math.log/2 on complex, real part

Percentage Accurate: 51.6% → 99.4%

Time: 9.9s

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: 51.6% 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 47.9%

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

   1. mul0-rgt 47.9%

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

   2. +-rgt-identity 47.9%

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

   3. metadata-eval 47.9%

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

   4. +-rgt-identity 47.9%

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

   5. times-frac 48.0%

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

   6. *-inverses 48.0%

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

   7. *-rgt-identity 48.0%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 26.9% 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 47.9%

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

   1. mul0-rgt 47.9%

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

   2. +-rgt-identity 47.9%

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

   3. metadata-eval 47.9%

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

   4. +-rgt-identity 47.9%

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

   5. times-frac 48.0%

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

   6. *-inverses 48.0%

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

   7. *-rgt-identity 48.0%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in re around 0 28.6%

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

5. Final simplification 28.6%

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

Alternative 3: 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 47.9%

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

   1. mul0-rgt 47.9%

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

   2. +-rgt-identity 47.9%

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

   3. metadata-eval 47.9%

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

   4. +-rgt-identity 47.9%

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

   5. times-frac 48.0%

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

   6. *-inverses 48.0%

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

   7. *-rgt-identity 48.0%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in re around 0 28.6%

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

   1. clear-num 28.6%

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

   2. inv-pow 28.6%

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

6. Applied egg-rr 28.6%

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

   1. unpow-1 28.6%

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

8. Simplified 28.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\log im}}} \]
9. Step-by-step derivation

   1. associate-/l* 28.6%

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

   2. add-sqr-sqrt 10.8%

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

   3. times-frac 10.8%

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

   4. metadata-eval 10.8%

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

   5. sqrt-div 10.8%

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

   6. add-exp-log 10.8%

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

   7. neg-log 10.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 2.2%

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

   10. sqr-neg 2.2%

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

   11. sqrt-unprod 2.2%

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

   12. add-sqr-sqrt 2.2%

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

   13. add-exp-log 2.2%

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

10. Applied egg-rr 2.2%

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

    1. associate-*r/ 2.2%

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

    2. *-commutative 2.2%

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

    3. associate-/l* 2.2%

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

    4. *-inverses 3.3%

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

    5. /-rgt-identity 3.3%

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

12. Simplified 3.3%

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

13. Final simplification 3.3%

    \[\leadsto \log im \]

Reproduce

herbie shell --seed 2023200 
(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))))