math.log10 on complex, real part

Percentage Accurate: 51.9% → 99.6%

Time: 18.9s

Alternatives: 7

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 51.9% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log 10}^{-3}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (log (hypot re im)) (cbrt (pow (log 10.0) -3.0))))

C

double code(double re, double im) {
	return log(hypot(re, im)) * cbrt(pow(log(10.0), -3.0));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) * Math.cbrt(Math.pow(Math.log(10.0), -3.0));
}

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) * cbrt((log(10.0) ^ -3.0)))
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Log[10.0], $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 99.2%

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

5. Applied egg-rr 99.2%

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

   1. frac-times 99.1%

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

   2. *-un-lft-identity 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. expm1-log1p-u 71.1%

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

   5. expm1-udef 71.1%

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

   6. log1p-udef 71.1%

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

   7. add-exp-log 99.1%

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

   8. +-commutative 99.1%

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

   9. associate--l+ 99.1%

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

   10. rem-cbrt-cube 98.9%

       \[\leadsto \frac{\color{blue}{\sqrt[3]{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}}}{\log 10} + \left(1 - 1\right) \]

   11. rem-cbrt-cube 97.9%

       \[\leadsto \frac{\sqrt[3]{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}}{\color{blue}{\sqrt[3]{{\log 10}^{3}}}} + \left(1 - 1\right) \]

   12. cbrt-div 98.9%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}{{\log 10}^{3}}}} + \left(1 - 1\right) \]

   13. div-inv 98.9%

       \[\leadsto \sqrt[3]{\color{blue}{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3} \cdot \frac{1}{{\log 10}^{3}}}} + \left(1 - 1\right) \]

   14. cbrt-prod 98.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}} \cdot \sqrt[3]{\frac{1}{{\log 10}^{3}}}} + \left(1 - 1\right) \]

   15. rem-cbrt-cube 98.6%

       \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \sqrt[3]{\frac{1}{{\log 10}^{3}}} + \left(1 - 1\right) \]

   16. metadata-eval 98.6%

       \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{\frac{1}{{\log 10}^{3}}} + \color{blue}{0} \]

   17. metadata-eval 98.6%

       \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{\frac{1}{{\log 10}^{3}}} + \color{blue}{\log 1} \]

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \sqrt[3]{{\log 10}^{-3}}, 0\right)} \]
8. Step-by-step derivation

   1. fma-udef 99.5%

      \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log 10}^{-3}} + 0} \]

   2. unpow1/3 99.5%

      \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \color{blue}{{\left({\log 10}^{-3}\right)}^{0.3333333333333333}} + 0 \]

   3. +-rgt-identity 99.5%

      \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot {\left({\log 10}^{-3}\right)}^{0.3333333333333333}} \]

   4. unpow1/3 99.5%

      \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \color{blue}{\sqrt[3]{{\log 10}^{-3}}} \]

9. Simplified 99.5%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log 10}^{-3}}} \]

10. Final simplification 99.5%

    \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log 10}^{-3}} \]

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(Float64(-log(hypot(re, im))) / 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 50.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. div-inv 98.6%

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

   2. frac-2neg 98.6%

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

   3. metadata-eval 98.6%

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

   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%

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

5. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{-1}{\log 0.1}} \]
6. 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%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

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 50.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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: 44.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5e-165) (/ (- (log (- re))) (log 0.1)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = -log(-re) / log(0.1);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-165) then
        tmp = -log(-re) / log(0.1d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = -Math.log(-re) / Math.log(0.1);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5e-165:
		tmp = -math.log(-re) / math.log(0.1)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5e-165)
		tmp = Float64(Float64(-log(Float64(-re))) / log(0.1));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5e-165)
		tmp = -log(-re) / log(0.1);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5e-165], N[((-N[Log[(-re)], $MachinePrecision]) / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.4999999999999999e-165

   1. Initial program 49.4%

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

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

      1. expm1-log1p-u 67.5%

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

      2. expm1-udef 67.5%

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

      3. log1p-udef 67.5%

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

      4. add-exp-log 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in re around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

