math.log10 on complex, real part

Percentage Accurate: 51.9% → 99.1%

Time: 7.4s

Alternatives: 8

Speedup: 0.8×

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: 51.9% 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

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

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

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 99.1

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

4. Applied rewrites 99.1%

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

Alternative 2: 25.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (fma (/ re im) (/ re im) (* 2.0 (log im)))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * fma((re / im), (re / im), (2.0 * log(im)))) / log(10.0);
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * fma(Float64(re / im), Float64(re / im), Float64(2.0 * log(im)))) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(2.0 * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. log-pow N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   6. lower-log.f64 54.5

      \[\leadsto \frac{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}{\log 10} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}{\log 10} \]

   10. lower-fma.f64 54.5

       \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   8. log-rec N/A

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

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right)}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right)}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right)}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right)}{\log 10} \]

   13. lower-log.f64 27.2

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 10} \]
8. Add Preprocessing

Alternative 3: 25.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{re}{im} \cdot \frac{re}{im}, -0.5, -\log im\right)}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (fma (* (/ re im) (/ re im)) -0.5 (- (log im))) (log 0.1)))

C

double code(double re, double im) {
	return fma(((re / im) * (re / im)), -0.5, -log(im)) / log(0.1);
}

Julia

function code(re, im)
	return Float64(fma(Float64(Float64(re / im) * Float64(re / im)), -0.5, Float64(-log(im))) / log(0.1))
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5 + (-N[Log[im], $MachinePrecision])), $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. log-pow N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   6. lower-log.f64 54.5

      \[\leadsto \frac{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}{\log 10} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}{\log 10} \]

   10. lower-fma.f64 54.5

       \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   8. log-rec N/A

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

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right)}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right)}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right)}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right)}{\log 10} \]

   13. lower-log.f64 27.2

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 10} \]
8. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\color{blue}{\log 10}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\log \color{blue}{\left(\frac{1}{\frac{1}{10}}\right)}} \]

   3. neg-log N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\color{blue}{\mathsf{neg}\left(\log \frac{1}{10}\right)}} \]

   4. lift-log.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\mathsf{neg}\left(\color{blue}{\log \frac{1}{10}}\right)} \]

   5. lower-neg.f64 27.2

      \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\color{blue}{-\log 0.1}} \]

9. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}{\color{blue}{-\log 0.1}} \]

10. Taylor expanded in im around inf

    \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log \frac{1}{10}} + \frac{\log \left(\frac{1}{im}\right)}{\log \frac{1}{10}}} \]
11. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{\frac{{re}^{2}}{{im}^{2}}}{\log \frac{1}{10}}} + \frac{\log \left(\frac{1}{im}\right)}{\log \frac{1}{10}} \]

    2. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}}{\log \frac{1}{10}}} + \frac{\log \left(\frac{1}{im}\right)}{\log \frac{1}{10}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{{re}^{2}}{{im}^{2}}}{\log \frac{1}{10}} + \frac{\log \left(\frac{1}{im}\right)}{\log \frac{1}{10}} \]

    4. div-add-rev N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{{re}^{2}}{{im}^{2}} + \log \left(\frac{1}{im}\right)}{\log \frac{1}{10}}} \]

    5. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{{re}^{2}}{{im}^{2}} + \log \left(\frac{1}{im}\right)}{\log \frac{1}{10}}} \]

12. Applied rewrites 27.2%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{re}{im} \cdot \frac{re}{im}, -0.5, -\log im\right)}{\log 0.1}} \]
13. Add Preprocessing

Alternative 4: 27.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

double code(double re, double im) {
	return log(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(im) / log(10.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
4. Step-by-step derivation

   1. lower-log.f64 28.5

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

5. Applied rewrites 28.5%

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

Alternative 5: 3.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \frac{\frac{re}{im} \cdot re}{im}}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (/ (* (/ re im) re) im)) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * (((re / im) * re) / 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 = (0.5d0 * (((re / im) * re) / im)) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * (((re / im) * re) / im)) / Math.log(10.0);
}

Python

def code(re, im):
	return (0.5 * (((re / im) * re) / im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * Float64(Float64(Float64(re / im) * re) / im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * (((re / im) * re) / im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. log-pow N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   6. lower-log.f64 54.5

      \[\leadsto \frac{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}{\log 10} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}{\log 10} \]

