math.log/1 on complex, real part

Percentage Accurate: 51.4% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 1.0×

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: 51.4% 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, 1.0× 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 51.1%

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

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

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 27.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im + ((re * 0.5d0) * (re / im))))
end function

Java

public static double code(double re, double im) {
	return Math.log((im + ((re * 0.5) * (re / im))));
}

Python

def code(re, im):
	return math.log((im + ((re * 0.5) * (re / im))))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = log((im + ((re * 0.5) * (re / im))));
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

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

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(im + \frac{1}{2} \cdot \frac{{re}^{2}}{im}\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{{re}^{2}}{im} \cdot \frac{1}{2}\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{{re}^{2} \cdot \frac{1}{2}}{im}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + {re}^{2} \cdot \frac{\frac{1}{2}}{im}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + {re}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{im}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2} \cdot 1}{im} \cdot {re}^{2}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2}}{im} \cdot {re}^{2}\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2} \cdot {re}^{2}}{im}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), im\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right)\right)\right) \]

   14. *-lowering-*.f64 23.3%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right)\right)\right) \]

7. Simplified 23.3%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\left(\frac{1}{2} \cdot re\right) \cdot re}{im}\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(\frac{1}{2} \cdot re\right) \cdot \frac{re}{im}\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot re\right), \left(\frac{re}{im}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \left(\frac{re}{im}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 25.1%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{/.f64}\left(re, im\right)\right)\right)\right) \]

9. Applied egg-rr 25.1%

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

10. Final simplification 25.1%

    \[\leadsto \log \left(im + \left(re \cdot 0.5\right) \cdot \frac{re}{im}\right) \]
11. Add Preprocessing

Alternative 3: 27.7% accurate, 2.0× 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 51.1%

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

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

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

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

   1. log-lowering-log.f64 25.4%

      \[\leadsto \mathsf{log.f64}\left(im\right) \]

7. Simplified 25.4%

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

Alternative 4: 2.9% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ re \cdot \frac{0.5}{\frac{im}{re}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (0.5d0 / (im / re))
end function

Java

public static double code(double re, double im) {
	return re * (0.5 / (im / re));
}

Python

def code(re, im):
	return re * (0.5 / (im / re))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = re * (0.5 / (im / re));
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

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

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\log im + \frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\log im, \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{re}^{2}}{{im}^{2}}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{1}{2} \cdot \frac{{re}^{2}}{im \cdot \color{blue}{im}}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{1}{2} \cdot \frac{\frac{{re}^{2}}{im}}{\color{blue}{im}}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{\frac{1}{2} \cdot \frac{{re}^{2}}{im}}{\color{blue}{im}}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{\frac{{re}^{2}}{im} \cdot \frac{1}{2}}{im}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{\frac{{re}^{2} \cdot \frac{1}{2}}{im}}{im}\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{{re}^{2} \cdot \frac{\frac{1}{2}}{im}}{im}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{{re}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{im}}{im}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{{re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)}{im}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right), \color{blue}{im}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}\right), im\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{im} \cdot {re}^{2}\right), im\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{im} \cdot {re}^{2}\right), im\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot {re}^{2}}{im}\right), im\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), im\right), im\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right), im\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right), im\right)\right) \]

   19. *-lowering-*.f64 22.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right), im\right)\right) \]

7. Simplified 22.4%

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

8. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot {re}^{2}}{\color{blue}{{im}^{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \left({\color{blue}{im}}^{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \left({im}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left({im}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left(im \cdot \color{blue}{im}\right)\right) \]

   7. *-lowering-*.f64 2.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

10. Simplified 2.6%

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

    1. pow2 N/A

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

    2. pow-to-exp N/A

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

    3. *-commutative N/A

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

    4. count-2 N/A

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

    5. flip-+ N/A

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

    6. +-inverses N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{0}{\log im - \log im}}} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{\frac{1}{2} \cdot 0}{\log im - \log im}}} \]

    8. +-inverses N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{\frac{1}{2} \cdot 0}{0}}} \]

    9. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{1}{2} \cdot \frac{0}{0}}} \]

    10. +-inverses N/A

        \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{1}{2} \cdot \frac{\log im \cdot \log im - \log im \cdot \log im}{0}}} \]

    11. +-inverses N/A

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

    12. flip-+ N/A

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

    13. distribute-lft-in N/A

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

    14. distribute-rgt-out N/A

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

    15. metadata-eval N/A

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

    16. pow-to-exp N/A

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

    17. unpow1 N/A

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

    18. associate-*r* N/A

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

    19. associate-*r/ N/A

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

    20. *-commutative N/A

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

    21. div-inv N/A

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

    22. associate-*l* N/A

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

12. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{im}{re}} \cdot re} \]

