math.log/1 on complex, real part

Percentage Accurate: 52.4% → 100.0%

Time: 4.7s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im))))
end function

Java

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

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im))))

Julia

function code(re, im)
	return log(sqrt(Float64(Float64(re * re) + Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im))));
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im))))
end function

Java

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

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im))))

Julia

function code(re, im)
	return log(sqrt(Float64(Float64(re * re) + Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im))));
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

   1. hypot-define 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 79.0%

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

   2. expm1-undefine 79.0%

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

   3. log1p-undefine 79.0%

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

   4. rem-exp-log 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \left(1 + \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) + -1 \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

   1. hypot-define 100.0%

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

3. Simplified 100.0%

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

Alternative 3: 27.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{1}{im}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-N[Log[N[(1.0 / im), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 47.8%

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

   1. hypot-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around inf 29.0%

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

6. Final simplification 29.0%

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

Alternative 4: 27.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

   1. hypot-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in re around 0 29.0%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))