math.log/1 on complex, real part

Average Accuracy: 50.3% → 100.0%

Time: 6.1s

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 50.3%

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

2. Simplified 100.0%

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

   [Start] 50.3	\[ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
   hypot-def [=>] 100.0	\[ \log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	41.8%
Cost	7629

\[\begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{-202} \lor \neg \left(im \leq 3.8 \cdot 10^{-137}\right) \land im \leq 5.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{im \cdot 0.5}{re} \cdot \frac{im}{re} - \log \left(\frac{-1}{re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Accuracy	41.9%
Cost	7628

\[\begin{array}{l} t_0 := \log \left(\frac{-1}{re}\right)\\ \mathbf{if}\;im \leq 2.7 \cdot 10^{-202}:\\ \;\;\;\;\frac{im \cdot 0.5}{re} \cdot \frac{im}{re} - t_0\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{0.5 \cdot \left(im \cdot im\right)}{re \cdot re} - t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 3
Accuracy	43.4%
Cost	7053

\[\begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{-171} \lor \neg \left(im \leq 3.8 \cdot 10^{-154}\right) \land im \leq 7.6 \cdot 10^{-114}:\\ \;\;\;\;-\log \left(\frac{-1}{re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 4
Accuracy	26.9%
Cost	6464

\[\log im \]

Alternative 5
Accuracy	2.2%
Cost	576

\[0.5 \cdot \left(re \cdot \frac{re}{im \cdot im}\right) \]

Alternative 6
Accuracy	2.5%
Cost	576

\[0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right) \]

Alternative 7
Accuracy	2.6%
Cost	576

\[0.5 \cdot \left(\frac{re}{im} \cdot \frac{re}{im}\right) \]

Reproduce

herbie shell --seed 2023133 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))