      2. unsub-neg 29.7%

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

   8. Simplified 29.7%

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

      1. associate--l- 29.7%

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

      2. sub-neg 29.7%

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

      3. frac-2neg 29.7%

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

      4. neg-log 29.7%

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

      5. clear-num 29.7%

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

      6. div-inv 29.7%

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

      7. metadata-eval 29.7%

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

      8. neg-log 29.7%

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

      9. metadata-eval 29.7%

         \[\leadsto 1 + \left(-\left(\frac{\log \left(re \cdot -1\right)}{\log \color{blue}{0.1}} + 1\right)\right) \]

   10. Applied egg-rr 29.7%

       \[\leadsto \color{blue}{1 + \left(-\left(\frac{\log \left(re \cdot -1\right)}{\log 0.1} + 1\right)\right)} \]
   11. Step-by-step derivation

       1. unsub-neg 29.7%

          \[\leadsto \color{blue}{1 - \left(\frac{\log \left(re \cdot -1\right)}{\log 0.1} + 1\right)} \]

       2. +-commutative 29.7%

          \[\leadsto 1 - \color{blue}{\left(1 + \frac{\log \left(re \cdot -1\right)}{\log 0.1}\right)} \]

       3. associate--r+ 29.7%

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

       4. metadata-eval 29.7%

          \[\leadsto \color{blue}{0} - \frac{\log \left(re \cdot -1\right)}{\log 0.1} \]

       5. neg-sub0 29.7%

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

       6. distribute-neg-frac 29.7%

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

       7. *-commutative 29.7%

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

       8. mul-1-neg 29.7%

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

   12. Simplified 29.7%

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

   if 2.4999999999999999e-165 < im

   1. Initial program 53.5%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 64.0%

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{-\log \left(-re\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5: 44.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5e-165) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-165) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5e-165:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5e-165)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5e-165)
		tmp = log(-re) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5e-165], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.4999999999999999e-165

   1. Initial program 49.4%

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

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

      1. expm1-log1p-u 67.5%

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

      2. expm1-udef 67.5%

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

      3. log1p-udef 67.5%

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

      4. add-exp-log 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in re around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

      2. unsub-neg 29.7%

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

   8. Simplified 29.7%

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

      1. sub-neg 29.7%

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

      2. metadata-eval 29.7%

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

      3. +-commutative 29.7%

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

      4. sub-neg 29.7%

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

      5. distribute-neg-frac 29.7%

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

      6. neg-log 29.7%

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

      7. clear-num 29.7%

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

      8. div-inv 29.7%

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

      9. metadata-eval 29.7%

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

   10. Applied egg-rr 29.7%

       \[\leadsto \color{blue}{-1 + \left(1 + \frac{\log \left(re \cdot -1\right)}{\log 10}\right)} \]
   11. Step-by-step derivation

       1. associate-+r+ 29.7%

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

       2. metadata-eval 29.7%

          \[\leadsto \color{blue}{0} + \frac{\log \left(re \cdot -1\right)}{\log 10} \]

       3. +-lft-identity 29.7%

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

       4. *-commutative 29.7%

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

       5. mul-1-neg 29.7%

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

   12. Simplified 29.7%

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

   if 2.4999999999999999e-165 < im

   1. Initial program 53.5%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 64.0%

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6: 3.0% 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) / 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 50.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 24.1%

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

   1. div-inv 24.0%

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

   2. frac-2neg 24.0%

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

   3. metadata-eval 24.0%

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

   4. neg-log 24.1%

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

   5. metadata-eval 24.1%

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

6. Applied egg-rr 24.1%

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

   1. *-commutative 24.1%

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

   2. associate-*l/ 24.1%

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

   3. neg-mul-1 24.1%

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

8. Simplified 24.1%

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

   1. add-sqr-sqrt 8.1%

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

   2. sqrt-unprod 8.5%

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

   3. sqr-neg 8.5%

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

   4. sqrt-unprod 0.4%

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

   5. add-sqr-sqrt 2.5%

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

   6. div-inv 2.5%

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

10. Applied egg-rr 2.5%

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

    1. associate-*r/ 2.5%

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

    2. *-rgt-identity 2.5%

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

12. Simplified 2.5%

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

13. Final simplification 2.5%

    \[\leadsto \frac{\log im}{\log 0.1} \]

Alternative 7: 28.2% 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 50.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 24.1%

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

5. Final simplification 24.1%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))