math.log10 on complex, real part

Percentage Accurate: 51.6% → 99.4%

Time: 10.2s

Alternatives: 7

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: 51.6% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* (log (hypot re im)) (/ 3.0 (log 0.1))) -0.3333333333333333))

C

double code(double re, double im) {
	return (log(hypot(re, im)) * (3.0 / log(0.1))) * -0.3333333333333333;
}

Java

public static double code(double re, double im) {
	return (Math.log(Math.hypot(re, im)) * (3.0 / Math.log(0.1))) * -0.3333333333333333;
}

Python

def code(re, im):
	return (math.log(math.hypot(re, im)) * (3.0 / math.log(0.1))) * -0.3333333333333333

Julia

function code(re, im)
	return Float64(Float64(log(hypot(re, im)) * Float64(3.0 / log(0.1))) * -0.3333333333333333)
end

MATLAB

function tmp = code(re, im)
	tmp = (log(hypot(re, im)) * (3.0 / log(0.1))) * -0.3333333333333333;
end

Wolfram

code[re_, im_] := N[(N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * N[(3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. distribute-neg-frac2 99.1%

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

8. Simplified 99.1%

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

   1. add-cbrt-cube 33.2%

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

   2. pow1/3 33.4%

      \[\leadsto \frac{\log \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)\right)}^{0.3333333333333333}\right)}}{-\log 0.1} \]

   3. log-pow 33.4%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \log \left(\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)\right)}}{-\log 0.1} \]

   4. pow3 33.4%

      \[\leadsto \frac{0.3333333333333333 \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}}{-\log 0.1} \]

   5. log-pow 99.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{-\log 0.1} \]

10. Applied egg-rr 99.2%

    \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{-\log 0.1} \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right)} \]
13. Step-by-step derivation

    1. *-commutative 99.4%

       \[\leadsto \color{blue}{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right) \cdot -0.3333333333333333} \]

14. Simplified 99.4%

    \[\leadsto \color{blue}{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right) \cdot -0.3333333333333333} \]
15. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / -math.log(0.1)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / -log(0.1);
end

Wolfram

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

Derivation

1. Initial program 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. distribute-neg-frac2 99.1%

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

8. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\log 0.1}} \]
9. 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 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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: 26.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \left(\frac{3}{\log 0.1} \cdot \log im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* -0.3333333333333333 (* (/ 3.0 (log 0.1)) (log im))))

C

double code(double re, double im) {
	return -0.3333333333333333 * ((3.0 / log(0.1)) * log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-0.3333333333333333d0) * ((3.0d0 / log(0.1d0)) * log(im))
end function

Java

public static double code(double re, double im) {
	return -0.3333333333333333 * ((3.0 / Math.log(0.1)) * Math.log(im));
}

Python

def code(re, im):
	return -0.3333333333333333 * ((3.0 / math.log(0.1)) * math.log(im))

Julia

function code(re, im)
	return Float64(-0.3333333333333333 * Float64(Float64(3.0 / log(0.1)) * log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = -0.3333333333333333 * ((3.0 / log(0.1)) * log(im));
end

Wolfram

code[re_, im_] := N[(-0.3333333333333333 * N[(N[(3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. distribute-neg-frac2 99.1%

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

8. Simplified 99.1%

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

   1. add-cbrt-cube 33.2%

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

   2. pow1/3 33.4%

      \[\leadsto \frac{\log \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)\right)}^{0.3333333333333333}\right)}}{-\log 0.1} \]

   3. log-pow 33.4%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \log \left(\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)\right)}}{-\log 0.1} \]

   4. pow3 33.4%

      \[\leadsto \frac{0.3333333333333333 \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}}{-\log 0.1} \]

   5. log-pow 99.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{-\log 0.1} \]

10. Applied egg-rr 99.2%

    \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{-\log 0.1} \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right)} \]
13. Step-by-step derivation

    1. *-commutative 99.4%

       \[\leadsto \color{blue}{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right) \cdot -0.3333333333333333} \]

14. Simplified 99.4%

    \[\leadsto \color{blue}{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{3}{\log 0.1}\right) \cdot -0.3333333333333333} \]

15. Taylor expanded in re around 0 24.9%

    \[\leadsto \color{blue}{\left(3 \cdot \frac{\log im}{\log 0.1}\right)} \cdot -0.3333333333333333 \]
16. Step-by-step derivation

    1. associate-*r/ 25.0%

       \[\leadsto \color{blue}{\frac{3 \cdot \log im}{\log 0.1}} \cdot -0.3333333333333333 \]

    2. *-commutative 25.0%

       \[\leadsto \frac{\color{blue}{\log im \cdot 3}}{\log 0.1} \cdot -0.3333333333333333 \]

    3. associate-*r/ 25.0%

       \[\leadsto \color{blue}{\left(\log im \cdot \frac{3}{\log 0.1}\right)} \cdot -0.3333333333333333 \]

17. Simplified 25.0%

    \[\leadsto \color{blue}{\left(\log im \cdot \frac{3}{\log 0.1}\right)} \cdot -0.3333333333333333 \]

18. Final simplification 25.0%

    \[\leadsto -0.3333333333333333 \cdot \left(\frac{3}{\log 0.1} \cdot \log im\right) \]
19. Add Preprocessing

Alternative 5: 26.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\log im \cdot 0.3333333333333333\right) \cdot \frac{-3}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* (log im) 0.3333333333333333) (/ -3.0 (log 0.1))))

C

double code(double re, double im) {
	return (log(im) * 0.3333333333333333) * (-3.0 / log(0.1));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (log(im) * 0.3333333333333333d0) * ((-3.0d0) / log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return (Math.log(im) * 0.3333333333333333) * (-3.0 / Math.log(0.1));
}

Python

def code(re, im):
	return (math.log(im) * 0.3333333333333333) * (-3.0 / math.log(0.1))

Julia

function code(re, im)
	return Float64(Float64(log(im) * 0.3333333333333333) * Float64(-3.0 / log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = (log(im) * 0.3333333333333333) * (-3.0 / log(0.1));
end

Wolfram

code[re_, im_] := N[(N[(N[Log[im], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. distribute-neg-frac2 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in re around 0 24.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{\log im}{\log 0.1}} \]
10. Step-by-step derivation

    1. neg-mul-1 24.9%

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

    2. distribute-neg-frac2 24.9%

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

11. Simplified 24.9%

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

13. Applied egg-rr 25.0%

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

    1. associate-*r/ 25.0%

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

    2. metadata-eval 25.0%

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

15. Simplified 25.0%

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

17. Applied egg-rr 25.0%

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log im\right)} \cdot \frac{-3}{\log 0.1} \]

18. Final simplification 25.0%

    \[\leadsto \left(\log im \cdot 0.3333333333333333\right) \cdot \frac{-3}{\log 0.1} \]
19. Add Preprocessing

Alternative 6: 26.6% 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 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. distribute-neg-frac2 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in re around 0 24.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{\log im}{\log 0.1}} \]
10. Step-by-step derivation

    1. neg-mul-1 24.9%

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

    2. distribute-neg-frac2 24.9%

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

11. Simplified 24.9%

    \[\leadsto \color{blue}{\frac{\log im}{-\log 0.1}} \]
12. Add Preprocessing

Alternative 7: 26.6% 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 48.9%

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

   1. +-commutative 48.9%

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

   2. +-commutative 48.9%

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

   3. sqr-neg 48.9%

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

   4. sqr-neg 48.9%

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

   5. 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. Taylor expanded in re around 0 24.9%

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

Reproduce

herbie shell --seed 2024117 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))