math.log/2 on complex, real part

Percentage Accurate: 52.7% → 99.2%

Time: 26.7s

Alternatives: 5

Speedup: 2.7×

Specification

Math

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

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)
    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: 52.7% accurate, 1.0× speedup

Math

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

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)
    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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \frac{\log \left(\mathsf{hypot}\left(e^{\log im\_m}, re\right)\right) \cdot \log base + 0}{{\log base}^{2}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m base)
 :precision binary64
 (/
  (+ (* (log (hypot (exp (log im_m)) re)) (log base)) 0.0)
  (pow (log base) 2.0)))

C

im_m = fabs(im);
double code(double re, double im_m, double base) {
	return ((log(hypot(exp(log(im_m)), re)) * log(base)) + 0.0) / pow(log(base), 2.0);
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m, double base) {
	return ((Math.log(Math.hypot(Math.exp(Math.log(im_m)), re)) * Math.log(base)) + 0.0) / Math.pow(Math.log(base), 2.0);
}

Python

im_m = math.fabs(im)
def code(re, im_m, base):
	return ((math.log(math.hypot(math.exp(math.log(im_m)), re)) * math.log(base)) + 0.0) / math.pow(math.log(base), 2.0)

Julia

im_m = abs(im)
function code(re, im_m, base)
	return Float64(Float64(Float64(log(hypot(exp(log(im_m)), re)) * log(base)) + 0.0) / (log(base) ^ 2.0))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m, base)
	tmp = ((log(hypot(exp(log(im_m)), re)) * log(base)) + 0.0) / (log(base) ^ 2.0);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[Exp[N[Log[im$95$m], $MachinePrecision]], $MachinePrecision] ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] / N[Power[N[Log[base], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{{im}^{2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   6. pow-to-exp N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{e^{\log im \cdot 2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   7. exp-lft-sqr N/A

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

   8. lift-*.f64 N/A

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

   9. lower-hypot.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log.f64 48.1

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

4. Applied rewrites 48.1%

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

5. Taylor expanded in im around 0

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
6. Step-by-step derivation

7. Applied rewrites 48.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \]

   2. metadata-eval 48.1

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]

9. Applied rewrites 48.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{\log base \cdot \log base + 0}} \]

    2. +-rgt-identity 48.1

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{\log base \cdot \log base}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{\log base \cdot \log base}} \]

    4. pow2 N/A

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

    5. lower-pow.f64 48.1

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

11. Applied rewrites 48.1%

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]
12. Add Preprocessing

Alternative 2: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \frac{0 + \log base \cdot \log \left(\mathsf{hypot}\left(im\_m, re\right)\right)}{{\log base}^{2}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m base)
 :precision binary64
 (/ (+ 0.0 (* (log base) (log (hypot im_m re)))) (pow (log base) 2.0)))

C

im_m = fabs(im);
double code(double re, double im_m, double base) {
	return (0.0 + (log(base) * log(hypot(im_m, re)))) / pow(log(base), 2.0);
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m, double base) {
	return (0.0 + (Math.log(base) * Math.log(Math.hypot(im_m, re)))) / Math.pow(Math.log(base), 2.0);
}

Python

im_m = math.fabs(im)
def code(re, im_m, base):
	return (0.0 + (math.log(base) * math.log(math.hypot(im_m, re)))) / math.pow(math.log(base), 2.0)

Julia

im_m = abs(im)
function code(re, im_m, base)
	return Float64(Float64(0.0 + Float64(log(base) * log(hypot(im_m, re)))) / (log(base) ^ 2.0))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m, base)
	tmp = (0.0 + (log(base) * log(hypot(im_m, re)))) / (log(base) ^ 2.0);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_, base_] := N[(N[(0.0 + N[(N[Log[base], $MachinePrecision] * N[Log[N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Log[base], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{{im}^{2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   6. pow-to-exp N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{e^{\log im \cdot 2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   7. exp-lft-sqr N/A

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

   8. lift-*.f64 N/A

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

   9. lower-hypot.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log.f64 48.1

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

4. Applied rewrites 48.1%

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

5. Taylor expanded in im around 0

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
6. Step-by-step derivation

7. Applied rewrites 48.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(\color{blue}{e^{\log im}}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0 \cdot 0} \]

   2. lift-log.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\color{blue}{\log im}}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0 \cdot 0} \]

   3. rem-exp-log 99.2

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

9. Applied rewrites 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(\color{blue}{im}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0 \cdot 0} \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. metadata-eval N/A

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

    4. +-rgt-identity 99.2

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

    5. lift-*.f64 N/A

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

    6. pow2 N/A

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

    7. lower-pow.f64 99.2

       \[\leadsto \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

11. Applied rewrites 99.2%

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}}} \]

12. Final simplification 99.2%

    \[\leadsto \frac{0 + \log base \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)}{{\log base}^{2}} \]
13. Add Preprocessing

Alternative 3: 53.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \frac{\log im\_m}{\log \left(\frac{1}{\frac{1}{base}}\right)} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m base)
 :precision binary64
 (/ (log im_m) (log (/ 1.0 (/ 1.0 base)))))

C

im_m = fabs(im);
double code(double re, double im_m, double base) {
	return log(im_m) / log((1.0 / (1.0 / base)));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8), intent (in) :: base
    code = log(im_m) / log((1.0d0 / (1.0d0 / base)))
end function

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m, base):
	return math.log(im_m) / math.log((1.0 / (1.0 / base)))

Julia

im_m = abs(im)
function code(re, im_m, base)
	return Float64(log(im_m) / log(Float64(1.0 / Float64(1.0 / base))))
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_, base_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[N[(1.0 / N[(1.0 / base), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

7. Applied rewrites 29.1%

   \[\leadsto \frac{\log im}{\log \left(\frac{1}{\frac{1}{base}}\right)} \]
8. Add Preprocessing

Alternative 4: 53.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \frac{\log im\_m}{\log base} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m base) :precision binary64 (/ (log im_m) (log base)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_, base_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

Alternative 5: 3.1% accurate, 562.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m base) :precision binary64 0.0)

C

im_m = fabs(im);
double code(double re, double im_m, double base) {
	return 0.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8), intent (in) :: base
    code = 0.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m, double base) {
	return 0.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m, base):
	return 0.0

Julia

im_m = abs(im)
function code(re, im_m, base)
	return 0.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m, base)
	tmp = 0.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_, base_] := 0.0

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in re around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

7. Applied rewrites 3.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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))))