math.log10 on complex, real part

Percentage Accurate: 51.6% → 98.9%

Time: 25.7s

Alternatives: 3

Speedup: 1.1×

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

real(8) function code(re, im)
    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: 51.6% 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

real(8) function code(re, im)
    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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \begin{array}{l} t_0 := {\log 10}^{-0.5}\\ \left(t\_0 \cdot t\_0\right) \cdot \log im\_m \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (let* ((t_0 (pow (log 10.0) -0.5))) (* (* t_0 t_0) (log im_m))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	double t_0 = pow(log(10.0), -0.5);
	return (t_0 * t_0) * log(im_m);
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: t_0
    t_0 = log(10.0d0) ** (-0.5d0)
    code = (t_0 * t_0) * log(im_m)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	double t_0 = Math.pow(Math.log(10.0), -0.5);
	return (t_0 * t_0) * Math.log(im_m);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	t_0 = math.pow(math.log(10.0), -0.5)
	return (t_0 * t_0) * math.log(im_m)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	t_0 = log(10.0) ^ -0.5
	return Float64(Float64(t_0 * t_0) * log(im_m))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	t_0 = log(10.0) ^ -0.5;
	tmp = (t_0 * t_0) * log(im_m);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := Block[{t$95$0 = N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 47.6%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 98.5

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

5. Simplified 98.5%

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

   1. lift-log.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   4. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log im} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log im} \]

   6. lower-/.f64 98.0

      \[\leadsto \color{blue}{\frac{1}{\log 10}} \cdot \log im \]

7. Applied egg-rr 98.0%

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

   1. lift-log.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\log 10}} \cdot \log im \]

   2. inv-pow N/A

      \[\leadsto \color{blue}{{\log 10}^{-1}} \cdot \log im \]

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 99.0

      \[\leadsto \left({\log 10}^{-0.5} \cdot {\log 10}^{\color{blue}{-0.5}}\right) \cdot \log im \]

9. Applied egg-rr 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(\mathsf{fma}\left(re\_m, re\_m \cdot \frac{0.5}{im\_m}, im\_m\right)\right)}{\log 10} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (log (fma re_m (* re_m (/ 0.5 im_m)) im_m)) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(fma(re_m, (re_m * (0.5 / im_m)), im_m)) / log(10.0);
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(fma(re_m, Float64(re_m * Float64(0.5 / im_m)), im_m)) / log(10.0))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(re$95$m * N[(re$95$m * N[(0.5 / im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 98.8

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

5. Simplified 98.8%

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

Reproduce

herbie shell --seed 2024218 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))