math.log10 on complex, real part

Percentage Accurate: 51.5% → 99.4%

Time: 11.5s

Alternatives: 3

Speedup: 1.0×

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: 51.5% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \frac{-3}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (log (cbrt (hypot re im))) (/ -3.0 (log 0.1))))

C

double code(double re, double im) {
	return log(cbrt(hypot(re, im))) * (-3.0 / log(0.1));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.cbrt(Math.hypot(re, im))) * (-3.0 / Math.log(0.1));
}

Julia

function code(re, im)
	return Float64(log(cbrt(hypot(re, im))) * Float64(-3.0 / log(0.1)))
end

Wolfram

code[re_, im_] := N[(N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

   1. +-commutative 56.4%

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

   2. +-commutative 56.4%

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

   3. sqr-neg 56.4%

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

   4. sqr-neg 56.4%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

   1. add-cube-cbrt 99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

   2. pow3 99.1%

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{3}\right)}}{\log 10} \]

   3. log-pow 99.0%

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

6. Applied egg-rr 99.0%

   \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]
7. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right) \cdot \frac{1}{\log 10}} \]

   2. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot 3\right)} \cdot \frac{1}{\log 10} \]

   3. associate-*l* 98.9%

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{1}{\log 10}\right)} \]

   4. frac-2neg 98.9%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \color{blue}{\frac{-1}{-\log 10}}\right) \]

   5. metadata-eval 98.9%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{\color{blue}{-1}}{-\log 10}\right) \]

   6. neg-log 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)}}\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log \color{blue}{0.1}}\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log 0.1}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \color{blue}{\frac{3 \cdot -1}{\log 0.1}} \]

   2. metadata-eval 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \frac{\color{blue}{-3}}{\log 0.1} \]

10. Simplified 99.4%

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \frac{-3}{\log 0.1}} \]

11. Final simplification 99.4%

    \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \frac{-3}{\log 0.1} \]
12. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× 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 56.4%

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

   1. +-commutative 56.4%

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

   2. +-commutative 56.4%

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

   3. sqr-neg 56.4%

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

   4. sqr-neg 56.4%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 27.4% accurate, 1.5× 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 56.4%

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

   1. +-commutative 56.4%

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

   2. +-commutative 56.4%

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

   3. sqr-neg 56.4%

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

   4. sqr-neg 56.4%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in re around 0 24.8%

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

6. Final simplification 24.8%

   \[\leadsto \frac{\log im}{\log 10} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))