math.log10 on complex, real part

Percentage Accurate: 50.8% → 99.4%

Time: 15.5s

Alternatives: 5

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: 50.8% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\log 10}^{-0.5}\\ t_0 \cdot \left(t_0 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (pow (log 10.0) -0.5))) (* t_0 (* t_0 (log (hypot re im))))))

C

double code(double re, double im) {
	double t_0 = pow(log(10.0), -0.5);
	return t_0 * (t_0 * log(hypot(re, im)));
}

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(Math.log(10.0), -0.5);
	return t_0 * (t_0 * Math.log(Math.hypot(re, im)));
}

Python

def code(re, im):
	t_0 = math.pow(math.log(10.0), -0.5)
	return t_0 * (t_0 * math.log(math.hypot(re, im)))

Julia

function code(re, im)
	t_0 = log(10.0) ^ -0.5
	return Float64(t_0 * Float64(t_0 * log(hypot(re, im))))
end

MATLAB

function tmp = code(re, im)
	t_0 = log(10.0) ^ -0.5;
	tmp = t_0 * (t_0 * log(hypot(re, im)));
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.7%

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

   1. +-commutative 53.7%

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

   2. +-commutative 53.7%

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

   3. sqr-neg 53.7%

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

   4. sqr-neg 53.7%

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

   5. sqr-neg 53.7%

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

3. Simplified 99.1%

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

   1. div-inv 98.6%

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

   2. *-commutative 98.6%

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

   3. inv-pow 98.6%

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

   4. sqr-pow 99.5%

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

   5. associate-*l* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

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 53.7%

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

   1. +-commutative 53.7%

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

   2. +-commutative 53.7%

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

   3. sqr-neg 53.7%

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

   4. sqr-neg 53.7%

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

   5. sqr-neg 53.7%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 27.3% 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 53.7%

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

   1. +-commutative 53.7%

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

   2. +-commutative 53.7%

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

   3. sqr-neg 53.7%

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

   4. sqr-neg 53.7%

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

   5. sqr-neg 53.7%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 28.7%

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

5. Final simplification 28.7%

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

Alternative 4: 7.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \log im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log im))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im)
end function

Java

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

Python

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

Julia

function code(re, im)
	return log(im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 53.7%

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

   1. +-commutative 53.7%

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

   2. +-commutative 53.7%

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

   3. sqr-neg 53.7%

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

   4. sqr-neg 53.7%

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

   5. sqr-neg 53.7%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 28.7%

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

6. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{\log im + 0} \]
7. Step-by-step derivation

   1. +-rgt-identity 6.9%

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

8. Simplified 6.9%

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

9. Final simplification 6.9%

   \[\leadsto \log im \]

Alternative 5: 11.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 53.7%

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

   1. +-commutative 53.7%

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

   2. +-commutative 53.7%

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

   3. sqr-neg 53.7%

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

   4. sqr-neg 53.7%

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

   5. sqr-neg 53.7%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 28.7%

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

   1. expm1-log1p-u_binary64 21.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log im}{\log 10}\right)\right)} \]

6. Applied rewrite-once 21.6%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log im}{\log 10}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p 28.7%

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

   2. div-inv 28.5%

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

   3. *-commutative 28.5%

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

   4. inv-pow 28.5%

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

   5. metadata-eval 28.5%

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

   6. pow-sqr 28.8%

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

   7. associate-*l* 28.8%

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

8. Applied egg-rr 28.8%

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

10. Applied egg-rr 5.7%

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

    1. *-inverses 11.7%

       \[\leadsto \color{blue}{1} \]

12. Simplified 11.7%

    \[\leadsto \color{blue}{1} \]

13. Final simplification 11.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))