math.log10 on complex, real part

Percentage Accurate: 51.7% → 99.3%

Time: 11.5s

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: 51.7% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

   1. +-commutative 48.6%

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

   2. +-commutative 48.6%

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

   3. sqr-neg 48.6%

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

   4. sqr-neg 48.6%

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

   5. sqr-neg 48.6%

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

   6. sqr-neg 48.6%

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

   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. Step-by-step derivation

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. metadata-eval 99.1%

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

   2. pow-flip 99.1%

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

   3. pow1/2 99.1%

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

   4. times-frac 99.0%

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

   5. *-un-lft-identity 99.0%

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

   6. add-sqr-sqrt 99.0%

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

   7. frac-2neg 99.0%

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

   8. neg-log 99.1%

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

   9. metadata-eval 99.1%

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

8. Applied egg-rr 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 31.6%

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

   2. pow1/3 31.7%

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

   3. log-pow 31.7%

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

   4. pow3 31.7%

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

   5. log-pow 99.1%

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

10. Applied egg-rr 99.1%

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

    1. distribute-lft-neg-in 99.1%

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

    2. associate-/l* 99.3%

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

    3. metadata-eval 99.3%

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

12. Applied egg-rr 99.3%

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

13. Final simplification 99.3%

    \[\leadsto -0.3333333333333333 \cdot \frac{3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1} \]
14. 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 48.6%

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

   1. +-commutative 48.6%

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

   2. +-commutative 48.6%

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

   3. sqr-neg 48.6%

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

   4. sqr-neg 48.6%

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

   5. sqr-neg 48.6%

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

   6. sqr-neg 48.6%

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

   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. 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 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   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. Final simplification 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\log 0.1} \]
10. 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 48.6%

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

   1. +-commutative 48.6%

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

   2. +-commutative 48.6%

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

   3. sqr-neg 48.6%

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

   4. sqr-neg 48.6%

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

   5. sqr-neg 48.6%

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

   6. sqr-neg 48.6%

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

   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. Final simplification 99.0%

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

Alternative 4: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

   1. +-commutative 48.6%

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

   2. +-commutative 48.6%

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

   3. sqr-neg 48.6%

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

   4. sqr-neg 48.6%

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

   5. sqr-neg 48.6%

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

   6. sqr-neg 48.6%

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

   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. Step-by-step derivation

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. metadata-eval 99.1%

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

   2. pow-flip 99.1%

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

   3. pow1/2 99.1%

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

   4. times-frac 99.0%

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

   5. *-un-lft-identity 99.0%

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

   6. add-sqr-sqrt 99.0%

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

   7. frac-2neg 99.0%

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

   8. neg-log 99.1%

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

   9. metadata-eval 99.1%

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

8. Applied egg-rr 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 31.6%

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

   2. pow1/3 31.7%

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

   3. log-pow 31.7%

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

   4. pow3 31.7%

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

   5. log-pow 99.1%

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

10. Applied egg-rr 99.1%

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

11. Taylor expanded in re around 0 27.8%

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

    1. *-commutative 27.8%

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

13. Simplified 27.8%

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

14. Final simplification 27.8%

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

Alternative 5: 27.6% 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 48.6%

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

   1. +-commutative 48.6%

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

   2. +-commutative 48.6%

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

   3. sqr-neg 48.6%

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

   4. sqr-neg 48.6%

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

   5. sqr-neg 48.6%

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

   6. sqr-neg 48.6%

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

   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. Taylor expanded in re around 0 27.8%

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

   1. frac-2neg 27.8%

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

   2. distribute-frac-neg 27.8%

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

   3. neg-log 27.8%

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

   4. metadata-eval 27.8%

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

7. Applied egg-rr 27.8%

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

   1. distribute-neg-frac2 27.8%

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

9. Simplified 27.8%

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

10. Final simplification 27.8%

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

Alternative 6: 27.7% 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 48.6%

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

   1. +-commutative 48.6%

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

   2. +-commutative 48.6%

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

   3. sqr-neg 48.6%

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

   4. sqr-neg 48.6%

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

   5. sqr-neg 48.6%

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

   6. sqr-neg 48.6%

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

   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. Taylor expanded in re around 0 27.8%

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

6. Final simplification 27.8%

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

Reproduce

herbie shell --seed 2024078 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))