math.log/2 on complex, real part

Percentage Accurate: 51.9% → 99.4%

Time: 10.9s

Alternatives: 6

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (/ (log (hypot re im)) (log base)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 49.7%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. +-rgt-identity 49.7%

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. metadata-eval 49.7%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   4. +-rgt-identity 49.7%

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

   5. times-frac 49.8%

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

   6. *-inverses 49.8%

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

   7. *-rgt-identity 49.8%

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

   8. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 9.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log base \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\log base \cdot \log im\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(re, im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= (log base) -2e-310) (* (log base) (log im)) (log (hypot re im))))

C

double code(double re, double im, double base) {
	double tmp;
	if (log(base) <= -2e-310) {
		tmp = log(base) * log(im);
	} else {
		tmp = log(hypot(re, im));
	}
	return tmp;
}

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (Math.log(base) <= -2e-310) {
		tmp = Math.log(base) * Math.log(im);
	} else {
		tmp = Math.log(Math.hypot(re, im));
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if math.log(base) <= -2e-310:
		tmp = math.log(base) * math.log(im)
	else:
		tmp = math.log(math.hypot(re, im))
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (log(base) <= -2e-310)
		tmp = Float64(log(base) * log(im));
	else
		tmp = log(hypot(re, im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (log(base) <= -2e-310)
		tmp = log(base) * log(im);
	else
		tmp = log(hypot(re, im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[N[Log[base], $MachinePrecision], -2e-310], N[(N[Log[base], $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision], N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 base) < -1.999999999999994e-310

   1. Initial program 49.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 49.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 49.7%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 49.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 49.7%

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

      5. times-frac 49.8%

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

      6. *-inverses 49.8%

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

      7. *-rgt-identity 49.8%

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

      8. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in re around 0 23.3%

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

      1. frac-2neg 23.3%

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

      2. div-inv 23.2%

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

      3. add-sqr-sqrt 23.1%

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

      4. sqrt-unprod 23.2%

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

      5. sqr-neg 23.2%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 2.7%

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

   6. Applied egg-rr 2.7%

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

      1. log-rec 2.7%

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

      2. associate-*r/ 2.7%

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

      3. *-rgt-identity 2.7%

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

      4. log-rec 2.7%

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

   8. Simplified 2.7%

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

      1. div-inv 2.7%

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

      2. add-sqr-sqrt 2.3%

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

      3. sqrt-unprod 22.4%

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

      4. sqr-neg 22.4%

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

      5. sqrt-unprod 20.0%

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

      6. add-sqr-sqrt 23.2%

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

      7. add-exp-log 0.0%

         \[\leadsto \log im \cdot \frac{1}{\color{blue}{e^{\log \log base}}} \]

      8. exp-neg 0.0%

         \[\leadsto \log im \cdot \color{blue}{e^{-\log \log base}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \log im \cdot e^{\color{blue}{\sqrt{-\log \log base} \cdot \sqrt{-\log \log base}}} \]

      10. sqrt-unprod 0.0%

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

      11. sqr-neg 0.0%

          \[\leadsto \log im \cdot e^{\sqrt{\color{blue}{\log \log base \cdot \log \log base}}} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \log im \cdot e^{\color{blue}{\sqrt{\log \log base} \cdot \sqrt{\log \log base}}} \]

      13. add-sqr-sqrt 0.0%

          \[\leadsto \log im \cdot e^{\color{blue}{\log \log base}} \]

      14. add-exp-log 4.2%

          \[\leadsto \log im \cdot \color{blue}{\log base} \]

   10. Applied egg-rr 4.2%

       \[\leadsto \color{blue}{\log im \cdot \log base} \]

   if -1.999999999999994e-310 < (log.f64 base)

   1. Initial program 49.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 49.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 49.7%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 49.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 49.7%

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

      5. times-frac 49.8%

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

      6. *-inverses 49.8%

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

      7. *-rgt-identity 49.8%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate-/r/ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. associate-/l* 99.5%

