math.log10 on complex, real part

Percentage Accurate: 51.7% → 99.1%

Time: 4.0s

Alternatives: 8

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \left(\frac{-1}{\log 100} - \frac{-1}{\log 0.01}\right) \cdot \left(-\log im\_m\right) \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (- (/ -1.0 (log 100.0)) (/ -1.0 (log 0.01))) (- (log im_m))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return ((-1.0 / log(100.0)) - (-1.0 / log(0.01))) * -log(im_m);
}

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m 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)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = (((-1.0d0) / log(100.0d0)) - ((-1.0d0) / log(0.01d0))) * -log(im_m)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return ((-1.0 / Math.log(100.0)) - (-1.0 / Math.log(0.01))) * -Math.log(im_m);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return ((-1.0 / math.log(100.0)) - (-1.0 / math.log(0.01))) * -math.log(im_m)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(Float64(-1.0 / log(100.0)) - Float64(-1.0 / log(0.01))) * Float64(-log(im_m)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = ((-1.0 / log(100.0)) - (-1.0 / log(0.01))) * -log(im_m);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(N[(-1.0 / N[Log[100.0], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Log[0.01], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Log[im$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

2. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\log 10} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

   3. lower-/.f64 98.3

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

4. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]

6. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\frac{-2}{\log 100} \cdot \left(-\log im\right)} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{\log 100}} \cdot \left(-\log im\right) \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(\log 100\right)}} \cdot \left(-\log im\right) \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{2}}{\mathsf{neg}\left(\log 100\right)} \cdot \left(-\log im\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1 - -1}}{\mathsf{neg}\left(\log 100\right)} \cdot \left(-\log im\right) \]

   5. lift-log.f64 N/A

      \[\leadsto \frac{1 - -1}{\mathsf{neg}\left(\color{blue}{\log 100}\right)} \cdot \left(-\log im\right) \]

   6. neg-log N/A

      \[\leadsto \frac{1 - -1}{\color{blue}{\log \left(\frac{1}{100}\right)}} \cdot \left(-\log im\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 - -1}{\log \color{blue}{\frac{1}{100}}} \cdot \left(-\log im\right) \]

   8. lift-log.f64 N/A

      \[\leadsto \frac{1 - -1}{\color{blue}{\log \frac{1}{100}}} \cdot \left(-\log im\right) \]

   9. div-sub N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\log \frac{1}{100}} - \frac{-1}{\log \frac{1}{100}}\right)} \cdot \left(-\log im\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\log \frac{1}{100}} - \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{\log \frac{1}{100}}\right) \cdot \left(-\log im\right) \]

   11. lift-log.f64 N/A

       \[\leadsto \left(\frac{1}{\log \frac{1}{100}} - \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\log \frac{1}{100}}}\right) \cdot \left(-\log im\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\log \frac{1}{100}} - \frac{\mathsf{neg}\left(1\right)}{\log \color{blue}{\left(\frac{1}{100}\right)}}\right) \cdot \left(-\log im\right) \]

   13. neg-log N/A

       \[\leadsto \left(\frac{1}{\log \frac{1}{100}} - \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\log 100\right)}}\right) \cdot \left(-\log im\right) \]

   14. lift-log.f64 N/A

       \[\leadsto \left(\frac{1}{\log \frac{1}{100}} - \frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\color{blue}{\log 100}\right)}\right) \cdot \left(-\log im\right) \]

   15. frac-2neg N/A

       \[\leadsto \left(\frac{1}{\log \frac{1}{100}} - \color{blue}{\frac{1}{\log 100}}\right) \cdot \left(-\log im\right) \]

   16. lower--.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{\log \frac{1}{100}} - \frac{1}{\log 100}\right)} \cdot \left(-\log im\right) \]

   17. frac-2neg N/A

       \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\log \frac{1}{100}\right)}} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   18. metadata-eval N/A

       \[\leadsto \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\log \frac{1}{100}\right)} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   19. lift-log.f64 N/A

       \[\leadsto \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\log \frac{1}{100}}\right)} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   20. neg-log N/A

       \[\leadsto \left(\frac{-1}{\color{blue}{\log \left(\frac{1}{\frac{1}{100}}\right)}} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   21. metadata-eval N/A

       \[\leadsto \left(\frac{-1}{\log \color{blue}{100}} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   22. lift-log.f64 N/A

       \[\leadsto \left(\frac{-1}{\color{blue}{\log 100}} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   23. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{-1}{\log 100}} - \frac{1}{\log 100}\right) \cdot \left(-\log im\right) \]

