math.log10 on complex, real part

Percentage Accurate: 52.3% → 99.1%

Time: 9.7s

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: 52.3% 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 54.2%

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

   1. lower-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: 26.7% 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 54.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. lower-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. lower-*.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. lower-/.f64 25.1

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

5. Simplified 25.1%

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

Alternative 3: 26.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\log 10 \cdot \frac{1}{\log im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (* (log 10.0) (/ 1.0 (log im)))))

C

double code(double re, double im) {
	return 1.0 / (log(10.0) * (1.0 / log(im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (log(10.0d0) * (1.0d0 / log(im)))
end function

Java

public static double code(double re, double im) {
	return 1.0 / (Math.log(10.0) * (1.0 / Math.log(im)));
}

Python

def code(re, im):
	return 1.0 / (math.log(10.0) * (1.0 / math.log(im)))

Julia

function code(re, im)
	return Float64(1.0 / Float64(log(10.0) * Float64(1.0 / log(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / (log(10.0) * (1.0 / log(im)));
end

Wolfram

code[re_, im_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] * N[(1.0 / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 25.6

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

5. Simplified 25.6%

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

   1. lift-log.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. clear-num N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 2}}{\frac{\log 10}{\log im}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 2}{\frac{\log 10}{\log im}}} \]

   6. metadata-eval N/A

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

   7. lower-/.f64 25.6

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

7. Applied egg-rr 25.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

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

   2. neg-log N/A

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

   3. lift-log.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 25.5

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

   10. lift-neg.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. metadata-eval N/A

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

   14. lift-log.f64 25.6

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

9. Applied egg-rr 25.6%

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

10. Final simplification 25.6%

    \[\leadsto \frac{1}{\log 10 \cdot \frac{1}{\log im}} \]
11. Add Preprocessing

Alternative 4: 26.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (/ (log 10.0) (log im))))

C

double code(double re, double im) {
	return 1.0 / (log(10.0) / log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (log(10.0d0) / log(im))
end function

Java

public static double code(double re, double im) {
	return 1.0 / (Math.log(10.0) / Math.log(im));
}

Python

def code(re, im):
	return 1.0 / (math.log(10.0) / math.log(im))

Julia

function code(re, im)
	return Float64(1.0 / Float64(log(10.0) / log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / (log(10.0) / log(im));
end

Wolfram

code[re_, im_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 25.6

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

5. Simplified 25.6%

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

   1. lift-log.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. clear-num N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 2}}{\frac{\log 10}{\log im}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot 2}{\frac{\log 10}{\log im}}} \]

   6. metadata-eval N/A

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

   7. lower-/.f64 25.6

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

7. Applied egg-rr 25.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
8. Add Preprocessing

Alternative 5: 26.8% 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 54.2%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 25.6

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

5. Simplified 25.6%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))