math.log/2 on complex, real part

Percentage Accurate: 52.7% → 99.5%

Time: 12.9s

Alternatives: 3

Speedup: 1.8×

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: 52.7% 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: 99.5% accurate, 1.8× 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 54.9%

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

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 2: 25.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{re}{im} \cdot -0.5, \frac{re}{im}, -\log im\right) \cdot \frac{-1}{\log base} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (* (fma (* (/ re im) -0.5) (/ re im) (- (log im))) (/ -1.0 (log base))))

C

double code(double re, double im, double base) {
	return fma(((re / im) * -0.5), (re / im), -log(im)) * (-1.0 / log(base));
}

Julia

function code(re, im, base)
	return Float64(fma(Float64(Float64(re / im) * -0.5), Float64(re / im), Float64(-log(im))) * Float64(-1.0 / log(base)))
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[(re / im), $MachinePrecision] * -0.5), $MachinePrecision] * N[(re / im), $MachinePrecision] + (-N[Log[im], $MachinePrecision])), $MachinePrecision] * N[(-1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

   3. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   5. mul0-rgt N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   6. metadata-eval N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. sub-neg N/A

      \[\leadsto \frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \left(\mathsf{neg}\left(0\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}\right) \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \color{blue}{\left(0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)} \]

   12. +-lft-identity N/A

       \[\leadsto \frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)} \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\left({\log base}^{-2} \cdot \log base\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\log base}^{-2} \cdot \log base\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\log base}^{-2} \cdot \log base\right)} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\log base}^{-2}} \cdot \log base\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right) \]

   4. pow-plus N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-div N/A

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

   8. unpow1 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-/r/ N/A

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

   11. *-rgt-identity N/A

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

   12. frac-2neg N/A

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

   13. div-inv N/A

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

   14. times-frac N/A

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in re around 0

   \[\leadsto \frac{-1}{\log base} \cdot \color{blue}{\left(-1 \cdot \log im + \frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. log-rec N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

      \[\leadsto \frac{-1}{\log base} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{re}{im}, \frac{re}{im}, \log \left(\frac{1}{im}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\log base} \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \frac{re}{im}}, \frac{re}{im}, \log \left(\frac{1}{im}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{-1}{\log base} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{re}{im}}, \frac{re}{im}, \log \left(\frac{1}{im}\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{-1}{\log base} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{re}{im}, \color{blue}{\frac{re}{im}}, \log \left(\frac{1}{im}\right)\right) \]

   12. log-rec N/A

       \[\leadsto \frac{-1}{\log base} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{re}{im}, \frac{re}{im}, \color{blue}{\mathsf{neg}\left(\log im\right)}\right) \]

   13. lower-neg.f64 N/A

       \[\leadsto \frac{-1}{\log base} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{re}{im}, \frac{re}{im}, \color{blue}{-\log im}\right) \]

   14. lower-log.f64 23.2

       \[\leadsto \frac{-1}{\log base} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{re}{im}, \frac{re}{im}, -\color{blue}{\log im}\right) \]

9. Applied rewrites 23.2%

   \[\leadsto \frac{-1}{\log base} \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{re}{im}, \frac{re}{im}, -\log im\right)} \]

10. Final simplification 23.2%

    \[\leadsto \mathsf{fma}\left(\frac{re}{im} \cdot -0.5, \frac{re}{im}, -\log im\right) \cdot \frac{-1}{\log base} \]
11. Add Preprocessing

Alternative 3: 27.2% accurate, 2.7× 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 54.9%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 25.0

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

5. Applied rewrites 25.0%

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

Reproduce

herbie shell --seed 2024264 
(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))))