math.log/1 on complex, real part

Percentage Accurate: 52.5% → 100.0%

Time: 8.3s

Alternatives: 9

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: 52.5% 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 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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: 27.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 im around inf

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

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

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

   2. +-commutative N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. associate-/l* N/A

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

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

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. /-lowering-/.f64 N/A

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

   11. /-lowering-/.f64 21.3%

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

7. Simplified 21.3%

   \[\leadsto \log \color{blue}{\left(im \cdot \left(0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right) + 1\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. fma-define N/A

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

   4. associate-*r* N/A

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

   5. div-inv N/A

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

   6. associate-*r* N/A

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

   7. associate-*l* N/A

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

   8. lft-mult-inverse N/A

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

   9. *-rgt-identity N/A

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

   10. remove-double-neg N/A

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

   11. distribute-rgt-neg-out N/A

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

   12. fmm-def N/A

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

   13. *-lft-identity N/A

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

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

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

   15. distribute-rgt-neg-out N/A

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

   16. clear-num N/A

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

   17. un-div-inv N/A

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

   18. distribute-neg-frac N/A

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

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

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

9. Applied egg-rr 21.7%

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

Alternative 3: 28.0% 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 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 22.1%

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

7. Simplified 22.1%

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

Alternative 4: 3.3% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{im}{\frac{0.5}{\frac{\frac{im}{re}}{re}}}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (/ im (/ 0.5 (/ (/ im re) re)))))

C

double code(double re, double im) {
	return 1.0 / (im / (0.5 / ((im / re) / re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (im / (0.5d0 / ((im / re) / re)))
end function

Java

public static double code(double re, double im) {
	return 1.0 / (im / (0.5 / ((im / re) / re)));
}

Python

def code(re, im):
	return 1.0 / (im / (0.5 / ((im / re) / re)))

Julia

function code(re, im)
	return Float64(1.0 / Float64(im / Float64(0.5 / Float64(Float64(im / re) / re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / (im / (0.5 / ((im / re) / re)));
end

Wolfram

code[re_, im_] := N[(1.0 / N[(im / N[(0.5 / N[(N[(im / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

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

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

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

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

   10. /-lowering-/.f64 20.5%

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

7. Simplified 20.5%

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

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. clear-num N/A

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

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

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

    3. associate-*r* N/A

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

    4. times-frac N/A

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

    5. metadata-eval N/A

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

    6. distribute-lft-neg-in N/A

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

    7. *-commutative N/A

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

    8. associate-/r/ N/A

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

    9. distribute-neg-frac N/A

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

    10. /-lowering-/.f64 N/A

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

    11. associate-/r/ N/A

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

    12. associate-*l/ N/A

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

    13. distribute-neg-frac N/A

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

    14. distribute-lft-neg-in N/A

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

    15. *-commutative N/A

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

    16. associate-/l* N/A

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

    17. clear-num N/A

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

    18. *-commutative N/A

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

    19. distribute-lft-neg-in N/A

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

    20. metadata-eval N/A

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

    21. un-div-inv N/A

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

12. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{\frac{1}{\frac{im}{\frac{re}{\frac{im}{re \cdot 0.5}}}}} \]
13. Step-by-step derivation

    1. associate-/r* N/A

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

    2. associate-/r/ N/A

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

    3. clear-num N/A

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

    4. associate-/r* N/A

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

    5. associate-*l/ N/A

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

    6. metadata-eval N/A

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

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

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

    8. associate-/r* N/A

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

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

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

    10. /-lowering-/.f64 3.2%

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

14. Applied egg-rr 3.2%

    \[\leadsto \frac{1}{\frac{im}{\color{blue}{\frac{0.5}{\frac{\frac{im}{re}}{re}}}}} \]
15. Add Preprocessing

Alternative 5: 3.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{re}{im \cdot \frac{2}{re}}}{im} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (/ re (* im (/ 2.0 re))) im))

C

double code(double re, double im) {
	return (re / (im * (2.0 / re))) / im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (re / (im * (2.0d0 / re))) / im
end function

