math.log/2 on complex, real part

Percentage Accurate: 51.8% → 99.4%

Time: 9.8s

Alternatives: 3

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: 51.8% 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, 1.1× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
(FPCore (re im_m base)
 :precision binary64
 (/ (log (hypot (exp (log im_m)) re)) (log base)))

C

im_m = fabs(im);
assert(re < im_m && im_m < base);
double code(double re, double im_m, double base) {
	return log(hypot(exp(log(im_m)), re)) / log(base);
}

Java

im_m = Math.abs(im);
assert re < im_m && im_m < base;
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);
}

Python

im_m = math.fabs(im)
[re, im_m, base] = sort([re, im_m, base])
def code(re, im_m, base):
	return math.log(math.hypot(math.exp(math.log(im_m)), re)) / math.log(base)

Julia

im_m = abs(im)
re, im_m, base = sort([re, im_m, base])
function code(re, im_m, base)
	return Float64(log(hypot(exp(log(im_m)), re)) / log(base))
end

MATLAB

im_m = abs(im);
re, im_m, base = num2cell(sort([re, im_m, base])){:}
function tmp = code(re, im_m, base)
	tmp = log(hypot(exp(log(im_m)), re)) / log(base);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
code[re_, im$95$m_, base_] := N[(N[Log[N[Sqrt[N[Exp[N[Log[im$95$m], $MachinePrecision]], $MachinePrecision] ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 54.8%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)}{\log base} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.8× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
(FPCore (re im_m base)
 :precision binary64
 (/ (log (hypot im_m re)) (log base)))

C

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

Java

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

Python

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

Julia

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

MATLAB

im_m = abs(im);
re, im_m, base = num2cell(sort([re, im_m, base])){:}
function tmp = code(re, im_m, base)
	tmp = log(hypot(im_m, re)) / log(base);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
code[re_, im$95$m_, base_] := N[(N[Log[N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. lower-hypot.f64 N/A

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

   6. lower-log.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 3: 77.3% accurate, 2.7× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
(FPCore (re im_m base) :precision binary64 (/ (log im_m) (log base)))

C

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

Fortran

im_m = abs(im)
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
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);
assert re < im_m && im_m < base;
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)
[re, im_m, base] = sort([re, im_m, base])
def code(re, im_m, base):
	return math.log(im_m) / math.log(base)

Julia

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

MATLAB

im_m = abs(im);
re, im_m, base = num2cell(sort([re, im_m, base])){:}
function tmp = code(re, im_m, base)
	tmp = log(im_m) / log(base);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
NOTE: re, im_m, and base should be sorted in increasing order before calling this function.
code[re_, im$95$m_, base_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in re around 0

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

   1. remove-double-neg N/A

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

   2. log-rec N/A

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

   3. lower-/.f64 N/A

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

   4. log-rec N/A

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

   5. remove-double-neg N/A

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

   6. lower-log.f64 N/A

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

   7. lower-log.f64 30.4

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

5. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2024329 
(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))))