math.log/2 on complex, real part

Percentage Accurate: 25.5% → 49.9%

Time: 9.2s

Alternatives: 4

Speedup: 3.4×

• Could not converge on a ground truth

  1. re = 1.7533746046869366e+195
  2. im = -1.7030234586554808e+258
  3. base = -1.7593880419529947e+99

Specification

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (/
 (+
  (* (log (sqrt (+ (* re re) (* im im)))) (log base))
  (* (atan2 im re) 0.0))
 (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 25.5% accurate, 1.0× speedup

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (/
 (+
  (* (log (sqrt (+ (* re re) (* im im)))) (log base))
  (* (atan2 im re) 0.0))
 (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 49.9% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\left|\log base\right|} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (*
 (copysign 1.0 (log base))
 (/ (log (hypot re im)) (fabs (log base)))))

C

double code(double re, double im, double base) {
	return copysign(1.0, log(base)) * (log(hypot(re, im)) / fabs(log(base)));
}

Java

public static double code(double re, double im, double base) {
	return Math.copySign(1.0, Math.log(base)) * (Math.log(Math.hypot(re, im)) / Math.abs(Math.log(base)));
}

Python

def code(re, im, base):
	return math.copysign(1.0, math.log(base)) * (math.log(math.hypot(re, im)) / math.fabs(math.log(base)))

Julia

function code(re, im, base)
	return Float64(copysign(1.0, log(base)) * Float64(log(hypot(re, im)) / abs(log(base))))
end

MATLAB

function tmp = code(re, im, base)
	tmp = (sign(log(base)) * abs(1.0)) * (log(hypot(re, im)) / abs(log(base)));
end

Wolfram

code[re_, im_, base_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Log[base], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Abs[N[Log[base], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}}{\log base \cdot \log base + 0 \cdot 0} \]

   6. sub-flip-reverse N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0}}{\log base \cdot \log base + 0 \cdot 0} \]

   7. --rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base \cdot \log base + 0 \cdot 0} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base \cdot \log base + \color{blue}{0}} \]

   13. +-rgt-identity N/A

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

   14. lift-*.f64 N/A

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

   15. sqr-abs-rev N/A

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

3. Applied rewrites 25.5%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}{\left|\log base\right|}} \]
4. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \color{blue}{\left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}}{\left|\log base\right|} \]

   2. lift-fma.f64 N/A

      \[\leadsto \mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}{\left|\log base\right|} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\left|\log base\right|} \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\left|\log base\right|} \]

   5. lift-hypot.f64 49.9%

      \[\leadsto \mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\left|\log base\right|} \]

5. Applied rewrites 49.9%

   \[\leadsto \mathsf{copysign}\left(1, \log base\right) \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\left|\log base\right|} \]
6. Add Preprocessing

Alternative 2: 49.9% accurate, 2.1× speedup

Math

\[\frac{1}{\log base} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

FPCore

(FPCore (re im base)
  :precision binary64
  (* (/ 1.0 (log base)) (log (hypot re im))))

C

double code(double re, double im, double base) {
	return (1.0 / log(base)) * log(hypot(re, im));
}

Java

public static double code(double re, double im, double base) {
	return (1.0 / Math.log(base)) * Math.log(Math.hypot(re, im));
}

Python

def code(re, im, base):
	return (1.0 / math.log(base)) * math.log(math.hypot(re, im))

Julia

function code(re, im, base)
	return Float64(Float64(1.0 / log(base)) * log(hypot(re, im)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = (1.0 / log(base)) * log(hypot(re, im));
end

Wolfram

code[re_, im_, base_] := N[(N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

   3. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   5. mul0-rgt N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   6. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)} \]

3. Applied rewrites 25.5%

   \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

      \[\leadsto \left(\color{blue}{{\left(\log base \cdot \log base\right)}^{-1}} \cdot \log base\right) \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left({\color{blue}{\left(\log base \cdot \log base\right)}}^{-1} \cdot \log base\right) \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   7. pow-prod-down N/A

      \[\leadsto \left(\color{blue}{\left({\log base}^{-1} \cdot {\log base}^{-1}\right)} \cdot \log base\right) \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   8. pow-prod-up N/A

      \[\leadsto \left(\color{blue}{{\log base}^{\left(-1 + -1\right)}} \cdot \log base\right) \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   9. metadata-eval N/A

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

   10. metadata-eval N/A

       \[\leadsto \left({\log base}^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}} \cdot \log base\right) \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   11. pow-plus N/A

       \[\leadsto \color{blue}{{\log base}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)}} \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto {\log base}^{\left(\color{blue}{-2} + 1\right)} \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto {\log base}^{\color{blue}{-1}} \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right) \]

   14. inv-pow N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 25.5%

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

5. Applied rewrites 25.5%

   \[\leadsto \color{blue}{\frac{1}{\log base} \cdot \log \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. lift-hypot.f64 49.9%

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

7. Applied rewrites 49.9%

   \[\leadsto \frac{1}{\log base} \cdot \log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)} \]
8. Add Preprocessing

Alternative 3: 49.6% accurate, 2.6× speedup

Math

\[-1 \cdot \frac{\log \left(\frac{1}{\mathsf{max}\left(\left|re\right|, \left|im\right|\right)}\right)}{\log base} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (* -1.0 (/ (log (/ 1.0 (fmax (fabs re) (fabs im)))) (log base))))

C

double code(double re, double im, double base) {
	return -1.0 * (log((1.0 / fmax(fabs(re), fabs(im)))) / log(base));
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = (-1.0d0) * (log((1.0d0 / fmax(abs(re), abs(im)))) / log(base))
end function

Java

public static double code(double re, double im, double base) {
	return -1.0 * (Math.log((1.0 / fmax(Math.abs(re), Math.abs(im)))) / Math.log(base));
}

Python

def code(re, im, base):
	return -1.0 * (math.log((1.0 / fmax(math.fabs(re), math.fabs(im)))) / math.log(base))

Julia

function code(re, im, base)
	return Float64(-1.0 * Float64(log(Float64(1.0 / fmax(abs(re), abs(im)))) / log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = -1.0 * (log((1.0 / max(abs(re), abs(im)))) / log(base));
end

Wolfram

code[re_, im_, base_] := N[(-1.0 * N[(N[Log[N[(1.0 / N[Max[N[Abs[re], $MachinePrecision], N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

2. Taylor expanded in im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-log.f64 13.4%

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

4. Applied rewrites 13.4%

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

Alternative 4: 49.6% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

2. Taylor expanded in im around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-log.f64 13.6%

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

4. Applied rewrites 13.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. mul-1-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. log-div N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. log-div N/A

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

   12. log-rec N/A

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

   13. remove-double-neg N/A

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

   14. lower-log.f64 13.4%

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

6. Applied rewrites 13.4%

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

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))