math.log10 on complex, real part

Average Error: 31.9 → 0.3

Time: 9.8s

Precision: binary64

Cost: 38848

Math

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

↓

\[\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (/ (pow (log 10.0) -0.5) (sqrt (log 10.0))) (log (hypot re im))))

C

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

↓

double code(double re, double im) {
	return (pow(log(10.0), -0.5) / sqrt(log(10.0))) * log(hypot(re, im));
}

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.pow(Math.log(10.0), -0.5) / Math.sqrt(Math.log(10.0))) * Math.log(Math.hypot(re, im));
}

Python

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

↓

def code(re, im):
	return (math.pow(math.log(10.0), -0.5) / math.sqrt(math.log(10.0))) * math.log(math.hypot(re, im))

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(Float64((log(10.0) ^ -0.5) / sqrt(log(10.0))) * log(hypot(re, im)))
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(10.0) ^ -0.5) / sqrt(log(10.0))) * log(hypot(re, im));
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[(N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9

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

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

3. Applied egg-rr 0.6

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

4. Applied egg-rr 0.6

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

5. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   Proof
   (*.f64 (/.f64 (pow.f64 (log.f64 10) -1/2) (sqrt.f64 (log.f64 10))) (log.f64 (hypot.f64 re im))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 (log.f64 10) -1/2) (/.f64 (sqrt.f64 (log.f64 10)) (log.f64 (hypot.f64 re im))))): 83 points increase in error, 14 points decrease in error

6. Final simplification 0.3

   \[\leadsto \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	26048

\[3 \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \]

Alternative 2
Error	0.4
Cost	19584

\[3 \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\mathsf{log1p}\left(999\right)} \]

Alternative 3
Error	0.6
Cost	19520

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

Alternative 4
Error	0.6
Cost	19456

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

Alternative 5
Error	36.1
Cost	13516

\[\begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\\ \mathbf{elif}\;re \leq -1.56 \cdot 10^{-83}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -5.1 \cdot 10^{-110}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 6
Error	36.1
Cost	13516

\[\begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log \left(-re\right)}}\\ \mathbf{elif}\;re \leq -3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 7
Error	36.1
Cost	13516

\[\begin{array}{l} t_0 := \log \left(\frac{-1}{re}\right)\\ \mathbf{if}\;re \leq -6.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{\log 10 \cdot \frac{-1}{t_0}}\\ \mathbf{elif}\;re \leq -2.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{t_0}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 8
Error	36.1
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\\ t_1 := \frac{\log im}{\log 10}\\ \mathbf{if}\;re \leq -2.3 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.6 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.7 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	36.1
Cost	13452

\[\begin{array}{l} t_0 := \frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\\ \mathbf{if}\;re \leq -2 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 10
Error	62.0
Cost	12992

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

Alternative 11
Error	46.6
Cost	12992

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

Reproduce

herbie shell --seed 2022328 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))