math.log10 on complex, real part

Percentage Accurate: 52.3% → 99.4%

Time: 6.3s

Alternatives: 7

Speedup: 0.7×

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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}^{-1} \cdot {\left({\log 10}^{-0.5}\right)}^{2} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (pow (pow (log (hypot im re)) -1.0) -1.0) (pow (pow (log 10.0) -0.5) 2.0)))

C

double code(double re, double im) {
	return pow(pow(log(hypot(im, re)), -1.0), -1.0) * pow(pow(log(10.0), -0.5), 2.0);
}

Java

public static double code(double re, double im) {
	return Math.pow(Math.pow(Math.log(Math.hypot(im, re)), -1.0), -1.0) * Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0);
}

Python

def code(re, im):
	return math.pow(math.pow(math.log(math.hypot(im, re)), -1.0), -1.0) * math.pow(math.pow(math.log(10.0), -0.5), 2.0)

Julia

function code(re, im)
	return Float64(((log(hypot(im, re)) ^ -1.0) ^ -1.0) * ((log(10.0) ^ -0.5) ^ 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = ((log(hypot(im, re)) ^ -1.0) ^ -1.0) * ((log(10.0) ^ -0.5) ^ 2.0);
end

Wolfram

code[re_, im_] := N[(N[Power[N[Power[N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. clear-num N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. div-inv N/A

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

   6. associate-/l* N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. lower-*.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{{\log 10}^{-1} \cdot {\left(\frac{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}{1}\right)}^{-1}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 99.5

      \[\leadsto {\color{blue}{\left({\log 10}^{-0.5}\right)}}^{2} \cdot {\left(\frac{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}{1}\right)}^{-1} \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{{\left({\log 10}^{-0.5}\right)}^{2}} \cdot {\left(\frac{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}{1}\right)}^{-1} \]

7. Final simplification 99.5%

   \[\leadsto {\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}^{-1} \cdot {\left({\log 10}^{-0.5}\right)}^{2} \]
8. Add Preprocessing

Alternative 2: 27.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log im \cdot {\left({\log 10}^{-0.5}\right)}^{2} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (log im) (pow (pow (log 10.0) -0.5) 2.0)))

C

double code(double re, double im) {
	return log(im) * pow(pow(log(10.0), -0.5), 2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) * ((log(10.0d0) ** (-0.5d0)) ** 2.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(im) * Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0);
}

Python

def code(re, im):
	return math.log(im) * math.pow(math.pow(math.log(10.0), -0.5), 2.0)

Julia

function code(re, im)
	return Float64(log(im) * ((log(10.0) ^ -0.5) ^ 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) * ((log(10.0) ^ -0.5) ^ 2.0);
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] * N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. clear-num N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. div-inv N/A

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

   6. associate-/l* N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. lower-*.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{{\log 10}^{-1} \cdot {\left(\frac{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}{1}\right)}^{-1}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 99.5

      \[\leadsto {\color{blue}{\left({\log 10}^{-0.5}\right)}}^{2} \cdot {\left(\frac{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}{1}\right)}^{-1} \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{{\left({\log 10}^{-0.5}\right)}^{2}} \cdot {\left(\frac{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}{1}\right)}^{-1} \]

7. Taylor expanded in re around 0

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

   1. lower-log.f64 26.7

      \[\leadsto {\left({\log 10}^{-0.5}\right)}^{2} \cdot \color{blue}{\log im} \]

9. Applied rewrites 26.7%

   \[\leadsto {\left({\log 10}^{-0.5}\right)}^{2} \cdot \color{blue}{\log im} \]

10. Final simplification 26.7%

    \[\leadsto \log im \cdot {\left({\log 10}^{-0.5}\right)}^{2} \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-1}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \frac{\log 10}{-1}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 / N[(N[(-1.0 / N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Log[10.0], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 50.9

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 99.1

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. remove-double-div N/A

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

   3. unpow-1 N/A

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

   4. lift-pow.f64 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow-1 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.1

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

   15. lift-hypot.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-hypot.f64 99.1

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

6. Applied rewrites 99.1%

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

7. Final simplification 99.1%

   \[\leadsto \frac{1}{\frac{-1}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \frac{\log 10}{-1}} \]
8. Add Preprocessing

Alternative 4: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 50.9

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 99.1

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

4. Applied rewrites 99.1%

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

Alternative 5: 25.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (fma (/ re im) (/ re im) (* (- -2.0) (log im)))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * fma((re / im), (re / im), (-(-2.0) * log(im)))) / log(10.0);
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * fma(Float64(re / im), Float64(re / im), Float64(Float64(-(-2.0)) * log(im)))) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[((--2.0) * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. pow-to-exp N/A

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

   5. rem-log-exp N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 50.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 50.9

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 24.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\color{blue}{\log im}\right) \cdot -2\right) \cdot 0.5}{\log 10} \]

7. Applied rewrites 24.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)} \cdot 0.5}{\log 10} \]

8. Final simplification 24.4%

   \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)}{\log 10} \]
9. Add Preprocessing

Alternative 6: 27.5% 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 50.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 26.6

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

5. Applied rewrites 26.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 26.6

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

7. Applied rewrites 26.6%

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

Alternative 7: 27.5% 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 50.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 26.6

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

5. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))