math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.1%

Time: 2.9s

Alternatives: 3

Speedup: 1.8×

• Could not converge on a ground truth

  1. re = 4.614906735260092e-21
  2. im = +nan.0

Specification

Math

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

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)
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: 51.4% accurate, 1.0× speedup

Math

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

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)
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, 1.1× speedup

Math

\[\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{2.302585092994046} \]

FPCore

(FPCore (re im)
  :precision binary64
  (/ (log (hypot re im)) 2.302585092994046))

C

double code(double re, double im) {
	return log(hypot(re, im)) / 2.302585092994046;
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / 2.302585092994046;
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / 2.302585092994046

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / 2.302585092994046)
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / 2.302585092994046;
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / 2.302585092994046), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. lift-sqrt.f64 N/A

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

   2. sqrt-fabs-rev N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sqr-neg-rev N/A

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

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

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

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

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

   13. lift-*.f64 N/A

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

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

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

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

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

   16. sqr-neg-rev N/A

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

   17. sqr-neg-rev N/A

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

   18. lower-hypot.f64 99.1%

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

3. Applied rewrites 99.1%

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

4. Evaluated real constant 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{2.302585092994046}} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.7× speedup

Math

\[\frac{\log \left(\mathsf{max}\left(\left|re\right|, \left|im\right|\right)\right)}{2.302585092994046} \]

FPCore

(FPCore (re im)
  :precision binary64
  (/ (log (fmax (fabs re) (fabs im))) 2.302585092994046))

C

double code(double re, double im) {
	return log(fmax(fabs(re), fabs(im))) / 2.302585092994046;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(fmax(abs(re), abs(im))) / 2.302585092994046d0
end function

Java

public static double code(double re, double im) {
	return Math.log(fmax(Math.abs(re), Math.abs(im))) / 2.302585092994046;
}

Python

def code(re, im):
	return math.log(fmax(math.fabs(re), math.fabs(im))) / 2.302585092994046

Julia

function code(re, im)
	return Float64(log(fmax(abs(re), abs(im))) / 2.302585092994046)
end

MATLAB

function tmp = code(re, im)
	tmp = log(max(abs(re), abs(im))) / 2.302585092994046;
end

Wolfram

code[re_, im_] := N[(N[Log[N[Max[N[Abs[re], $MachinePrecision], N[Abs[im], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.302585092994046), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

2. Evaluated real constant 51.4%

   \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{2.302585092994046}} \]

3. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{2.302585092994046} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\frac{2592480341699211}{1125899906842624}} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\frac{2592480341699211}{1125899906842624}} \]

   3. lower-/.f64 27.9%

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{2.302585092994046} \]

5. Applied rewrites 27.9%

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{2.302585092994046} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\frac{2592480341699211}{1125899906842624}}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   5. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   6. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   7. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   8. log-rec N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   9. remove-double-neg N/A

      \[\leadsto \log im \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   10. lower-log.f64 N/A

       \[\leadsto \log im \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   11. lower-log.f64 N/A

       \[\leadsto \log im \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1125899906842624}{2592480341699211}\right) \]

7. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\log im \cdot 0.43429448190325176} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\log im \cdot \frac{1125899906842624}{2592480341699211}} \]

   2. metadata-eval N/A

      \[\leadsto \log im \cdot \color{blue}{\frac{1}{\frac{2592480341699211}{1125899906842624}}} \]

   3. metadata-eval N/A

      \[\leadsto \log im \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-2592480341699211}{1125899906842624}\right)}} \]

   4. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{\log im}{\mathsf{neg}\left(\frac{-2592480341699211}{1125899906842624}\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{\mathsf{neg}\left(\frac{-2592480341699211}{1125899906842624}\right)}} \]

   6. metadata-eval 27.9%

      \[\leadsto \frac{\log im}{\color{blue}{2.302585092994046}} \]

9. Applied rewrites 27.9%

   \[\leadsto \color{blue}{\frac{\log im}{2.302585092994046}} \]
10. Add Preprocessing

Alternative 3: 97.8% accurate, 1.8× speedup

Math

\[\log \left(\mathsf{max}\left(\left|re\right|, \left|im\right|\right)\right) \cdot 0.43429448190325176 \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (log (fmax (fabs re) (fabs im))) 0.43429448190325176))

C

double code(double re, double im) {
	return log(fmax(fabs(re), fabs(im))) * 0.43429448190325176;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(fmax(abs(re), abs(im))) * 0.43429448190325176d0
end function

Java

public static double code(double re, double im) {
	return Math.log(fmax(Math.abs(re), Math.abs(im))) * 0.43429448190325176;
}

Python

def code(re, im):
	return math.log(fmax(math.fabs(re), math.fabs(im))) * 0.43429448190325176

Julia

function code(re, im)
	return Float64(log(fmax(abs(re), abs(im))) * 0.43429448190325176)
end

MATLAB

function tmp = code(re, im)
	tmp = log(max(abs(re), abs(im))) * 0.43429448190325176;
end

Wolfram

code[re_, im_] := N[(N[Log[N[Max[N[Abs[re], $MachinePrecision], N[Abs[im], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.43429448190325176), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

2. Evaluated real constant 51.4%

   \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{2.302585092994046}} \]

3. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{2.302585092994046} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\frac{2592480341699211}{1125899906842624}} \]

   2. lower-log.f64 N/A

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\frac{2592480341699211}{1125899906842624}} \]

   3. lower-/.f64 27.9%

      \[\leadsto \frac{-1 \cdot \log \left(\frac{1}{im}\right)}{2.302585092994046} \]

5. Applied rewrites 27.9%

   \[\leadsto \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{2.302585092994046} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\frac{2592480341699211}{1125899906842624}}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   5. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   6. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   7. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   8. log-rec N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)\right) \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   9. remove-double-neg N/A

      \[\leadsto \log im \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   10. lower-log.f64 N/A

       \[\leadsto \log im \cdot \frac{1}{\frac{2592480341699211}{1125899906842624}} \]

   11. lower-log.f64 N/A

       \[\leadsto \log im \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1125899906842624}{2592480341699211}\right) \]

7. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\log im \cdot 0.43429448190325176} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))