   24. frac-2neg N/A

       \[\leadsto \left(\frac{-1}{\log 100} - \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\log 100\right)}}\right) \cdot \left(-\log im\right) \]

   25. metadata-eval N/A

       \[\leadsto \left(\frac{-1}{\log 100} - \frac{\color{blue}{-1}}{\mathsf{neg}\left(\log 100\right)}\right) \cdot \left(-\log im\right) \]

   26. lift-log.f64 N/A

       \[\leadsto \left(\frac{-1}{\log 100} - \frac{-1}{\mathsf{neg}\left(\color{blue}{\log 100}\right)}\right) \cdot \left(-\log im\right) \]

   27. neg-log N/A

       \[\leadsto \left(\frac{-1}{\log 100} - \frac{-1}{\color{blue}{\log \left(\frac{1}{100}\right)}}\right) \cdot \left(-\log im\right) \]

   28. metadata-eval N/A

       \[\leadsto \left(\frac{-1}{\log 100} - \frac{-1}{\log \color{blue}{\frac{1}{100}}}\right) \cdot \left(-\log im\right) \]

   29. lift-log.f64 N/A

       \[\leadsto \left(\frac{-1}{\log 100} - \frac{-1}{\color{blue}{\log \frac{1}{100}}}\right) \cdot \left(-\log im\right) \]

8. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\left(\frac{-1}{\log 100} - \frac{-1}{\log 0.01}\right)} \cdot \left(-\log im\right) \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(\mathsf{hypot}\left(re\_m, im\_m\right)\right)}{-\log 0.1} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (log (hypot re_m im_m)) (- (log 0.1))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(hypot(re_m, im_m)) / -log(0.1);
}

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(Math.hypot(re_m, im_m)) / -Math.log(0.1);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(math.hypot(re_m, im_m)) / -math.log(0.1)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(hypot(re_m, im_m)) / Float64(-log(0.1)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(hypot(re_m, im_m)) / -log(0.1);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[N[Sqrt[re$95$m ^ 2 + im$95$m ^ 2], $MachinePrecision]], $MachinePrecision] / (-N[Log[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. add-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(im \cdot im\right)\right)}}\right)}{\log 10} \]

   4. sub-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im \cdot im\right)\right)\right)\right)}}\right)}{\log 10} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   7. sqr-abs-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   8. sqr-neg-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}\right)}{\log 10} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   13. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   14. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   15. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   16. sqr-abs-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   17. lower-hypot.f64 99.1

       \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)}} \]

   2. lower-neg.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{-\left(\mathsf{neg}\left(\log 10\right)\right)}} \]

   3. lift-log.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\left(\mathsf{neg}\left(\color{blue}{\log 10}\right)\right)} \]

   4. neg-log N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   5. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   6. metadata-eval 99.0

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\log \color{blue}{0.1}} \]

5. Applied rewrites 99.0%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{-\log 0.1}} \]
6. Add Preprocessing

Alternative 3: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(\mathsf{hypot}\left(re\_m, im\_m\right)\right)}{\log 10} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log (hypot re_m im_m)) (log 10.0)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(hypot(re_m, im_m)) / log(10.0);
}

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(Math.hypot(re_m, im_m)) / Math.log(10.0);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(math.hypot(re_m, im_m)) / math.log(10.0)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(hypot(re_m, im_m)) / log(10.0))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(hypot(re_m, im_m)) / log(10.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[N[Sqrt[re$95$m ^ 2 + im$95$m ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. add-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(im \cdot im\right)\right)}}\right)}{\log 10} \]

   4. sub-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im \cdot im\right)\right)\right)\right)}}\right)}{\log 10} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   7. sqr-abs-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   8. sqr-neg-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}\right)}{\log 10} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   13. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   14. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   15. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   16. sqr-abs-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   17. lower-hypot.f64 99.1

       \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
4. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(im\_m \cdot 2\right) - \log 2}{\log 10} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (- (log (* im_m 2.0)) (log 2.0)) (log 10.0)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (log((im_m * 2.0)) - log(2.0)) / log(10.0);
}

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m 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)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = (log((im_m * 2.0d0)) - log(2.0d0)) / log(10.0d0)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (Math.log((im_m * 2.0)) - Math.log(2.0)) / Math.log(10.0);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (math.log((im_m * 2.0)) - math.log(2.0)) / math.log(10.0)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(log(Float64(im_m * 2.0)) - log(2.0)) / log(10.0))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (log((im_m * 2.0)) - log(2.0)) / log(10.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(N[Log[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

2. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\log 10} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

   3. lower-/.f64 98.3

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

4. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\log 10} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. lift-log.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   4. neg-log N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{im}}\right)}{\log 10} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{im}}\right)}{\log 10} \]

   6. frac-2neg N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(im\right)}}\right)}{\log 10} \]

   7. distribute-frac-neg N/A

      \[\leadsto \frac{\log \left(\frac{1}{\mathsf{neg}\left(\frac{1}{\mathsf{neg}\left(im\right)}\right)}\right)}{\log 10} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)}}\right)}{\log 10} \]

   9. remove-double-div N/A

      \[\leadsto \frac{\log \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right)}{\log 10} \]

   10. *-lft-identity N/A

       \[\leadsto \frac{\log \left(1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right)\right)}{\log 10} \]

   11. *-commutative N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot 1\right)}{\log 10} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot \frac{2}{2}\right)}{\log 10} \]

   13. associate-*r/ N/A

       \[\leadsto \frac{\log \left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot 2}{2}\right)}{\log 10} \]

   14. log-div N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot 2\right) - \color{blue}{\log 2}}{\log 10} \]

   15. lower-unsound--.f64 N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot 2\right) - \color{blue}{\log 2}}{\log 10} \]

   16. lower-unsound-log.f64 N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot 2\right) - \log \color{blue}{2}}{\log 10} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot 2\right) - \log 2}{\log 10} \]

   18. mul-1-neg N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(-1 \cdot im\right)\right) \cdot 2\right) - \log 2}{\log 10} \]

   19. *-commutative N/A

       \[\leadsto \frac{\log \left(\left(\mathsf{neg}\left(im \cdot -1\right)\right) \cdot 2\right) - \log 2}{\log 10} \]

   20. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\log \left(\left(im \cdot \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot 2\right) - \log 2}{\log 10} \]

   21. metadata-eval N/A

       \[\leadsto \frac{\log \left(\left(im \cdot 1\right) \cdot 2\right) - \log 2}{\log 10} \]

   22. *-rgt-identity N/A

       \[\leadsto \frac{\log \left(im \cdot 2\right) - \log 2}{\log 10} \]

   23. lower-unsound-log.f64 98.2

       \[\leadsto \frac{\log \left(im \cdot 2\right) - \log 2}{\log 10} \]

6. Applied rewrites 98.2%

   \[\leadsto \frac{\log \left(im \cdot 2\right) - \color{blue}{\log 2}}{\log 10} \]
7. Add Preprocessing

Alternative 5: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\frac{-2}{\log 0.01}}{\frac{1}{\log im\_m}} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (/ -2.0 (log 0.01)) (/ 1.0 (log im_m))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (-2.0 / log(0.01)) / (1.0 / log(im_m));
}

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m 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)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = ((-2.0d0) / log(0.01d0)) / (1.0d0 / log(im_m))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (-2.0 / Math.log(0.01)) / (1.0 / Math.log(im_m));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (-2.0 / math.log(0.01)) / (1.0 / math.log(im_m))

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(-2.0 / log(0.01)) / Float64(1.0 / log(im_m)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (-2.0 / log(0.01)) / (1.0 / log(im_m));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(-2.0 / N[Log[0.01], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. add-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(im \cdot im\right)\right)}}\right)}{\log 10} \]

   4. sub-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im \cdot im\right)\right)\right)\right)}}\right)}{\log 10} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   7. sqr-abs-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   8. sqr-neg-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}\right)}{\log 10} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   13. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   14. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   15. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   16. sqr-abs-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   17. lower-hypot.f64 99.1

       \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

5. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\frac{\frac{-2}{\log 0.01}}{\frac{2}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

6. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\color{blue}{\frac{-1}{\log \left(\frac{1}{im}\right)}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{-1}{\color{blue}{\log \left(\frac{1}{im}\right)}}} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{-1}{\log \left(\frac{1}{im}\right)}} \]

   3. lower-/.f64 98.2

      \[\leadsto \frac{\frac{-2}{\log 0.01}}{\frac{-1}{\log \left(\frac{1}{im}\right)}} \]

8. Applied rewrites 98.2%

   \[\leadsto \frac{\frac{-2}{\log 0.01}}{\color{blue}{\frac{-1}{\log \left(\frac{1}{im}\right)}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{-1}{\color{blue}{\log \left(\frac{1}{im}\right)}}} \]

   2. frac-2neg N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{1}{\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{im}\right)}\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{1}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}} \]

   5. lift-log.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{1}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{1}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}} \]

