math.log10 on complex, real part

Average Error: 32.7 → 0.5

Time: 11.7s

Precision: binary64

Cost: 38912

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\log 10}\\ \frac{1}{t_0} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{t_0} \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (log 10.0)))) (* (/ 1.0 t_0) (/ (log (hypot re im)) t_0))))

C

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

↓

double code(double re, double im) {
	double t_0 = sqrt(log(10.0));
	return (1.0 / t_0) * (log(hypot(re, im)) / t_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) {
	double t_0 = Math.sqrt(Math.log(10.0));
	return (1.0 / t_0) * (Math.log(Math.hypot(re, im)) / t_0);
}

Python

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

↓

def code(re, im):
	t_0 = math.sqrt(math.log(10.0))
	return (1.0 / t_0) * (math.log(math.hypot(re, im)) / t_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)
	t_0 = sqrt(log(10.0))
	return Float64(Float64(1.0 / t_0) * Float64(log(hypot(re, im)) / t_0))
end

MATLAB

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

↓

function tmp = code(re, im)
	t_0 = sqrt(log(10.0));
	tmp = (1.0 / t_0) * (log(hypot(re, im)) / t_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_] := Block[{t$95$0 = N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 32.7

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

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)): 144 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]

4. Final simplification 0.5

   \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]

Alternatives

Alternative 1
Error	0.6
Cost	19520

\[\frac{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1} \]

Alternative 2
Error	0.6
Cost	19456

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

Alternative 3
Error	37.0
Cost	13636

\[\begin{array}{l} \mathbf{if}\;im \leq 5.79327615567453 \cdot 10^{-213}:\\ \;\;\;\;\frac{\log \left(\frac{-0.5}{\frac{\frac{re}{im}}{im}} - re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 1.0885147118789056 \cdot 10^{-64}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \mathbf{elif}\;im \leq 3.118274195317058 \cdot 10^{-38}:\\ \;\;\;\;\frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log im}}\\ \end{array} \]

Alternative 4
Error	36.8
Cost	13516

\[\begin{array}{l} t_0 := -\frac{\log im}{\log 0.1}\\ t_1 := \frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{if}\;im \leq 5.79327615567453 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.0885147118789056 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.118274195317058 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	36.8
Cost	13516

\[\begin{array}{l} t_0 := \frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{if}\;im \leq 5.79327615567453 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.0885147118789056 \cdot 10^{-64}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \mathbf{elif}\;im \leq 3.118274195317058 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log im}}\\ \end{array} \]

Alternative 6
Error	36.8
Cost	13516

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;im \leq 5.79327615567453 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{t_0}}\\ \mathbf{elif}\;im \leq 1.0885147118789056 \cdot 10^{-64}:\\ \;\;\;\;-\frac{\log im}{\log 0.1}\\ \mathbf{elif}\;im \leq 3.118274195317058 \cdot 10^{-38}:\\ \;\;\;\;\frac{-t_0}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log im}}\\ \end{array} \]

Alternative 7
Error	36.8
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log 10}\\ t_1 := -\frac{\log im}{\log 0.1}\\ \mathbf{if}\;im \leq 5.79327615567453 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.0885147118789056 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.118274195317058 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	46.7
Cost	13056

\[-\frac{\log im}{\log 0.1} \]

Alternative 9
Error	46.7
Cost	12992

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

Alternative 10
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022295 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))