math.log/1 on complex, real part

Percentage Accurate: 52.2% → 100.0%

Time: 7.6s

Alternatives: 5

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: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 53.0%

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

   1. accelerator-lowering-hypot.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 26.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(im + \frac{re \cdot \left(\frac{re \cdot -0.125}{im \cdot \frac{im}{re}} + 0.5\right)}{\frac{im}{re}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (+ im (/ (* re (+ (/ (* re -0.125) (* im (/ im re))) 0.5)) (/ im re)))))

C

double code(double re, double im) {
	return log((im + ((re * (((re * -0.125) / (im * (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 * (((re * (-0.125d0)) / (im * (im / re))) + 0.5d0)) / (im / re))))
end function

Java

public static double code(double re, double im) {
	return Math.log((im + ((re * (((re * -0.125) / (im * (im / re))) + 0.5)) / (im / re))));
}

Python

def code(re, im):
	return math.log((im + ((re * (((re * -0.125) / (im * (im / re))) + 0.5)) / (im / re))))

Julia

function code(re, im)
	return log(Float64(im + Float64(Float64(re * Float64(Float64(Float64(re * -0.125) / Float64(im * Float64(im / re))) + 0.5)) / Float64(im / re))))
end

MATLAB

function tmp = code(re, im)
	tmp = log((im + ((re * (((re * -0.125) / (im * (im / re))) + 0.5)) / (im / re))));
end

Wolfram

code[re_, im_] := N[Log[N[(im + N[(N[(re * N[(N[(N[(re * -0.125), $MachinePrecision] / N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(im + {re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}\right) + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{3}} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2} \cdot \left(\frac{-1}{8} \cdot {re}^{2}\right)}{{im}^{3}} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   5. cube-mult N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2} \cdot \left(\frac{-1}{8} \cdot {re}^{2}\right)}{im \cdot \left(im \cdot im\right)} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2} \cdot \left(\frac{-1}{8} \cdot {re}^{2}\right)}{im \cdot {im}^{2}} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2}}{im} \cdot \frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + {re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2}}{im} \cdot \frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + {re}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{im}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2}}{im} \cdot \frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + {re}^{2} \cdot \frac{\frac{1}{2}}{im}\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2}}{im} \cdot \frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{{re}^{2} \cdot \frac{1}{2}}{im}\right)\right)\right) \]

   11. associate-*l/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2}}{im} \cdot \frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{{re}^{2}}{im} \cdot \frac{1}{2}\right)\right)\right) \]

   12. distribute-lft-out N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{{re}^{2}}{im} \cdot \left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left(\frac{{re}^{2}}{im}\right), \left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({re}^{2}\right), im\right), \left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{1}{2}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(re \cdot re\right), im\right), \left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, re\right), im\right), \left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}} + \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, re\right), im\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{2}}\right), \frac{1}{2}\right)\right)\right)\right) \]

5. Simplified 21.3%

   \[\leadsto \log \color{blue}{\left(im + \frac{re \cdot re}{im} \cdot \left(\frac{re \cdot \left(re \cdot -0.125\right)}{im \cdot im} + 0.5\right)\right)} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot \frac{re}{im}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{8}\right)\right), \mathsf{*.f64}\left(im, im\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left(\frac{re}{im} \cdot re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{8}\right)\right), \mathsf{*.f64}\left(im, im\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{re}{im}\right), re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{8}\right)\right), \mathsf{*.f64}\left(im, im\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 21.3%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{8}\right)\right), \mathsf{*.f64}\left(im, im\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

7. Applied egg-rr 21.3%

   \[\leadsto \log \left(im + \color{blue}{\left(\frac{re}{im} \cdot re\right)} \cdot \left(\frac{re \cdot \left(re \cdot -0.125\right)}{im \cdot im} + 0.5\right)\right) \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{re}{im} \cdot \left(re \cdot \left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im} + \frac{1}{2}\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{1}{\frac{im}{re}} \cdot \left(re \cdot \left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im} + \frac{1}{2}\right)\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{1 \cdot \left(re \cdot \left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im} + \frac{1}{2}\right)\right)}{\frac{im}{re}}\right)\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{re \cdot \left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im} + \frac{1}{2}\right)}{\frac{im}{re}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\left(re \cdot \left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im} + \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im} + \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{re \cdot \left(re \cdot \frac{-1}{8}\right)}{im \cdot im}\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{re}{im} \cdot \frac{re \cdot \frac{-1}{8}}{im}\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{1}{\frac{im}{re}} \cdot \frac{re \cdot \frac{-1}{8}}{im}\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

   10. frac-times N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{1 \cdot \left(re \cdot \frac{-1}{8}\right)}{\frac{im}{re} \cdot im}\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

   11. *-lft-identity N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{re \cdot \frac{-1}{8}}{\frac{im}{re} \cdot im}\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(re \cdot \frac{-1}{8}\right), \left(\frac{im}{re} \cdot im\right)\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{8}\right), \left(\frac{im}{re} \cdot im\right)\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(\frac{im}{re}\right), im\right)\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(im, re\right), im\right)\right), \frac{1}{2}\right)\right), \left(\frac{im}{re}\right)\right)\right)\right) \]

   16. /-lowering-/.f64 24.4%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(im, re\right), im\right)\right), \frac{1}{2}\right)\right), \mathsf{/.f64}\left(im, re\right)\right)\right)\right) \]

9. Applied egg-rr 24.4%

   \[\leadsto \log \left(im + \color{blue}{\frac{re \cdot \left(\frac{re \cdot -0.125}{\frac{im}{re} \cdot im} + 0.5\right)}{\frac{im}{re}}}\right) \]

10. Final simplification 24.4%

    \[\leadsto \log \left(im + \frac{re \cdot \left(\frac{re \cdot -0.125}{im \cdot \frac{im}{re}} + 0.5\right)}{\frac{im}{re}}\right) \]
11. Add Preprocessing

Alternative 3: 27.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in re around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. *-inverses N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

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

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

   14. associate-*r/ N/A

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

   15. associate-*r/ N/A

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

   16. unpow2 N/A

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

   17. times-frac N/A

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

   18. *-inverses N/A

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

   19. associate-*l/ N/A

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

5. Simplified 23.9%

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

   1. +-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. /-lowering-/.f64 25.2%

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

7. Applied egg-rr 25.2%

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

8. Final simplification 25.2%

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

Alternative 4: 27.4% 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 53.0%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{im}\right) \]
4. Step-by-step derivation

5. Simplified 25.6%

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

Alternative 5: 0.0% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{0}{0} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (/ 0.0 0.0)))

C

double code(double re, double im) {
	return 0.5 * (0.0 / 0.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (0.0d0 / 0.0d0)
end function

Java

public static double code(double re, double im) {
	return 0.5 * (0.0 / 0.0);
}

Python

def code(re, im):
	return 0.5 * (0.0 / 0.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(0.0 / 0.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (0.0 / 0.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(0.0 / 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. pow1/2 N/A

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

   2. pow-to-exp N/A

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

   3. rem-log-exp N/A

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 53.0%

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

4. Applied egg-rr 53.0%

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

5. Taylor expanded in re around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 30.7%

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

7. Simplified 30.7%

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

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

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

   2. log-prod N/A

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

   3. flip-+ N/A

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

   4. +-inverses N/A

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

   5. +-inverses N/A

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

   6. /-lowering-/.f64 0.0%

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

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{0}{0} \cdot 0.5} \]

10. Final simplification 0.0%

    \[\leadsto 0.5 \cdot \frac{0}{0} \]
11. Add Preprocessing

Reproduce

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