math.log/2 on complex, real part

Percentage Accurate: 52.3% → 99.3%

Time: 9.8s

Alternatives: 5

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: 52.3% 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: 99.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im base)
 :precision binary64
 (* (/ 1.0 (log base)) (log (hypot re im))))

C

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

Java

public static double code(double re, double im, double base) {
	return (1.0 / Math.log(base)) * Math.log(Math.hypot(re, im));
}

Python

def code(re, im, base):
	return (1.0 / math.log(base)) * math.log(math.hypot(re, im))

Julia

function code(re, im, base)
	return Float64(Float64(1.0 / log(base)) * log(hypot(re, im)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = (1.0 / log(base)) * log(hypot(re, im));
end

Wolfram

code[re_, im_, base_] := N[(N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

   1. fma-define 51.5%

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

   2. +-commutative 51.5%

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

   3. mul0-rgt 51.5%

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

   4. mul0-rgt 51.5%

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

   5. fma-define 51.5%

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

3. Simplified 99.3%

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

   1. clear-num 99.3%

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

   2. hypot-define 51.6%

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

   3. associate-/r/ 51.6%

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

   4. hypot-define 99.4%

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

6. Applied egg-rr 99.4%

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

Alternative 2: 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 51.5%

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

   1. fma-define 51.5%

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

   2. +-commutative 51.5%

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

   3. mul0-rgt 51.5%

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

   4. mul0-rgt 51.5%

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

   5. fma-define 51.5%

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

3. Simplified 99.3%

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

Alternative 3: 27.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double re, double im, double base) {
	return (-1.0 / log((1.0 / base))) * 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 = ((-1.0d0) / log((1.0d0 / base))) * log(im)
end function

Java

public static double code(double re, double im, double base) {
	return (-1.0 / Math.log((1.0 / base))) * Math.log(im);
}

Python

def code(re, im, base):
	return (-1.0 / math.log((1.0 / base))) * math.log(im)

Julia

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

MATLAB

function tmp = code(re, im, base)
	tmp = (-1.0 / log((1.0 / base))) * log(im);
end

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. fma-define 51.5%

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

   2. +-commutative 51.5%

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

   3. mul0-rgt 51.5%

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

   4. mul0-rgt 51.5%

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

   5. fma-define 51.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in re around 0 26.0%

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

   1. clear-num 26.0%

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

   2. associate-/r/ 26.1%

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

7. Applied egg-rr 26.1%

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

8. Taylor expanded in base around inf 26.1%

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

Alternative 4: 27.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\log base} \cdot \log im \end{array} \]

FPCore

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

C

double code(double re, double im, double base) {
	return (1.0 / log(base)) * 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 = (1.0d0 / log(base)) * log(im)
end function

Java

public static double code(double re, double im, double base) {
	return (1.0 / Math.log(base)) * Math.log(im);
}

Python

def code(re, im, base):
	return (1.0 / math.log(base)) * math.log(im)

Julia

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

MATLAB

function tmp = code(re, im, base)
	tmp = (1.0 / log(base)) * log(im);
end

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. fma-define 51.5%

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

   2. +-commutative 51.5%

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

   3. mul0-rgt 51.5%

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

   4. mul0-rgt 51.5%

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

   5. fma-define 51.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in re around 0 26.0%

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

   1. clear-num 26.0%

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

   2. associate-/r/ 26.1%

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

7. Applied egg-rr 26.1%

   \[\leadsto \color{blue}{\frac{1}{\log base} \cdot \log im} \]
8. Add Preprocessing

Alternative 5: 27.2% 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 51.5%

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

   1. fma-define 51.5%

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

   2. +-commutative 51.5%

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

   3. mul0-rgt 51.5%

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

   4. mul0-rgt 51.5%

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

   5. fma-define 51.5%

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

3. Simplified 99.3%

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

5. Taylor expanded in re around 0 26.0%

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

Reproduce

herbie shell --seed 2024096 
(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))))