math.log/2 on complex, real part

Percentage Accurate: 51.2% → 99.4%

Time: 10.2s

Alternatives: 7

Speedup: 1.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(re, im, base)
	return Float64(log(hypot(im, re)) / log(base))
end

MATLAB

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

Wolfram

code[re_, im_, base_] := N[(N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 2: 25.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.8%

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

Alternative 3: 26.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im, base)
	return Float64(log(im) / log(base))
end

MATLAB

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

Wolfram

code[re_, im_, base_] := N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 29.9

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

5. Applied rewrites 29.9%

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

Alternative 4: 3.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{\frac{im}{re}}{re} \cdot \left(2 \cdot im\right)}}{\log base} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im, base):
	return (1.0 / (((im / re) / re) * (2.0 * im))) / math.log(base)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.8%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.4%

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

13. Applied rewrites 3.4%

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

14. Final simplification 3.4%

    \[\leadsto \frac{\frac{1}{\frac{\frac{im}{re}}{re} \cdot \left(2 \cdot im\right)}}{\log base} \]
15. Add Preprocessing

Alternative 5: 3.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.8%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.4%

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

13. Applied rewrites 3.3%

    \[\leadsto \frac{\frac{\frac{re \cdot 0.5}{im} \cdot re}{im}}{\log base} \]

14. Final simplification 3.3%

    \[\leadsto \frac{\frac{\frac{0.5 \cdot re}{im} \cdot re}{im}}{\log base} \]
15. Add Preprocessing

Alternative 6: 3.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.8%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.4%

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

12. Final simplification 3.4%

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

Alternative 7: 3.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.8%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.4%

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

13. Applied rewrites 3.1%

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

14. Final simplification 3.1%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))