math.log10 on complex, real part

Percentage Accurate: 51.9% → 99.0%

Time: 7.2s

Alternatives: 9

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: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 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 54.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 3: 25.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 99.1

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-log.f64 23.7

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

7. Applied rewrites 23.7%

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

8. Final simplification 23.7%

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

Alternative 4: 25.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 23.7%

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

Alternative 5: 27.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg-rev N/A

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

   3. lift-log.f64 N/A

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

   4. neg-log N/A

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

   5. metadata-eval N/A

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

   6. lift-log.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 25.4

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

7. Applied rewrites 25.4%

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

8. Final simplification 25.4%

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

Alternative 6: 27.1% 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 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

Alternative 7: 3.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (((re / im) * re) * (-0.5d0)) / (im * log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return (((re / im) * re) * -0.5) / (im * Math.log(0.1));
}

Python

def code(re, im):
	return (((re / im) * re) * -0.5) / (im * math.log(0.1))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (((re / im) * re) * -0.5) / (im * log(0.1));
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 23.7%

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

9. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log 10}} \]
10. Step-by-step derivation

11. Applied rewrites 3.3%

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

13. Applied rewrites 3.3%

    \[\leadsto \frac{\left(\frac{re}{im} \cdot re\right) \cdot -0.5}{im \cdot \color{blue}{\log 0.1}} \]
14. Add Preprocessing

Alternative 8: 3.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 / log(10.0d0)) * (re * (re / (im * im)))
end function

Java

public static double code(double re, double im) {
	return (0.5 / Math.log(10.0)) * (re * (re / (im * im)));
}

Python

def code(re, im):
	return (0.5 / math.log(10.0)) * (re * (re / (im * im)))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 / log(10.0)) * (re * (re / (im * im)));
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 23.7%

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

9. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log 10}} \]
10. Step-by-step derivation

11. Applied rewrites 3.3%

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

13. Applied rewrites 2.9%

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

Alternative 9: 2.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return ((0.5 * 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 = ((0.5d0 * re) * re) / ((im * im) * log(10.0d0))
end function

Java

public static double code(double re, double im) {
	return ((0.5 * re) * re) / ((im * im) * Math.log(10.0));
}

Python

def code(re, im):
	return ((0.5 * re) * re) / ((im * im) * math.log(10.0))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = ((0.5 * re) * re) / ((im * im) * log(10.0));
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 23.7%

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

9. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log 10}} \]
10. Step-by-step derivation

11. Applied rewrites 3.3%

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

13. Applied rewrites 2.6%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))