math.log10 on complex, real part

Average Error: 32.5 → 0.6

Time: 10.3s

Precision: binary64

Cost: 19456

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

↓

\[\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

↓

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

↓

double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

↓

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

↓

def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

↓

function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

↓

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

2. Simplified 0.6

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
   Proof
   (/.f64 (log.f64 (hypot.f64 re im)) (log.f64 10)): 0 points increase in error, 0 points decrease in error
   (/.f64 (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))))) (log.f64 10)): 117 points increase in error, 0 points decrease in error

3. Final simplification 0.6

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]

Alternatives

Alternative 1
Error	36.3
Cost	13900

\[\begin{array}{l} \mathbf{if}\;re \leq -9.567785861119571 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{elif}\;re \leq -1.7901068031272898 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -1.828719783023698 \cdot 10^{-125}:\\ \;\;\;\;\frac{\log \left(\frac{-0.5}{\frac{\frac{re}{im}}{im}} - re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\log im}{\mathsf{log1p}\left(9\right)} + -1\right)\\ \end{array} \]

Alternative 2
Error	36.4
Cost	13644

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -9.567785861119571 \cdot 10^{-29}:\\ \;\;\;\;\frac{-t_0}{\log 0.1}\\ \mathbf{elif}\;re \leq -1.7901068031272898 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -1.828719783023698 \cdot 10^{-125}:\\ \;\;\;\;\frac{t_0}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\log im}{\mathsf{log1p}\left(9\right)} + -1\right)\\ \end{array} \]

Alternative 3
Error	36.4
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;re \leq -9.567785861119571 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.7901068031272898 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.828719783023698 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	36.4
Cost	13452

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;re \leq -9.567785861119571 \cdot 10^{-29}:\\ \;\;\;\;\frac{-t_0}{\log 0.1}\\ \mathbf{elif}\;re \leq -1.7901068031272898 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.828719783023698 \cdot 10^{-125}:\\ \;\;\;\;\frac{t_0}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	47.0
Cost	12992

\[\frac{\log im}{\log 10} \]

Reproduce

herbie shell --seed 2022296 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))