math.log/1 on complex, real part

Percentage Accurate: 52.4% → 100.0%

Time: 4.1s

Alternatives: 8

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 55.1%

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

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

   4. lift-*.f64 N/A

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

   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: 25.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 23.9

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

8. Applied rewrites 23.9%

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

Alternative 3: 27.4% 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 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

Alternative 4: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 23.9

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

8. Applied rewrites 23.9%

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

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 2.9%

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

13. Applied rewrites 3.3%

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

Alternative 5: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 23.9

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

8. Applied rewrites 23.9%

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

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 2.9%

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

13. Applied rewrites 3.3%

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

Alternative 6: 3.3% accurate, 3.8× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 23.9

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

8. Applied rewrites 23.9%

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

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 2.9%

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

13. Applied rewrites 3.3%

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

Alternative 7: 3.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 23.9

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

8. Applied rewrites 23.9%

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

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 2.9%

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

13. Applied rewrites 2.9%

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

Alternative 8: 3.0% accurate, 4.6× speedup

Math

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

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(Float64(0.5 / Float64(im * im)) * re) * 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 55.1%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 23.9

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

8. Applied rewrites 23.9%

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

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 2.9%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))