math.log/2 on complex, real part

Average Error: 31.5 → 0.4

Time: 16.6s

Precision: binary64

Cost: 19456

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

↓

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

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

↓

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

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

↓

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

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

↓

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

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

↓

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

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

↓

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

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

↓

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

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

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

Derivation

1. Initial program 31.5

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]
   Proof
   (/.f64 (log.f64 (hypot.f64 re im)) (log.f64 base)): 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 base)): 138 points increase in error, 0 points decrease in error
   (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) 1)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (Rewrite<= *-inverses_binary64 (/.f64 (log.f64 base) (log.f64 base)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (*.f64 (log.f64 base) (log.f64 base)))): 25 points increase in error, 8 points decrease in error
   (/.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) 0)) (*.f64 (log.f64 base) (log.f64 base))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (Rewrite<= mul0-rgt_binary64 (*.f64 (atan2.f64 im re) 0))) (*.f64 (log.f64 base) (log.f64 base))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (*.f64 (atan2.f64 im re) 0)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (log.f64 base) (log.f64 base)) 0))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (*.f64 (atan2.f64 im re) 0)) (+.f64 (*.f64 (log.f64 base) (log.f64 base)) (Rewrite<= metadata-eval (*.f64 0 0)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	35.7
Cost	13900

\[\begin{array}{l} t_0 := \log \left(\frac{1}{base}\right)\\ \mathbf{if}\;re \leq -0.0021608679603808663:\\ \;\;\;\;\frac{-\log \left(-re\right)}{t_0}\\ \mathbf{elif}\;re \leq -1.435001089059972 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;re \leq -1.766577241023185 \cdot 10^{-108}:\\ \;\;\;\;\frac{\log \left(\frac{-0.5}{\frac{\frac{re}{im}}{im}} - re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{t_0}\\ \end{array} \]

Alternative 2
Error	35.7
Cost	13580

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;re \leq -0.0021608679603808663:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.435001089059972 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;re \leq -1.766577241023185 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log \left(\frac{1}{base}\right)}\\ \end{array} \]

Alternative 3
Error	35.7
Cost	13580

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ t_1 := \log \left(\frac{1}{base}\right)\\ \mathbf{if}\;re \leq -0.0021608679603808663:\\ \;\;\;\;\frac{-t_0}{t_1}\\ \mathbf{elif}\;re \leq -1.435001089059972 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;re \leq -1.766577241023185 \cdot 10^{-108}:\\ \;\;\;\;\frac{t_0}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{t_1}\\ \end{array} \]

Alternative 4
Error	35.7
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log base}\\ t_1 := \frac{\log im}{\log base}\\ \mathbf{if}\;re \leq -0.0021608679603808663:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.435001089059972 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.766577241023185 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	35.7
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;re \leq -0.0021608679603808663:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.435001089059972 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;re \leq -1.766577241023185 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 6
Error	46.6
Cost	12992

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

Reproduce

herbie shell --seed 2022317 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))