math.log10 on complex, real part

Percentage Accurate: 52.9% → 99.7%

Time: 13.4s

Alternatives: 6

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: 52.9% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (log1p (expm1 (/ 1.0 (log 10.0)))))))

C

double code(double re, double im) {
	return log(pow(hypot(re, im), log1p(expm1((1.0 / log(10.0))))));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(Math.hypot(re, im), Math.log1p(Math.expm1((1.0 / Math.log(10.0))))));
}

Python

def code(re, im):
	return math.log(math.pow(math.hypot(re, im), math.log1p(math.expm1((1.0 / math.log(10.0))))))

Julia

function code(re, im)
	return log((hypot(re, im) ^ log1p(expm1(Float64(1.0 / log(10.0))))))
end

Wolfram

code[re_, im_] := N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.2%

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

   1. div-inv 52.0%

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

   2. add-log-exp 52.0%

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

   3. exp-to-pow 51.9%

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

   4. hypot-define 98.5%

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

   5. frac-2neg 98.5%

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

   6. metadata-eval 98.5%

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

   7. neg-log 99.0%

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

   8. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

   1. metadata-eval 99.0%

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

   2. metadata-eval 99.0%

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

   3. neg-log 98.5%

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

   4. frac-2neg 98.5%

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

   5. log1p-expm1-u 99.7%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}}\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}}\right) \]
7. Add Preprocessing

Alternative 2: 27.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\log 10}^{-2}} \cdot \log im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in re around 0 23.7%

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

   1. clear-num 23.7%

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

   2. associate-/r/ 23.6%

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

5. Applied egg-rr 23.6%

   \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log im} \]
6. Step-by-step derivation

   1. log1p-expm1-u 23.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)} \cdot \log im \]

   2. add-sqr-sqrt 23.8%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)}\right)} \cdot \log im \]

   3. sqrt-unprod 23.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)}} \cdot \log im \]

   4. log1p-expm1-u 23.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\log 10}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)} \cdot \log im \]

   5. inv-pow 23.8%

      \[\leadsto \sqrt{\color{blue}{{\log 10}^{-1}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)} \cdot \log im \]

   6. log1p-expm1-u 23.6%

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

   7. inv-pow 23.6%

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

   8. pow-prod-up 23.8%

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

   9. metadata-eval 23.8%

      \[\leadsto \sqrt{{\log 10}^{\color{blue}{-2}}} \cdot \log im \]

7. Applied egg-rr 23.8%

   \[\leadsto \color{blue}{\sqrt{{\log 10}^{-2}}} \cdot \log im \]
8. Add Preprocessing

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 52.2%

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

   1. +-commutative 52.2%

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

   2. +-commutative 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. sqr-neg 52.2%

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

   6. sqr-neg 52.2%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

Alternative 4: 27.0% accurate, 1.5× 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 52.2%

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

3. Taylor expanded in re around 0 23.7%

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

   1. frac-2neg 23.7%

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

   2. neg-log 23.7%

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

   3. metadata-eval 23.7%

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

   4. distribute-frac-neg 23.7%

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

5. Applied egg-rr 23.7%

   \[\leadsto \color{blue}{-\frac{\log im}{\log 0.1}} \]
6. Step-by-step derivation

   1. clear-num 23.7%

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

7. Applied egg-rr 23.7%

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

8. Final simplification 23.7%

   \[\leadsto \frac{-1}{\frac{\log 0.1}{\log im}} \]
9. Add Preprocessing

Alternative 5: 27.0% accurate, 1.5× 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 52.2%

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

3. Taylor expanded in re around 0 23.7%

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

   1. frac-2neg 23.7%

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

   2. neg-log 23.7%

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

   3. metadata-eval 23.7%

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

   4. distribute-frac-neg 23.7%

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

5. Applied egg-rr 23.7%

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

6. Final simplification 23.7%

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

Alternative 6: 27.0% 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 52.2%

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

3. Taylor expanded in re around 0 23.7%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))