math.log/1 on complex, real part

Percentage Accurate: 51.7% → 100.0%

Time: 2.0s

Alternatives: 3

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 51.7% 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

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

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 51.9%

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

   1. expm1-log1p-u 51.9%

      \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)} \]

   2. expm1-udef 32.4%

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

   3. log1p-udef 32.4%

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

   4. add-exp-log 32.4%

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

   5. hypot-def 74.0%

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

3. Applied egg-rr 74.0%

   \[\leadsto \log \color{blue}{\left(\left(1 + \mathsf{hypot}\left(re, im\right)\right) - 1\right)} \]
4. Step-by-step derivation

   1. +-commutative 74.0%

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

   2. associate--l+ 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-rgt-identity 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 43.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.5e-157) (log (- re)) (log im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5e-157) {
		tmp = log(-re);
	} else {
		tmp = log(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.5d-157)) then
        tmp = log(-re)
    else
        tmp = log(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.5e-157) {
		tmp = Math.log(-re);
	} else {
		tmp = Math.log(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.5e-157:
		tmp = math.log(-re)
	else:
		tmp = math.log(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.5e-157)
		tmp = log(Float64(-re));
	else
		tmp = log(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.5e-157)
		tmp = log(-re);
	else
		tmp = log(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.5e-157], N[Log[(-re)], $MachinePrecision], N[Log[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5000000000000002e-157

   1. Initial program 55.0%

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]

   2. Taylor expanded in re around -inf 70.9%

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

      1. mul-1-neg 70.9%

         \[\leadsto \log \color{blue}{\left(-re\right)} \]

   4. Simplified 70.9%

      \[\leadsto \log \color{blue}{\left(-re\right)} \]

   if -3.5000000000000002e-157 < re

   1. Initial program 50.1%

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]

   2. Taylor expanded in re around 0 39.4%

      \[\leadsto \log \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 3: 27.5% 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 51.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]

2. Taylor expanded in re around 0 31.5%

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

3. Final simplification 31.5%

   \[\leadsto \log im \]

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))