math.log/1 on complex, real part

Percentage Accurate: 52.4% → 100.0%

Time: 7.8s

Alternatives: 12

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 26.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. times-frac N/A

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

   8. /-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 26.4

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

7. Applied rewrites 26.4%

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

8. Final simplification 26.4%

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

Alternative 3: 27.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 26.2

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

5. Applied rewrites 26.2%

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

Alternative 4: 3.3% accurate, 2.8× 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 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 3.6%

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

12. Applied rewrites 3.6%

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

Alternative 5: 3.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{2}{re} \cdot im}{re} \cdot im} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (* (/ (* (/ 2.0 re) im) re) im)))

C

double code(double re, double im) {
	return 1.0 / ((((2.0 / re) * im) / re) * im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / ((((2.0d0 / re) * im) / re) * im)
end function

Java

public static double code(double re, double im) {
	return 1.0 / ((((2.0 / re) * im) / re) * im);
}

Python

def code(re, im):
	return 1.0 / ((((2.0 / re) * im) / re) * im)

Julia

function code(re, im)
	return Float64(1.0 / Float64(Float64(Float64(Float64(2.0 / re) * im) / re) * im))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / ((((2.0 / re) * im) / re) * im);
end

Wolfram

code[re_, im_] := N[(1.0 / N[(N[(N[(N[(2.0 / re), $MachinePrecision] * im), $MachinePrecision] / re), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 3.6%

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

12. Applied rewrites 3.6%

    \[\leadsto \frac{1}{\frac{\frac{2}{re} \cdot im}{re} \cdot im} \]
13. Add Preprocessing

Alternative 6: 3.3% accurate, 3.5× speedup

Math

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

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(Float64(Float64(re / im) * re) * -0.5) / Float64(-im))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 3.6%

    \[\leadsto \frac{-0.5 \cdot \left(\frac{re}{im} \cdot re\right)}{-im} \]

11. Final simplification 3.6%

    \[\leadsto \frac{\left(\frac{re}{im} \cdot re\right) \cdot -0.5}{-im} \]
12. Add Preprocessing

Alternative 7: 3.3% accurate, 3.8× speedup

Math

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

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 3.6%

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

11. Final simplification 3.6%

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

Alternative 8: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.6%

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

Alternative 9: 3.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 3.6%

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

12. Applied rewrites 3.2%

    \[\leadsto \frac{-0.5 \cdot re}{im \cdot im} \cdot \left(-re\right) \]

13. Final simplification 3.2%

    \[\leadsto \left(-re\right) \cdot \frac{-0.5 \cdot re}{im \cdot im} \]
14. Add Preprocessing

Alternative 10: 3.0% accurate, 4.3× speedup

Math

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

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(Float64(-0.5 / Float64(im * im)) * re) * Float64(-re))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 3.6%

    \[\leadsto \frac{0.5 \cdot \frac{re}{im}}{-im} \cdot \left(-re\right) \]

11. Taylor expanded in re around 0

    \[\leadsto \left(\frac{-1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot \left(-re\right) \]
12. Step-by-step derivation

13. Applied rewrites 3.2%

    \[\leadsto \left(\frac{-0.5}{im \cdot im} \cdot re\right) \cdot \left(-re\right) \]
14. Add Preprocessing

Alternative 11: 2.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 2.8%

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

Alternative 12: 2.8% accurate, 4.6× 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 55.2%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.6%

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

10. Applied rewrites 2.8%

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

11. Final simplification 2.8%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))