math.log10 on complex, real part

Percentage Accurate: 51.2% → 99.0%

Time: 8.5s

Alternatives: 4

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: 51.2% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{0 - \log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (- 0.0 (log 0.1))))

C

double code(double re, double im) {
	return log(hypot(re, im)) / (0.0 - log(0.1));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / (0.0 - Math.log(0.1));
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / (0.0 - math.log(0.1))

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / Float64(0.0 - log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / (0.0 - log(0.1));
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

   1. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right) \]

   3. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{1}{10}\right)\right)\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{10}\right)\right)\right)\right) \]

   5. metadata-eval 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\frac{1}{10}\right)\right)\right) \]

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{0 - \log 0.1} \]
8. Add Preprocessing

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 54.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

Alternative 3: 26.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{0 - \log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (- 0.0 (log 0.1))))

C

double code(double re, double im) {
	return log(im) / (0.0 - log(0.1));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / (0.0d0 - log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return Math.log(im) / (0.0 - Math.log(0.1));
}

Python

def code(re, im):
	return math.log(im) / (0.0 - math.log(0.1))

Julia

function code(re, im)
	return Float64(log(im) / Float64(0.0 - log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) / (0.0 - log(0.1));
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[(0.0 - N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 25.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 25.7%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \left(\frac{1}{\frac{1}{10}}\right)\right) \]

   2. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \left(\mathsf{neg}\left(\log \frac{1}{10}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{10}\right)\right)\right)\right) \]

   4. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right)\right) \]

   5. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right) \]

   6. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{neg.f64}\left(\log \left(\frac{1}{10}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{neg.f64}\left(\log \frac{1}{10}\right)\right) \]

   8. log-lowering-log.f64 25.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\frac{1}{10}\right)\right)\right) \]

9. Applied egg-rr 25.7%

   \[\leadsto \frac{\log im}{\color{blue}{-\log 0.1}} \]

10. Final simplification 25.7%

    \[\leadsto \frac{\log im}{0 - \log 0.1} \]
11. Add Preprocessing

Alternative 4: 26.6% 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 54.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 25.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 25.7%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))