math.log/1 on complex, real part

Percentage Accurate: 52.2% → 100.0%

Time: 4.4s

Alternatives: 4

Speedup: 0.6×

Specification

Math

\[\begin{array}{l} \\ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im))))
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im))));
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im))))

Julia

function code(re, im)
	return log(sqrt(Float64(Float64(re * re) + Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im))));
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im))))
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im))));
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im))))

Julia

function code(re, im)
	return log(sqrt(Float64(Float64(re * re) + Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im))));
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (log (hypot re im)))

C

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

Java

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

Python

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

Julia

function code(re, im)
	return log(hypot(re, im))
end

MATLAB

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

Wolfram

code[re_, im_] := N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-hypot.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 27.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.0%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. associate-*l* N/A

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

   10. /-rgt-identity N/A

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

   11. times-frac N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-lowering-*.f64 N/A

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

   15. /-lowering-/.f64 25.2

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

5. Simplified 25.2%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(re, 0.5 \cdot \frac{re}{im}, im\right)\right)} \]
6. Add Preprocessing

Alternative 3: 27.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im)
end function

Java

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

Python

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

Julia

function code(re, im)
	return log(im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. log-lowering-log.f64 25.6

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

5. Simplified 25.6%

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

Alternative 4: 0.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \frac{0}{0} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 0.0 0.0))

C

double code(double re, double im) {
	return 0.0 / 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0 / 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0 / 0.0;
}

Python

def code(re, im):
	return 0.0 / 0.0

Julia

function code(re, im)
	return Float64(0.0 / 0.0)
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0 / 0.0;
end

Wolfram

code[re_, im_] := N[(0.0 / 0.0), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 N/A

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

   2. pow-to-exp N/A

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

   3. rem-log-exp N/A

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

   4. *-lowering-*.f64 N/A

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

   5. log-lowering-log.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 53.0

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

4. Applied egg-rr 53.0%

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

5. Taylor expanded in re around 0

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

   1. log-lowering-log.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 30.7

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

7. Simplified 30.7%

   \[\leadsto \color{blue}{\log \left(im \cdot im\right)} \cdot 0.5 \]
8. Step-by-step derivation

   1. log-prod N/A

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

   2. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\log im \cdot \log im - \log im \cdot \log im}{\log im - \log im}} \cdot \frac{1}{2} \]

   3. +-inverses N/A

      \[\leadsto \frac{\log im \cdot \log im - \log im \cdot \log im}{\color{blue}{0}} \cdot \frac{1}{2} \]

   4. +-inverses N/A

      \[\leadsto \frac{\log im \cdot \log im - \log im \cdot \log im}{\color{blue}{\log im \cdot \log im - \log im \cdot \log im}} \cdot \frac{1}{2} \]

   5. associate-*l/ N/A

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

   6. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \frac{1}{2}}{\log im \cdot \log im - \log im \cdot \log im} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{\log im \cdot \log im - \log im \cdot \log im} \]

   8. +-inverses N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-inverses N/A

       \[\leadsto \frac{\color{blue}{0}}{\log im \cdot \log im - \log im \cdot \log im} \]

   11. +-inverses 0.0

       \[\leadsto \frac{0}{\color{blue}{0}} \]

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{0}{0}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))