math.log/2 on complex, real part

Percentage Accurate: 52.8% → 99.1%

Time: 8.1s

Alternatives: 3

Speedup: 2.7×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im, base)
use fmin_fmax_functions
    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: 52.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im, base)
use fmin_fmax_functions
    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: 99.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m, base] = \mathsf{sort}([re_m, im_m, base])\\ \\ \frac{\mathsf{fma}\left(\frac{\frac{0.5 \cdot re\_m}{im\_m}}{im\_m}, re\_m, \log im\_m\right)}{\log base} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
(FPCore (re_m im_m base)
 :precision binary64
 (/ (fma (/ (/ (* 0.5 re_m) im_m) im_m) re_m (log im_m)) (log base)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m && im_m < base);
double code(double re_m, double im_m, double base) {
	return fma((((0.5 * re_m) / im_m) / im_m), re_m, log(im_m)) / log(base);
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m, base = sort([re_m, im_m, base])
function code(re_m, im_m, base)
	return Float64(fma(Float64(Float64(Float64(0.5 * re_m) / im_m) / im_m), re_m, log(im_m)) / log(base))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_, base_] := N[(N[(N[(N[(N[(0.5 * re$95$m), $MachinePrecision] / im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision] * re$95$m + N[Log[im$95$m], $MachinePrecision]), $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. mul0-rgt N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   4. +-rgt-identity 49.4

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   8. +-rgt-identity 49.4

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

   9. lower-/.f64 N/A

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

   10. +-rgt-identity N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   17. +-rgt-identity N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-log.f64 28.3

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

7. Applied rewrites 28.3%

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

9. Applied rewrites 29.9%

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

Alternative 2: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m, base] = \mathsf{sort}([re_m, im_m, base])\\ \\ \frac{\log im\_m}{\log base} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
(FPCore (re_m im_m base) :precision binary64 (/ (log im_m) (log base)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m && im_m < base);
double code(double re_m, double im_m, double base) {
	return log(im_m) / log(base);
}

Fortran

im_m =     private
re_m =     private
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re_m, im_m, base)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8), intent (in) :: base
    code = log(im_m) / log(base)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m && im_m < base;
public static double code(double re_m, double im_m, double base) {
	return Math.log(im_m) / Math.log(base);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m, base] = sort([re_m, im_m, base])
def code(re_m, im_m, base):
	return math.log(im_m) / math.log(base)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m, base = sort([re_m, im_m, base])
function code(re_m, im_m, base)
	return Float64(log(im_m) / log(base))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m, base = num2cell(sort([re_m, im_m, base])){:}
function tmp = code(re_m, im_m, base)
	tmp = log(im_m) / log(base);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_, base_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
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 31.1

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

5. Applied rewrites 31.1%

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

Alternative 3: 3.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m, base] = \mathsf{sort}([re_m, im_m, base])\\ \\ \frac{\left(\frac{0.5}{im\_m \cdot im\_m} \cdot re\_m\right) \cdot re\_m}{\log base} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
(FPCore (re_m im_m base)
 :precision binary64
 (/ (* (* (/ 0.5 (* im_m im_m)) re_m) re_m) (log base)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m && im_m < base);
double code(double re_m, double im_m, double base) {
	return (((0.5 / (im_m * im_m)) * re_m) * re_m) / log(base);
}

Fortran

im_m =     private
re_m =     private
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re_m, im_m, base)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8), intent (in) :: base
    code = (((0.5d0 / (im_m * im_m)) * re_m) * re_m) / log(base)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m && im_m < base;
public static double code(double re_m, double im_m, double base) {
	return (((0.5 / (im_m * im_m)) * re_m) * re_m) / Math.log(base);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m, base] = sort([re_m, im_m, base])
def code(re_m, im_m, base):
	return (((0.5 / (im_m * im_m)) * re_m) * re_m) / math.log(base)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m, base = sort([re_m, im_m, base])
function code(re_m, im_m, base)
	return Float64(Float64(Float64(Float64(0.5 / Float64(im_m * im_m)) * re_m) * re_m) / log(base))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m, base = num2cell(sort([re_m, im_m, base])){:}
function tmp = code(re_m, im_m, base)
	tmp = (((0.5 / (im_m * im_m)) * re_m) * re_m) / log(base);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m, im_m, and base should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_, base_] := N[(N[(N[(N[(0.5 / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision] * re$95$m), $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. mul0-rgt N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   4. +-rgt-identity 49.4

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   8. +-rgt-identity 49.4

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

   9. lower-/.f64 N/A

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

   10. +-rgt-identity N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   17. +-rgt-identity N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-log.f64 28.3

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

7. Applied rewrites 28.3%

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

8. Taylor expanded in re around inf

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

10. Applied rewrites 2.9%

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

Reproduce

herbie shell --seed 2025010 
(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))))