math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.7%

Time: 7.1s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

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

Python

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

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * 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]

Initial Program: 51.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

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

Python

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

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * 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]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (pow (pow (log 10.0) -0.5) 2.0))))

C

double code(double re, double im) {
	return log(pow(hypot(re, im), pow(pow(log(10.0), -0.5), 2.0)));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(Math.hypot(re, im), Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0)));
}

Python

def code(re, im):
	return math.log(math.pow(math.hypot(re, im), math.pow(math.pow(math.log(10.0), -0.5), 2.0)))

Julia

function code(re, im)
	return log((hypot(re, im) ^ ((log(10.0) ^ -0.5) ^ 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = log((hypot(re, im) ^ ((log(10.0) ^ -0.5) ^ 2.0)));
end

Wolfram

code[re_, im_] := N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. Step-by-step derivation

   1. add-log-exp 99.1%

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

   2. div-inv 98.5%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 99.0%

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

   7. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1}\right)}\right)} \]
6. Step-by-step derivation

   1. metadata-eval 99.0%

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

   2. metadata-eval 99.0%

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

   3. neg-log 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. add-sqr-sqrt 98.5%

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

   7. frac-times 99.7%

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

   8. pow2 99.7%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left(\frac{1}{\sqrt{\log 10}}\right)}^{2}\right)}}\right) \]

   9. inv-pow 99.7%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\color{blue}{\left({\left(\sqrt{\log 10}\right)}^{-1}\right)}}^{2}\right)}\right) \]

   10. sqrt-pow2 99.7%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\color{blue}{\left({\log 10}^{\left(\frac{-1}{2}\right)}\right)}}^{2}\right)}\right) \]

   11. metadata-eval 99.7%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{\color{blue}{-0.5}}\right)}^{2}\right)}\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]

8. Final simplification 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \]

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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.hypot(re, im)) / Math.log(10.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 27.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / log(10.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 27.4%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]

5. Final simplification 27.4%

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

Reproduce

herbie shell --seed 2023222 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))