Java

public static double code(double re, double im) {
	return (re / (im * (2.0 / re))) / im;
}

Python

def code(re, im):
	return (re / (im * (2.0 / re))) / im

Julia

function code(re, im)
	return Float64(Float64(re / Float64(im * Float64(2.0 / re))) / im)
end

MATLAB

function tmp = code(re, im)
	tmp = (re / (im * (2.0 / re))) / im;
end

Wolfram

code[re_, im_] := N[(N[(re / N[(im * N[(2.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

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

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

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

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

   10. /-lowering-/.f64 20.5%

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

7. Simplified 20.5%

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

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. clear-num N/A

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

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

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

    3. associate-*r* N/A

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

    4. times-frac N/A

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

    5. metadata-eval N/A

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

    6. distribute-lft-neg-in N/A

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

    7. *-commutative N/A

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

    8. associate-/r/ N/A

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

    9. distribute-neg-frac N/A

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

    10. /-lowering-/.f64 N/A

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

    11. associate-/r/ N/A

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

    12. associate-*l/ N/A

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

    13. distribute-neg-frac N/A

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

    14. distribute-lft-neg-in N/A

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

    15. *-commutative N/A

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

    16. associate-/l* N/A

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

    17. clear-num N/A

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

    18. *-commutative N/A

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

    19. distribute-lft-neg-in N/A

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

    20. metadata-eval N/A

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

    21. un-div-inv N/A

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

12. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{\frac{1}{\frac{im}{\frac{re}{\frac{im}{re \cdot 0.5}}}}} \]
13. Step-by-step derivation

    1. clear-num N/A

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

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

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

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

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

    4. div-inv N/A

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

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

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

    6. *-commutative N/A

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

    7. associate-/r* N/A

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

    8. metadata-eval N/A

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

    9. /-lowering-/.f64 3.2%

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

14. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{\frac{\frac{re}{im \cdot \frac{2}{re}}}{im}} \]
15. Add Preprocessing

Alternative 6: 3.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

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

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

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

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

   10. /-lowering-/.f64 20.5%

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

7. Simplified 20.5%

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

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. *-commutative N/A

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

    2. times-frac N/A

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

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

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

    4. associate-/l* N/A

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

    5. clear-num N/A

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

    6. div-inv N/A

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

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

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

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

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

    9. /-lowering-/.f64 3.2%

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

12. Applied egg-rr 3.2%

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

Alternative 7: 3.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

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

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

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

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

   10. /-lowering-/.f64 20.5%

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

7. Simplified 20.5%

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

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. associate-/l* N/A

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

    2. times-frac N/A

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

    3. associate-*r* N/A

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

    4. metadata-eval N/A

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

    5. times-frac N/A

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

    6. *-commutative N/A

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

    7. neg-mul-1 N/A

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

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

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

    9. associate-/l* N/A

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

    10. metadata-eval N/A

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

    11. frac-2neg N/A

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

    12. clear-num N/A

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

    13. un-div-inv N/A

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

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

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

    15. /-lowering-/.f64 N/A

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

    16. /-lowering-/.f64 3.2%

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

12. Applied egg-rr 3.2%

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

Alternative 8: 3.0% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

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

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

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

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

   10. /-lowering-/.f64 20.5%

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

7. Simplified 20.5%

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

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. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. associate-/l* N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. *-commutative N/A

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

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

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

    8. clear-num N/A

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

    9. un-div-inv N/A

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

    10. /-lowering-/.f64 N/A

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

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

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

    12. *-lowering-*.f64 2.8%

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

12. Applied egg-rr 2.8%

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

13. Final simplification 2.8%

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

Alternative 9: 2.8% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

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

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

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

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

   10. /-lowering-/.f64 20.5%

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

7. Simplified 20.5%

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

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. *-commutative N/A

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

    2. associate-/l* N/A

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

    3. *-commutative N/A

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

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

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

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

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

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

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

    7. *-lowering-*.f64 2.6%

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

12. Applied egg-rr 2.6%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))