math.log/2 on complex, real part

Percentage Accurate: 52.4% → 99.2%

Time: 16.3s

Alternatives: 3

Speedup: 1.1×

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: 52.4% 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: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{{\log base}^{2}} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/ (+ (* (log (hypot re im)) (log base)) 0.0) (pow (log base) 2.0)))

C

double code(double re, double im, double base) {
	return ((log(hypot(re, im)) * log(base)) + 0.0) / pow(log(base), 2.0);
}

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.hypot(re, im)) * Math.log(base)) + 0.0) / Math.pow(Math.log(base), 2.0);
}

Python

def code(re, im, base):
	return ((math.log(math.hypot(re, im)) * math.log(base)) + 0.0) / math.pow(math.log(base), 2.0)

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(hypot(re, im)) * log(base)) + 0.0) / (log(base) ^ 2.0))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(hypot(re, im)) * log(base)) + 0.0) / (log(base) ^ 2.0);
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] / N[Power[N[Log[base], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in im around 0

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
6. Step-by-step derivation

7. Applied rewrites 99.3%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. +-rgt-identity 99.3

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

   7. lift-pow.f64 99.3

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

9. Applied rewrites 99.3%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]
10. Add Preprocessing

Alternative 2: 27.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
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 19.6

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

5. Applied rewrites 19.6%

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

7. Applied rewrites 19.6%

   \[\leadsto \frac{\log im}{\log \left(\frac{1}{\frac{1}{base}}\right)} \]
8. Add Preprocessing

Alternative 3: 27.6% 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 53.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
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 19.6

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

5. Applied rewrites 19.6%

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

Reproduce

herbie shell --seed 2024231 
(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))))