13. Final simplification 3.1%

    \[\leadsto re \cdot \frac{0.5}{\frac{im}{re}} \]
14. Add Preprocessing

Alternative 5: 2.9% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 / im) * (re * re)
end function

Java

public static double code(double re, double im) {
	return (0.5 / im) * (re * re);
}

Python

def code(re, im):
	return (0.5 / im) * (re * re)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 / im) * (re * re);
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

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

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\log im + \frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\log im, \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{re}^{2}}{{im}^{2}}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{1}{2} \cdot \frac{{re}^{2}}{im \cdot \color{blue}{im}}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{1}{2} \cdot \frac{\frac{{re}^{2}}{im}}{\color{blue}{im}}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{\frac{1}{2} \cdot \frac{{re}^{2}}{im}}{\color{blue}{im}}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{\frac{{re}^{2}}{im} \cdot \frac{1}{2}}{im}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{\frac{{re}^{2} \cdot \frac{1}{2}}{im}}{im}\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{{re}^{2} \cdot \frac{\frac{1}{2}}{im}}{im}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{{re}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{im}}{im}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\frac{{re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)}{im}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right), \color{blue}{im}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}\right), im\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{im} \cdot {re}^{2}\right), im\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{im} \cdot {re}^{2}\right), im\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot {re}^{2}}{im}\right), im\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), im\right), im\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right), im\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right), im\right)\right) \]

   19. *-lowering-*.f64 22.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right), im\right)\right) \]

7. Simplified 22.4%

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

8. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot {re}^{2}}{\color{blue}{{im}^{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \left({\color{blue}{im}}^{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \left({im}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left({im}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left(im \cdot \color{blue}{im}\right)\right) \]

   7. *-lowering-*.f64 2.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

10. Simplified 2.6%

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

    1. pow2 N/A

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

    2. pow-to-exp N/A

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

    3. *-commutative N/A

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

    4. count-2 N/A

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

    5. flip-+ N/A

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

    6. +-inverses N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{0}{\log im - \log im}}} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{\frac{1}{2} \cdot 0}{\log im - \log im}}} \]

    8. +-inverses N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{\frac{1}{2} \cdot 0}{0}}} \]

    9. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{1}{2} \cdot \frac{0}{0}}} \]

    10. +-inverses N/A

        \[\leadsto \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{e^{\frac{1}{2} \cdot \frac{\log im \cdot \log im - \log im \cdot \log im}{0}}} \]

    11. +-inverses N/A

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

    12. flip-+ N/A

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

    13. distribute-lft-in N/A

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

    14. distribute-rgt-out N/A

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

    15. metadata-eval N/A

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

    16. pow-to-exp N/A

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

    17. unpow1 N/A

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

    18. associate-*l/ N/A

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

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

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{im}\right), \color{blue}{\left(re \cdot re\right)}\right) \]

    20. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, im\right), \left(\color{blue}{re} \cdot re\right)\right) \]

    21. *-lowering-*.f64 3.0%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, im\right), \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right) \]

12. Applied egg-rr 3.0%

    \[\leadsto \color{blue}{\frac{0.5}{im} \cdot \left(re \cdot re\right)} \]
13. Add Preprocessing

Alternative 6: 0.0% accurate, 69.0× 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 51.1%

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

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

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

   1. pow1/2 N/A

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

   2. pow-to-exp N/A

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

   3. rem-log-exp N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\log \left(re \cdot re + im \cdot im\right), \color{blue}{\frac{1}{2}}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(re \cdot re + im \cdot im\right)\right), \frac{1}{2}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(re \cdot re\right), \left(im \cdot im\right)\right)\right), \frac{1}{2}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot im\right)\right)\right), \frac{1}{2}\right) \]

   8. *-lowering-*.f64 51.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, im\right)\right)\right), \frac{1}{2}\right) \]

6. Applied egg-rr 51.1%

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

7. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\log \left({im}^{2}\right)}, \frac{1}{2}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left({im}^{2}\right)\right), \frac{1}{2}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(im \cdot im\right)\right), \frac{1}{2}\right) \]

   3. *-lowering-*.f64 25.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(im, im\right)\right), \frac{1}{2}\right) \]

9. Simplified 25.1%

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

    1. log-prod N/A

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

    2. flip-+ N/A

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

    3. +-inverses N/A

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

    4. +-inverses N/A

       \[\leadsto \frac{0}{0} \cdot \frac{1}{2} \]

    5. associate-*l/ N/A

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

    6. metadata-eval N/A

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

    7. /-lowering-/.f64 0.0%

       \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{0}\right) \]

11. Applied egg-rr 0.0%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))