math.log/1 on complex, real part

Percentage Accurate: 51.5% → 100.0%

Time: 8.3s

Alternatives: 7

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: 51.5% 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 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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: 27.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(im \cdot \left(0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (* im (+ (* 0.5 (* re (/ (/ re im) im))) 1.0))))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return Math.log((im * ((0.5 * (re * ((re / im) / im))) + 1.0)));
}

Python

def code(re, im):
	return math.log((im * ((0.5 * (re * ((re / im) / im))) + 1.0)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 25.5%

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

7. Simplified 25.5%

   \[\leadsto \log \color{blue}{\left(im \cdot \left(0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right) + 1\right)\right)} \]
8. Add Preprocessing

Alternative 3: 26.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 24.2%

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

7. Simplified 24.2%

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

8. Final simplification 24.2%

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

Alternative 4: 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 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 26.1%

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

7. Simplified 26.1%

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

Alternative 5: 3.3% accurate, 23.0× speedup

Math

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

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 / im) / Float64(Float64(im / 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 / im), $MachinePrecision] / N[(N[(im / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 24.2%

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

7. Simplified 24.2%

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

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 2.8%

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

10. Simplified 2.8%

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

    1. frac-2neg N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-neg-in N/A

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

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

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

    5. times-frac N/A

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

    6. distribute-frac-neg2 N/A

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

    7. associate-*r/ N/A

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

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

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

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

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

    10. distribute-frac-neg N/A

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

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

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

    12. *-commutative N/A

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

    13. distribute-rgt-neg-in N/A

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

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

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

    15. metadata-eval N/A

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

    16. neg-sub0 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{2}\right), im\right), \left(0 - re\right)\right), im\right) \]

    17. --lowering--.f64 3.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{2}\right), im\right), \mathsf{\_.f64}\left(0, re\right)\right), im\right) \]

12. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{\frac{\frac{re \cdot -0.5}{im} \cdot \left(0 - re\right)}{im}} \]
13. Step-by-step derivation

    1. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{re \cdot \frac{-1}{2}}{im} \cdot \left(0 - re\right)\right)}{\color{blue}{\mathsf{neg}\left(im\right)}} \]

    2. sub0-neg N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. remove-double-neg N/A

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

    5. associate-/r/ N/A

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

    6. associate-/r* N/A

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

    7. frac-times N/A

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

    8. sub0-neg N/A

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

    9. *-commutative N/A

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

    10. clear-num N/A

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

    11. un-div-inv N/A

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

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

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{0 - im}\right), \color{blue}{\left(\frac{\frac{im}{re}}{re}\right)}\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{0 - im}\right), \left(\frac{\frac{\color{blue}{im}}{re}}{re}\right)\right) \]

    14. sub0-neg N/A

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

    15. frac-2neg N/A

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

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

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

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

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

    18. /-lowering-/.f64 3.5%

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

14. Applied egg-rr 3.5%

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

Alternative 6: 3.3% accurate, 23.0× speedup

Math

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

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 / im) * Float64(Float64(re * 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 / im), $MachinePrecision] * N[(N[(re * 0.5), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 24.2%

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

7. Simplified 24.2%

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

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 2.8%

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

10. Simplified 2.8%

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

    1. associate-*r* N/A

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

    2. times-frac N/A

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

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

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

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

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

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

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

    6. /-lowering-/.f64 3.5%

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

12. Applied egg-rr 3.5%

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

13. Final simplification 3.5%

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

Alternative 7: 3.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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 24.2%

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

7. Simplified 24.2%

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

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 2.8%

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

10. Simplified 2.8%

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

    1. frac-2neg N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-neg-in N/A

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

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

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

    5. times-frac N/A

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

    6. distribute-frac-neg2 N/A

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

    7. associate-*r/ N/A

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

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

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

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

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

    10. distribute-frac-neg N/A

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

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

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

    12. *-commutative N/A

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

    13. distribute-rgt-neg-in N/A

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

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

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

    15. metadata-eval N/A

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

    16. neg-sub0 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{2}\right), im\right), \left(0 - re\right)\right), im\right) \]

    17. --lowering--.f64 3.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{2}\right), im\right), \mathsf{\_.f64}\left(0, re\right)\right), im\right) \]

12. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{\frac{\frac{re \cdot -0.5}{im} \cdot \left(0 - re\right)}{im}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

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

    2. associate-/l* N/A

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

    3. frac-times N/A

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

    4. distribute-rgt-neg-in N/A

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

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

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

    6. associate-*r/ N/A

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

    7. *-commutative N/A

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

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

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. associate-*r* N/A

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

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

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

    13. clear-num N/A

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

    14. associate-*l/ N/A

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

    15. metadata-eval N/A

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

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

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

    17. clear-num N/A

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

    18. associate-/r* N/A

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

    19. clear-num N/A

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

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

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

    21. /-lowering-/.f64 3.5%

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

14. Applied egg-rr 3.5%

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

15. Final simplification 3.5%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))