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

      2. add-sqr-sqrt 99.1%

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

      3. times-frac 99.0%

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

      4. metadata-eval 99.0%

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

      5. sqrt-div 99.1%

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

      6. add-exp-log 98.6%

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

      7. exp-neg 98.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 14.2%

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

      10. sqr-neg 14.2%

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

      11. sqrt-unprod 14.2%

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

      12. add-sqr-sqrt 14.2%

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

      13. add-exp-log 14.2%

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

   9. Applied egg-rr 14.2%

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

       1. associate-*r/ 14.2%

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

       2. *-commutative 14.2%

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

       3. associate-/l* 14.2%

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

       4. *-inverses 14.2%

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

       5. /-rgt-identity 14.2%

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

   11. Simplified 14.2%

       \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log base \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\log base \cdot \log im\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(re, im\right)\right)\\ \end{array} \]

Alternative 3: 43.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= im 8.5e-57) (/ (log (- re)) (log base)) (/ (log im) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if (im <= 8.5e-57) {
		tmp = log(-re) / log(base);
	} else {
		tmp = log(im) / log(base);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (im <= 8.5d-57) then
        tmp = log(-re) / log(base)
    else
        tmp = log(im) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (im <= 8.5e-57) {
		tmp = Math.log(-re) / Math.log(base);
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if im <= 8.5e-57:
		tmp = math.log(-re) / math.log(base)
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (im <= 8.5e-57)
		tmp = Float64(log(Float64(-re)) / log(base));
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (im <= 8.5e-57)
		tmp = log(-re) / log(base);
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[im, 8.5e-57], N[(N[Log[(-re)], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.49999999999999955e-57

   1. Initial program 53.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 53.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 53.7%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 53.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 53.7%

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

      5. times-frac 53.8%

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

      6. *-inverses 53.8%

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

      7. *-rgt-identity 53.8%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around -inf 30.9%

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

      1. associate-*r/ 30.9%

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

      2. mul-1-neg 30.9%

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

   6. Simplified 30.9%

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

      1. expm1-log1p-u 20.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\right)\right)} \]

      2. expm1-udef 20.7%

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

      3. neg-log 20.7%

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

      4. frac-2neg 20.7%

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

      5. metadata-eval 20.7%

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

      6. remove-double-div 20.7%

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

   8. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left(-re\right)}{\log base}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 20.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(-re\right)}{\log base}\right)\right)} \]

      2. expm1-log1p 30.9%

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

   10. Simplified 30.9%

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

   if 8.49999999999999955e-57 < im

   1. Initial program 41.0%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 41.0%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 41.0%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 41.0%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 41.0%

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

      5. times-frac 40.9%

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

      6. *-inverses 40.9%

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

      7. *-rgt-identity 40.9%

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

      8. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in re around 0 77.6%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 4: 31.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(re, im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= im -2e-310) (log (hypot re im)) (/ (log im) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if (im <= -2e-310) {
		tmp = log(hypot(re, im));
	} else {
		tmp = log(im) / log(base);
	}
	return tmp;
}

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (im <= -2e-310) {
		tmp = Math.log(Math.hypot(re, im));
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if im <= -2e-310:
		tmp = math.log(math.hypot(re, im))
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (im <= -2e-310)
		tmp = log(hypot(re, im));
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (im <= -2e-310)
		tmp = log(hypot(re, im));
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[im, -2e-310], N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.999999999999994e-310

   1. Initial program 53.4%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 53.4%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 53.4%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 53.4%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 53.4%

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

      5. times-frac 53.5%

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

      6. *-inverses 53.5%

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

      7. *-rgt-identity 53.5%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate-/r/ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. associate-/l* 99.5%

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

      2. add-sqr-sqrt 45.8%

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

      3. times-frac 45.8%

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

      4. metadata-eval 45.8%

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

      5. sqrt-div 45.8%

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

      6. add-exp-log 45.6%

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

      7. exp-neg 45.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 6.6%

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

      10. sqr-neg 6.6%

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

      11. sqrt-unprod 6.6%

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

      12. add-sqr-sqrt 6.6%

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

      13. add-exp-log 6.6%

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

   9. Applied egg-rr 6.6%

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

       1. associate-*r/ 6.6%

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

       2. *-commutative 6.6%

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

       3. associate-/l* 6.6%

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

       4. *-inverses 7.5%

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

       5. /-rgt-identity 7.5%

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

   11. Simplified 7.5%

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

   if -1.999999999999994e-310 < im

   1. Initial program 45.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 45.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 45.7%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 45.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 45.7%

