math.log10 on complex, real part

Average Accuracy: 49.7% → 99.3%

Time: 13.6s

Precision: binary64

Cost: 25984

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (/ -3.0 (log 0.1)) (log (cbrt (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 (-3.0 / log(0.1)) * log(cbrt(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 (-3.0 / Math.log(0.1)) * Math.log(Math.cbrt(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(-3.0 / log(0.1)) * log(cbrt(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[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

2. Simplified 99.1%

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

   [Start] 49.7	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   hypot-def [=>] 99.1	\[ \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}} \]

4. Simplified 99.0%

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

   [Start] 99.0	\[ \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1} \]
   distribute-lft-neg-out [=>] 99.0	\[ \color{blue}{-\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log 0.1}} \]
   associate-*r/ [=>] 99.0	\[ -\color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 1}{\log 0.1}} \]
   *-rgt-identity [=>] 99.0	\[ -\frac{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 0.1} \]
   distribute-neg-frac [=>] 99.0	\[ \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \frac{-\color{blue}{\left(\log \left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)}}{\log 0.1} \]

6. Simplified 99.0%

   \[\leadsto \frac{-\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 0.1} \]
   Proof

   [Start] 99.0	\[ \frac{-\left(\log \left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)}{\log 0.1} \]
   log-pow [=>] 99.0	\[ \frac{-\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)} + \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)}{\log 0.1} \]
   distribute-lft1-in [=>] 99.0	\[ \frac{-\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 0.1} \]
   metadata-eval [=>] 99.0	\[ \frac{-\color{blue}{3} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}{\log 0.1} \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{-3}{\log 0.1} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)} \]

8. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	19584

\[\frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

Alternative 2
Accuracy	99.0%
Cost	19520

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

Alternative 3
Accuracy	99.1%
Cost	19456

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

Alternative 4
Accuracy	43.6%
Cost	13517

\[\begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+15} \lor \neg \left(re \leq -620000000000\right) \land re \leq -2.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \]

Alternative 5
Accuracy	43.5%
Cost	13517

\[\begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\frac{-\log 0.1}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \leq -16600000000 \lor \neg \left(re \leq -2.7 \cdot 10^{-170}\right):\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \end{array} \]

Alternative 6
Accuracy	43.6%
Cost	13453

\[\begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+15} \lor \neg \left(re \leq -11600000000\right) \land re \leq -2.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 7
Accuracy	43.6%
Cost	13453

\[\begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+15} \lor \neg \left(re \leq -122000000000\right) \land re \leq -2.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 8
Accuracy	3.0%
Cost	12992

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

Alternative 9
Accuracy	27.5%
Cost	12992

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

Alternative 10
Accuracy	2.5%
Cost	7104

\[-0.5 \cdot \left(\frac{im}{re} \cdot \frac{\frac{im}{re}}{\log 0.1}\right) \]

Reproduce

herbie shell --seed 2023122 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))