math.log/2 on complex, real part

Percentage Accurate: 51.5% → 99.4%

Time: 10.9s

Alternatives: 4

Speedup: 2.0×

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: 51.5% 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: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\left(1 + \log base\right) + -1} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. mul0-rgt 55.1%

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

   2. +-rgt-identity 55.1%

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

   3. associate-/l* 55.2%

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

   4. hypot-def 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

3. Simplified 99.4%

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

   1. pow2 99.4%

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

   2. pow1 99.4%

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

   3. pow-div 99.4%

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

   4. metadata-eval 99.4%

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

   5. pow1 99.4%

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

   6. expm1-log1p-u 50.3%

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log base\right)\right)}} \]

5. Applied egg-rr 50.3%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log base\right)\right)}} \]
6. Step-by-step derivation

   1. expm1-udef 50.2%

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{e^{\mathsf{log1p}\left(\log base\right)} - 1}} \]

   2. log1p-udef 50.2%

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

   3. rem-exp-log 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. mul0-rgt 55.1%

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

   2. +-rgt-identity 55.1%

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

   3. associate-/l* 55.2%

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

   4. hypot-def 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. associate-/l* 99.4%

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

   8. *-inverses 99.4%

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

   9. /-rgt-identity 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 28.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

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(im) / log(base)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. mul0-rgt 55.1%

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

   2. +-rgt-identity 55.1%

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

   3. associate-/l* 55.2%

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

   4. hypot-def 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. associate-/l* 99.4%

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

   8. *-inverses 99.4%

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

   9. /-rgt-identity 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in re around 0 25.6%

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

5. Final simplification 25.6%

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

Alternative 4: 4.0% accurate, 6.2× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (log im))

C

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

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(im)
end function

Java

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

Python

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

Julia

function code(re, im, base)
	return log(im)
end

MATLAB

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

Wolfram

code[re_, im_, base_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 55.1%

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

   1. mul0-rgt 55.1%

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

   2. +-rgt-identity 55.1%

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

   3. associate-/l* 55.2%

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

   4. hypot-def 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. associate-/l* 99.4%

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

   8. *-inverses 99.4%

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

   9. /-rgt-identity 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in re around 0 25.6%

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

   1. log1p-expm1-u 25.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log im}{\log base}\right)\right)} \]

6. Applied egg-rr 25.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log im}{\log base}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-udef 25.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\frac{\log im}{\log base}} - 1}\right) \]

   2. div-inv 25.5%

      \[\leadsto \mathsf{log1p}\left(e^{\color{blue}{\log im \cdot \frac{1}{\log base}}} - 1\right) \]

   3. exp-to-pow 25.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{im}^{\left(\frac{1}{\log base}\right)}} - 1\right) \]

8. Applied egg-rr 25.5%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{{im}^{\left(\frac{1}{\log base}\right)} - 1}\right) \]

10. Applied egg-rr 3.5%

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

    1. *-commutative 3.5%

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

    2. associate-/l* 3.5%

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

    3. *-inverses 3.5%

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

    4. /-rgt-identity 3.5%

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

12. Simplified 3.5%

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

13. Final simplification 3.5%

    \[\leadsto \log im \]

Reproduce

herbie shell --seed 2023321 
(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))))