math.log/2 on complex, real part

Percentage Accurate: 52.2% → 99.4%

Time: 10.7s

Alternatives: 3

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: 52.2% 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 47.0%

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

   1. mul0-rgt 47.0%

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

   2. +-rgt-identity 47.0%

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

   3. metadata-eval 47.0%

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

   4. +-rgt-identity 47.0%

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

   5. times-frac 47.0%

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

   6. *-inverses 47.0%

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

   7. *-rgt-identity 47.0%

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

   8. hypot-def 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 25.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{re}{im} \cdot \frac{re}{im}\right) \cdot -0.5 - \log im}{-\log base} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/ (- (* (* (/ re im) (/ re im)) -0.5) (log im)) (- (log base))))

C

double code(double re, double im, double base) {
	return ((((re / im) * (re / im)) * -0.5) - log(im)) / -log(base);
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((((re / im) * (re / im)) * (-0.5d0)) - log(im)) / -log(base)
end function

Java

public static double code(double re, double im, double base) {
	return ((((re / im) * (re / im)) * -0.5) - Math.log(im)) / -Math.log(base);
}

Python

def code(re, im, base):
	return ((((re / im) * (re / im)) * -0.5) - math.log(im)) / -math.log(base)

Julia

function code(re, im, base)
	return Float64(Float64(Float64(Float64(Float64(re / im) * Float64(re / im)) * -0.5) - log(im)) / Float64(-log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((((re / im) * (re / im)) * -0.5) - log(im)) / -log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - N[Log[im], $MachinePrecision]), $MachinePrecision] / (-N[Log[base], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 47.0%

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

   1. mul0-rgt 47.0%

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

   2. +-rgt-identity 47.0%

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

   3. metadata-eval 47.0%

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

   4. +-rgt-identity 47.0%

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

   5. times-frac 47.0%

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

   6. *-inverses 47.0%

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

   7. *-rgt-identity 47.0%

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

   8. hypot-def 99.3%

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

3. Simplified 99.3%

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

   1. add-log-exp 99.0%

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

   2. add-cube-cbrt 97.5%

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

   3. log-prod 97.5%

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

   4. div-inv 97.5%

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

   5. exp-to-pow 97.5%

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

   6. div-inv 97.5%

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

   7. exp-to-pow 97.6%

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

5. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log base}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log base}\right)}}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log base}\right)}}\right)} \]
6. Step-by-step derivation

   1. log-prod 97.5%

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

   2. count-2 97.5%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log base}\right)}}\right)} + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log base}\right)}}\right) \]

   3. distribute-lft1-in 97.5%

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log base}\right)}}\right)} \]

   4. metadata-eval 97.5%

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

   5. hypot-def 45.8%

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

   6. unpow2 45.8%

      \[\leadsto 3 \cdot \log \left(\sqrt[3]{{\left(\sqrt{\color{blue}{{re}^{2}} + im \cdot im}\right)}^{\left(\frac{1}{\log base}\right)}}\right) \]

   7. unpow2 45.8%

      \[\leadsto 3 \cdot \log \left(\sqrt[3]{{\left(\sqrt{{re}^{2} + \color{blue}{{im}^{2}}}\right)}^{\left(\frac{1}{\log base}\right)}}\right) \]

   8. +-commutative 45.8%

      \[\leadsto 3 \cdot \log \left(\sqrt[3]{{\left(\sqrt{\color{blue}{{im}^{2} + {re}^{2}}}\right)}^{\left(\frac{1}{\log base}\right)}}\right) \]

   9. unpow2 45.8%

      \[\leadsto 3 \cdot \log \left(\sqrt[3]{{\left(\sqrt{\color{blue}{im \cdot im} + {re}^{2}}\right)}^{\left(\frac{1}{\log base}\right)}}\right) \]

   10. unpow2 45.8%

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

   11. hypot-def 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in base around 0 46.3%

   \[\leadsto 3 \cdot \color{blue}{\log \left({\left(e^{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}}\right)}^{0.3333333333333333}\right)} \]
9. Step-by-step derivation

   1. log-pow 46.6%

      \[\leadsto 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}}\right)\right)} \]

   2. rem-log-exp 46.9%

      \[\leadsto 3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}}\right) \]

   3. unpow2 46.9%

      \[\leadsto 3 \cdot \left(0.3333333333333333 \cdot \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + {im}^{2}}\right)}{\log base}\right) \]

   4. unpow2 46.9%

      \[\leadsto 3 \cdot \left(0.3333333333333333 \cdot \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log base}\right) \]

   5. hypot-def 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l/ 99.1%

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

10. Simplified 99.1%

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

    1. *-commutative 99.1%

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

    2. frac-2neg 99.1%

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

    3. associate-*l/ 99.2%

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

    4. distribute-rgt-neg-in 99.2%

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

    5. metadata-eval 99.2%

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

12. Applied egg-rr 99.2%

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

13. Taylor expanded in re around 0 24.7%

    \[\leadsto \frac{\color{blue}{-1 \cdot \log im + -0.5 \cdot \frac{{re}^{2}}{{im}^{2}}}}{-\log base} \]
14. Step-by-step derivation

    1. neg-mul-1 24.7%

       \[\leadsto \frac{\color{blue}{\left(-\log im\right)} + -0.5 \cdot \frac{{re}^{2}}{{im}^{2}}}{-\log base} \]

    2. +-commutative 24.7%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{{re}^{2}}{{im}^{2}} + \left(-\log im\right)}}{-\log base} \]

    3. unsub-neg 24.7%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{{re}^{2}}{{im}^{2}} - \log im}}{-\log base} \]

    4. *-commutative 24.7%

       \[\leadsto \frac{\color{blue}{\frac{{re}^{2}}{{im}^{2}} \cdot -0.5} - \log im}{-\log base} \]

    5. unpow2 24.7%

       \[\leadsto \frac{\frac{\color{blue}{re \cdot re}}{{im}^{2}} \cdot -0.5 - \log im}{-\log base} \]

    6. unpow2 24.7%

       \[\leadsto \frac{\frac{re \cdot re}{\color{blue}{im \cdot im}} \cdot -0.5 - \log im}{-\log base} \]

    7. times-frac 28.0%

       \[\leadsto \frac{\color{blue}{\left(\frac{re}{im} \cdot \frac{re}{im}\right)} \cdot -0.5 - \log im}{-\log base} \]

15. Simplified 28.0%

    \[\leadsto \frac{\color{blue}{\left(\frac{re}{im} \cdot \frac{re}{im}\right) \cdot -0.5 - \log im}}{-\log base} \]

16. Final simplification 28.0%

    \[\leadsto \frac{\left(\frac{re}{im} \cdot \frac{re}{im}\right) \cdot -0.5 - \log im}{-\log base} \]

Alternative 3: 27.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

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) / log(base)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_, base_] := N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

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

   1. mul0-rgt 47.0%

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

   2. +-rgt-identity 47.0%

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

   3. metadata-eval 47.0%

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

   4. +-rgt-identity 47.0%

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

   5. times-frac 47.0%

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

   6. *-inverses 47.0%

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

   7. *-rgt-identity 47.0%

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

   8. hypot-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in re around 0 29.3%

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

5. Final simplification 29.3%

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

Reproduce

herbie shell --seed 2023201 
(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))))