math.log/2 on complex, real part

Percentage Accurate: 52.0% → 99.4%

Time: 9.0s

Alternatives: 3

Speedup: 2.0×

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

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: 52.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in base around 0 46.0%

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

   1. +-commutative 46.0%

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

   2. unpow2 46.0%

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

   3. unpow2 46.0%

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

   4. hypot-def 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 43.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= re -4.9e-45)
   (/ (- (log (/ -1.0 re))) (log base))
   (/ (log im) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if (re <= -4.9e-45) {
		tmp = -log((-1.0 / re)) / log(base);
	} else {
		tmp = log(im) / log(base);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (re <= (-4.9d-45)) then
        tmp = -log(((-1.0d0) / re)) / log(base)
    else
        tmp = log(im) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (re <= -4.9e-45) {
		tmp = -Math.log((-1.0 / re)) / Math.log(base);
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if re <= -4.9e-45:
		tmp = -math.log((-1.0 / re)) / math.log(base)
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (re <= -4.9e-45)
		tmp = Float64(Float64(-log(Float64(-1.0 / re))) / log(base));
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (re <= -4.9e-45)
		tmp = -log((-1.0 / re)) / log(base);
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[re, -4.9e-45], N[((-N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]) / N[Log[base], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.8999999999999998e-45

   1. Initial program 37.1%

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

   2. Taylor expanded in re around -inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. mul-1-neg 76.6%

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

   4. Simplified 76.6%

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

   if -4.8999999999999998e-45 < re

   1. Initial program 49.1%

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

   2. Taylor expanded in re around 0 29.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 3: 27.4% 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 45.9%

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

2. Taylor expanded in re around 0 25.6%

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

3. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2023187 
(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))))