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

      5. times-frac 45.7%

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

      6. *-inverses 45.7%

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

      7. *-rgt-identity 45.7%

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

      8. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in re around 0 53.7%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(re, im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 5: 7.8% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (log (hypot re im)))

C

double code(double re, double im, double base) {
	return log(hypot(re, im));
}

Java

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

Python

def code(re, im, base):
	return math.log(math.hypot(re, im))

Julia

function code(re, im, base)
	return log(hypot(re, im))
end

MATLAB

function tmp = code(re, im, base)
	tmp = log(hypot(re, im));
end

Wolfram

code[re_, im_, base_] := N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 49.7%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. +-rgt-identity 49.7%

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. metadata-eval 49.7%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   4. +-rgt-identity 49.7%

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

   5. times-frac 49.8%

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

   6. *-inverses 49.8%

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

   7. *-rgt-identity 49.8%

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

   8. hypot-def 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.4%

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

5. Applied egg-rr 99.4%

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

   1. associate-/r/ 99.4%

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

7. Applied egg-rr 99.4%

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

   1. associate-/l* 99.6%

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

   2. add-sqr-sqrt 49.5%

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

   3. times-frac 49.5%

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

   4. metadata-eval 49.5%

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

   5. sqrt-div 49.6%

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

   6. add-exp-log 49.3%

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

   7. exp-neg 49.3%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 7.1%

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

   10. sqr-neg 7.1%

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

   11. sqrt-unprod 7.1%

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

   12. add-sqr-sqrt 7.1%

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

   13. add-exp-log 7.1%

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

9. Applied egg-rr 7.1%

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

    1. associate-*r/ 7.1%

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

    2. *-commutative 7.1%

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

    3. associate-/l* 7.1%

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

    4. *-inverses 7.9%

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

    5. /-rgt-identity 7.9%

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

11. Simplified 7.9%

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

12. Final simplification 7.9%

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

Alternative 6: 3.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \log im \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 (log im))

C

double code(double re, double im, double base) {
	return log(im);
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = log(im)
end function

Java

public static double code(double re, double im, double base) {
	return Math.log(im);
}

Python

def code(re, im, base):
	return math.log(im)

Julia

function code(re, im, base)
	return log(im)
end

MATLAB

function tmp = code(re, im, base)
	tmp = log(im);
end

Wolfram

code[re_, im_, base_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 49.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 49.7%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. +-rgt-identity 49.7%

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. metadata-eval 49.7%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   4. +-rgt-identity 49.7%

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

   5. times-frac 49.8%

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

   6. *-inverses 49.8%

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

   7. *-rgt-identity 49.8%

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

   8. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in re around 0 25.6%

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

   1. frac-2neg 25.6%

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

   2. div-inv 25.6%

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

   3. add-sqr-sqrt 11.5%

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

   4. sqrt-unprod 13.4%

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

   5. sqr-neg 13.4%

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

   6. sqrt-prod 1.8%

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

   7. add-sqr-sqrt 3.1%

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

6. Applied egg-rr 3.1%

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

   1. log-rec 3.1%

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

   2. associate-*r/ 3.1%

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

   3. *-rgt-identity 3.1%

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

   4. log-rec 3.1%

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

8. Simplified 3.1%

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

   1. add-sqr-sqrt 2.7%

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

   2. add-sqr-sqrt 1.6%

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

   3. sqrt-unprod 13.4%

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

   4. sqr-neg 13.4%

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

   5. sqrt-unprod 11.8%

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

   6. add-sqr-sqrt 13.9%

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

   7. *-un-lft-identity 13.9%

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

   8. times-frac 13.9%

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

10. Applied egg-rr 2.6%

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

    1. associate-*r/ 2.6%

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

    2. *-commutative 2.6%

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

    3. associate-/l* 2.6%

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

    4. *-inverses 3.7%

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

    5. /-rgt-identity 3.7%

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

12. Simplified 3.7%

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

13. Final simplification 3.7%

    \[\leadsto \log im \]

Reproduce

herbie shell --seed 2023195 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))