math.log10 on complex, real part

Percentage Accurate: 51.0% → 99.0%

Time: 9.5s

Alternatives: 8

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: 51.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 rewrites 99.0%

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

   1. metadata-eval N/A

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

   2. neg-log N/A

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

   3. lift-log.f64 N/A

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

   4. lower-neg.f64 99.1

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

6. Applied rewrites 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{-\log 0.1}} \]
7. 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 55.6%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 rewrites 99.0%

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

Alternative 3: 26.6% accurate, 1.0× speedup

Math

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

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(Float64(re * 0.5) / im), im))))
end

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. 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 31.0

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

5. Applied rewrites 31.0%

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

   1. metadata-eval N/A

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

   2. neg-log N/A

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

   3. lift-log.f64 N/A

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

   4. lower-neg.f64 31.0

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

7. Applied rewrites 31.0%

   \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}{\color{blue}{-\log 0.1}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. remove-double-neg N/A

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

   6. lift-log.f64 N/A

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

   7. frac-2neg N/A

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

   8. distribute-frac-neg N/A

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

   9. clear-num N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 31.0

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

9. Applied rewrites 31.0%

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

Alternative 4: 26.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. 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 31.0

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

5. Applied rewrites 31.0%

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

   1. metadata-eval N/A

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

   2. neg-log N/A

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

   3. lift-log.f64 N/A

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

   4. lower-neg.f64 31.0

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

7. Applied rewrites 31.0%

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

Alternative 5: 26.6% 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 55.6%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. 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 31.0

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

5. Applied rewrites 31.0%

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

Alternative 6: 26.8% 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 55.6%

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

3. Taylor expanded in re around 0

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

   1. 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 31.2

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

5. Applied rewrites 31.2%

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

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. lift-log.f64 N/A

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

   11. neg-log N/A

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

   12. metadata-eval N/A

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

   13. lift-log.f64 N/A

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

   14. lower-/.f64 31.3

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

7. Applied rewrites 31.3%

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

Alternative 7: 26.8% 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 55.6%

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

3. Taylor expanded in re around 0

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

   1. 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 31.2

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

5. Applied rewrites 31.2%

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

   1. lift-log.f64 N/A

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

   2. metadata-eval N/A

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

   3. neg-log N/A

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

   4. lift-log.f64 N/A

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

   5. distribute-frac-neg2 N/A

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

   6. distribute-frac-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 31.3

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

7. Applied rewrites 31.3%

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

8. Final simplification 31.3%

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

Alternative 8: 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 55.6%

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

3. Taylor expanded in re around 0

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

   1. 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 31.2

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

5. Applied rewrites 31.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))