math.log10 on complex, real part

Percentage Accurate: 51.9% → 99.1%

Time: 10.0s

Alternatives: 5

Speedup: 0.8×

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: 51.9% 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: 99.1% accurate, 0.8× 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 52.6%

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

   1. accelerator-lowering-hypot.f64 99.0

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

4. Applied egg-rr 99.0%

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

Alternative 2: 27.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ -1.0 (/ (log 0.1) (log (fma re (* re (/ 0.5 im)) im)))))

C

double code(double re, double im) {
	return -1.0 / (log(0.1) / log(fma(re, (re * (0.5 / im)), im)));
}

Julia

function code(re, im)
	return Float64(-1.0 / Float64(log(0.1) / log(fma(re, Float64(re * Float64(0.5 / im)), im))))
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[N[(re * N[(re * N[(0.5 / im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

3. Taylor expanded in re around 0

   \[\leadsto \frac{\log \color{blue}{\left(im + \frac{1}{2} \cdot \frac{{re}^{2}}{im}\right)}}{\log 10} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{im} + im\right)}}{\log 10} \]

   2. *-lft-identity N/A

      \[\leadsto \frac{\log \left(\frac{1}{2} \cdot \frac{\color{blue}{1 \cdot {re}^{2}}}{im} + im\right)}{\log 10} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\log \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im\right)}{\log 10} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im\right)}{\log 10} \]

   5. unpow2 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot \color{blue}{\left(re \cdot re\right)} + im\right)}{\log 10} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right) \cdot re} + im\right)}{\log 10} \]

   7. *-commutative N/A

      \[\leadsto \frac{\log \left(\color{blue}{re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right)} + im\right)}{\log 10} \]

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 29.7

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\frac{0.5}{im}}, im\right)\right)}{\log 10} \]

5. Simplified 29.7%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}}{\log 10} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{\log 10}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{\log 10}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}\right)} \]

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

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{\log 10}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}\right)}} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\log 10\right)}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}}} \]

   6. neg-log N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\log \left(\frac{1}{10}\right)}}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-1}{\frac{\log \color{blue}{\frac{1}{10}}}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\log \frac{1}{10}}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}}} \]

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

      \[\leadsto \frac{-1}{\frac{\color{blue}{\log \frac{1}{10}}}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}} \]

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

       \[\leadsto \frac{-1}{\frac{\log \frac{1}{10}}{\color{blue}{\log \left(re \cdot \left(re \cdot \frac{\frac{1}{2}}{im}\right) + im\right)}}} \]

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. /-lowering-/.f64 29.7

       \[\leadsto \frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\frac{0.5}{im}}, im\right)\right)}} \]

7. Applied egg-rr 29.7%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}}} \]
8. Add Preprocessing

Alternative 3: 27.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (fma re (* re (/ 0.5 im)) im)) (log 10.0)))

C

double code(double re, double im) {
	return log(fma(re, (re * (0.5 / im)), im)) / log(10.0);
}

Julia

function code(re, im)
	return Float64(log(fma(re, Float64(re * Float64(0.5 / im)), im)) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[Log[N[(re * N[(re * N[(0.5 / im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

3. Taylor expanded in re around 0

   \[\leadsto \frac{\log \color{blue}{\left(im + \frac{1}{2} \cdot \frac{{re}^{2}}{im}\right)}}{\log 10} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{im} + im\right)}}{\log 10} \]

   2. *-lft-identity N/A

      \[\leadsto \frac{\log \left(\frac{1}{2} \cdot \frac{\color{blue}{1 \cdot {re}^{2}}}{im} + im\right)}{\log 10} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\log \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im\right)}{\log 10} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im\right)}{\log 10} \]

   5. unpow2 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot \color{blue}{\left(re \cdot re\right)} + im\right)}{\log 10} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right) \cdot re} + im\right)}{\log 10} \]

   7. *-commutative N/A

      \[\leadsto \frac{\log \left(\color{blue}{re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right)} + im\right)}{\log 10} \]

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 29.7

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\frac{0.5}{im}}, im\right)\right)}{\log 10} \]

5. Simplified 29.7%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}}{\log 10} \]
6. Add Preprocessing

Alternative 4: 27.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ -1.0 (/ (log 0.1) (log im))))

C

double code(double re, double im) {
	return -1.0 / (log(0.1) / log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) / (log(0.1d0) / log(im))
end function

Java

public static double code(double re, double im) {
	return -1.0 / (Math.log(0.1) / Math.log(im));
}

Python

def code(re, im):
	return -1.0 / (math.log(0.1) / math.log(im))

Julia

function code(re, im)
	return Float64(-1.0 / Float64(log(0.1) / log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0 / (log(0.1) / log(im));
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

3. Taylor expanded in re around 0

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

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

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

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

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

   3. log-lowering-log.f64 29.3

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

5. Simplified 29.3%

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

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{\log 10}{\log im}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{\log 10}{\log im}\right)} \]

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

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{\log 10}{\log im}\right)}} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\log 10\right)}{\log im}}} \]

   6. neg-log N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\log \left(\frac{1}{10}\right)}}{\log im}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-1}{\frac{\log \color{blue}{\frac{1}{10}}}{\log im}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\log \frac{1}{10}}{\log im}}} \]

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

      \[\leadsto \frac{-1}{\frac{\color{blue}{\log \frac{1}{10}}}{\log im}} \]

   10. log-lowering-log.f64 29.3

       \[\leadsto \frac{-1}{\frac{\log 0.1}{\color{blue}{\log im}}} \]

7. Applied egg-rr 29.3%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log im}}} \]
8. Add Preprocessing

Alternative 5: 27.4% accurate, 1.1× 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 52.6%

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

3. Taylor expanded in re around 0

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

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

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

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

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

   3. log-lowering-log.f64 29.3

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

5. Simplified 29.3%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))