   7. log-rec N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)}} \]

   8. lift-log.f64 N/A

      \[\leadsto \frac{\frac{-2}{\log \frac{1}{100}}}{\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)}} \]

   9. remove-double-neg 98.2

      \[\leadsto \frac{\frac{-2}{\log 0.01}}{\frac{1}{\log im}} \]

10. Applied rewrites 98.2%

    \[\leadsto \frac{\frac{-2}{\log 0.01}}{\frac{1}{\color{blue}{\log im}}} \]
11. Add Preprocessing

Alternative 6: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{-\log im\_m}{\log 0.1} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (- (log im_m)) (log 0.1)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return -log(im_m) / log(0.1);
}

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m 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)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = -log(im_m) / log(0.1d0)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return -Math.log(im_m) / Math.log(0.1);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return -math.log(im_m) / math.log(0.1)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(-log(im_m)) / log(0.1))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = -log(im_m) / log(0.1);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[((-N[Log[im$95$m], $MachinePrecision]) / N[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

2. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\log 10} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

   3. lower-/.f64 98.3

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

4. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]

6. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{-\log im}{\log 0.1}} \]
7. Add Preprocessing

Alternative 7: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log im\_m}{\log 10} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log im_m) (log 10.0)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(im_m) / log(10.0);
}

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m 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)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(im_m) / log(10.0d0)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(im_m) / Math.log(10.0);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(im_m) / math.log(10.0)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(im_m) / log(10.0))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(im_m) / log(10.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

2. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\log 10} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

   3. lower-/.f64 98.3

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10} \]

4. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]

6. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
7. Add Preprocessing

Alternative 8: 10.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log re\_m}{\log 10} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log re_m) (log 10.0)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(re_m) / log(10.0);
}

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m 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)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(re_m) / log(10.0d0)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(re_m) / Math.log(10.0);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(re_m) / math.log(10.0)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(re_m) / log(10.0))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(re_m) / log(10.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[re$95$m], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. add-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(im \cdot im\right)\right)}}\right)}{\log 10} \]

   4. sub-flip N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im \cdot im\right)\right)\right)\right)}}\right)}{\log 10} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   7. sqr-abs-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   8. sqr-neg-rev N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}\right)}{\log 10} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   13. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|im\right|\right)\right)\right)\right)}}\right)}{\log 10} \]

   14. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left(\mathsf{neg}\left(\left|im\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|im\right|\right)\right)}}\right)}{\log 10} \]

   15. sqr-neg-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{\left|im\right| \cdot \left|im\right|}}\right)}{\log 10} \]

   16. sqr-abs-rev N/A

       \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   17. lower-hypot.f64 99.1

       \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)}} \]

   2. lower-neg.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{-\left(\mathsf{neg}\left(\log 10\right)\right)}} \]

   3. lift-log.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\left(\mathsf{neg}\left(\color{blue}{\log 10}\right)\right)} \]

   4. neg-log N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   5. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   6. metadata-eval 99.0

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\log \color{blue}{0.1}} \]

5. Applied rewrites 99.0%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{-\log 0.1}} \]

6. Taylor expanded in re around inf

   \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \frac{1}{10}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{re}\right)}{\color{blue}{\log \frac{1}{10}}} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{re}\right)}{\log \color{blue}{\frac{1}{10}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{re}\right)}{\log \frac{1}{10}} \]

   4. lower-log.f64 10.4

      \[\leadsto \frac{\log \left(\frac{1}{re}\right)}{\log 0.1} \]

8. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log 0.1}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\log \left(\frac{1}{re}\right)}{\color{blue}{\log \frac{1}{10}}} \]

   2. frac-2neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\color{blue}{\mathsf{neg}\left(\log \frac{1}{10}\right)}} \]

   3. lift-log.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\mathsf{neg}\left(\log \frac{1}{10}\right)} \]

   4. neg-log N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\log \left(\frac{1}{\frac{1}{10}}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\log 10} \]

   6. lift-log.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\color{blue}{\log 10}} \]

   8. lift-log.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\log 10} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{1}{re}\right)\right)}{\log 10} \]

   10. log-rec N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log re\right)\right)\right)}{\log 10} \]

   11. remove-double-neg N/A

       \[\leadsto \frac{\log re}{\log \color{blue}{10}} \]

   12. lower-log.f64 10.4

       \[\leadsto \frac{\log re}{\log \color{blue}{10}} \]

10. Applied rewrites 10.4%

    \[\leadsto \frac{\log re}{\color{blue}{\log 10}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))