   10. lower-fma.f64 54.5

       \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   8. log-rec N/A

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

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right)}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right)}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right)}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right)}{\log 10} \]

   13. lower-log.f64 27.2

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 10} \]

8. Taylor expanded in re around inf

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

10. Applied rewrites 3.3%

    \[\leadsto \frac{0.5 \cdot \left(\frac{re}{im} \cdot \color{blue}{\frac{re}{im}}\right)}{\log 10} \]
11. Step-by-step derivation

12. Applied rewrites 3.3%

    \[\leadsto \frac{0.5 \cdot \frac{\frac{re}{im} \cdot re}{im}}{\log 10} \]
13. Add Preprocessing

Alternative 6: 3.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(\frac{re}{im} \cdot \frac{re}{im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (* (/ re im) (/ re im))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * ((re / im) * (re / 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 = (0.5d0 * ((re / im) * (re / im))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * ((re / im) * (re / im))) / Math.log(10.0);
}

Python

def code(re, im):
	return (0.5 * ((re / im) * (re / im))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * Float64(Float64(re / im) * Float64(re / im))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * ((re / im) * (re / im))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. log-pow N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   6. lower-log.f64 54.5

      \[\leadsto \frac{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}{\log 10} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}{\log 10} \]

   10. lower-fma.f64 54.5

       \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   8. log-rec N/A

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

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right)}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right)}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right)}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right)}{\log 10} \]

   13. lower-log.f64 27.2

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 10} \]

8. Taylor expanded in re around inf

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

10. Applied rewrites 3.3%

    \[\leadsto \frac{0.5 \cdot \left(\frac{re}{im} \cdot \color{blue}{\frac{re}{im}}\right)}{\log 10} \]
11. Add Preprocessing

Alternative 7: 3.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (* re (/ (/ re im) im))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * (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 = (0.5d0 * (re * ((re / im) / im))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * (re * ((re / im) / im))) / Math.log(10.0);
}

Python

def code(re, im):
	return (0.5 * (re * ((re / im) / im))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * Float64(re * Float64(Float64(re / im) / im))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * (re * ((re / im) / im))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(re * N[(N[(re / im), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. log-pow N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   6. lower-log.f64 54.5

      \[\leadsto \frac{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}{\log 10} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}{\log 10} \]

   10. lower-fma.f64 54.5

       \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   8. log-rec N/A

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

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right)}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right)}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right)}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right)}{\log 10} \]

   13. lower-log.f64 27.2

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 10} \]

8. Taylor expanded in re around inf

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

10. Applied rewrites 3.3%

    \[\leadsto \frac{0.5 \cdot \left(\frac{re}{im} \cdot \color{blue}{\frac{re}{im}}\right)}{\log 10} \]
11. Step-by-step derivation

12. Applied rewrites 3.3%

    \[\leadsto \frac{0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{\color{blue}{im}}\right)}{\log 10} \]
13. Add Preprocessing

Alternative 8: 3.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(re \cdot \frac{re}{im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (* re (/ re (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * (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 = (0.5d0 * (re * (re / (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 * (re * (re / (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return (0.5 * (re * (re / (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 * Float64(re * Float64(re / Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * (re * (re / (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(re * N[(re / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. log-pow N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   6. lower-log.f64 54.5

      \[\leadsto \frac{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log 10} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}{\log 10} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}{\log 10} \]

   10. lower-fma.f64 54.5

       \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log 10} \]

   8. log-rec N/A

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

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right)}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right)}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right)}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right)}{\log 10} \]

   13. lower-log.f64 27.2

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 27.2%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 10} \]

8. Taylor expanded in re around inf

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

10. Applied rewrites 3.3%

    \[\leadsto \frac{0.5 \cdot \left(\frac{re}{im} \cdot \color{blue}{\frac{re}{im}}\right)}{\log 10} \]
11. Step-by-step derivation

12. Applied rewrites 3.0%

    \[\leadsto \frac{0.5 \cdot \left(re \cdot \frac{re}{\color{blue}{im \cdot im}}\right)}{\log 10} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))