math.log/1 on complex, real part

Percentage Accurate: 51.2% → 100.0%

Time: 4.4s

Alternatives: 6

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

4. Applied rewrites 100.0%

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

Alternative 2: 25.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

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

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

   3. remove-double-neg N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

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

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

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

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. log-rec N/A

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

   14. distribute-lft-neg-out N/A

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

   15. fp-cancel-sign-sub N/A

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

8. Applied rewrites 23.4%

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

Alternative 3: 27.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

Alternative 4: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 = (0.5d0 / im) * ((re / im) * re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

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

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

   3. remove-double-neg N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

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

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

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

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. log-rec N/A

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

   14. distribute-lft-neg-out N/A

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

   15. fp-cancel-sign-sub N/A

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

8. Applied rewrites 23.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.0%

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

13. Applied rewrites 3.2%

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

Alternative 5: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 = (0.5d0 * re) * ((re / im) / im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

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

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

   3. remove-double-neg N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

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

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

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

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. log-rec N/A

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

   14. distribute-lft-neg-out N/A

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

   15. fp-cancel-sign-sub N/A

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

8. Applied rewrites 23.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.0%

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

13. Applied rewrites 3.2%

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

Alternative 6: 3.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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 = ((0.5d0 / (im * im)) * re) * re
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 25.3

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

5. Applied rewrites 25.3%

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

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

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

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

   3. remove-double-neg N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

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

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

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

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. log-rec N/A

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

   14. distribute-lft-neg-out N/A

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

   15. fp-cancel-sign-sub N/A

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

8. Applied rewrites 23.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.0%

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

Reproduce

herbie shell --seed 2024356 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))