

Alternative 1
Error	36.6
Cost	6924

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 5.79327615567453 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.0885147118789056 \cdot 10^{-64}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;im \leq 3.118274195317058 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Error	46.5
Cost	6464

\[\log im \]

Reproduce

herbie shell --seed 2022295 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))