math.log10 on complex, real part

Percentage Accurate: 50.8% → 99.5%

Time: 8.0s

Alternatives: 4

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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (/ (pow (log 10.0) -0.5) (sqrt (log 10.0))) (log (hypot re im))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. hypot-def 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. Step-by-step derivation

   1. *-un-lft-identity 99.1%

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

   2. add-sqr-sqrt 99.1%

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

   3. times-frac 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.1%

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

   2. un-div-inv 99.2%

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

   3. pow1/2 99.2%

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

   4. pow-flip 99.2%

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

   5. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
8. Step-by-step derivation

   1. associate-/r/ 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 26.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return log((im ^ sqrt((log(10.0) ^ -2.0))))
end

MATLAB

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

Wolfram

code[re_, im_] := N[Log[N[Power[im, N[Sqrt[N[Power[N[Log[10.0], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.0%

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

   1. hypot-def 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. Taylor expanded in re around 0 25.7%

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

   1. add-log-exp 25.7%

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

   2. div-inv 25.6%

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

   3. exp-to-pow 25.5%

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

6. Applied egg-rr 25.5%

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

   1. add-sqr-sqrt 25.8%

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

   2. sqrt-unprod 25.5%

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

   3. inv-pow 25.5%

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

   4. inv-pow 25.5%

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

   5. pow-prod-up 25.8%

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

   6. metadata-eval 25.8%

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

8. Applied egg-rr 25.8%

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

9. Final simplification 25.8%

   \[\leadsto \log \left({im}^{\left(\sqrt{{\log 10}^{-2}}\right)}\right) \]

Alternative 3: 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 50.0%

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

   1. hypot-def 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. Final simplification 99.1%

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

Alternative 4: 26.7% 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 50.0%

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

   1. hypot-def 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. Taylor expanded in re around 0 25.7%

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

5. Final simplification 25.7%

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

Reproduce

herbie shell --seed 2023240 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))