math.log/2 on complex, real part

Percentage Accurate: 51.8% → 99.4%

Time: 10.7s

Alternatives: 6

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: 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, 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 51.6%

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

   1. mul0-rgt 51.6%

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

   2. +-rgt-identity 51.6%

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

   3. metadata-eval 51.6%

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

   4. +-rgt-identity 51.6%

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

   5. times-frac 51.7%

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

   6. *-inverses 51.7%

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

   7. *-rgt-identity 51.7%

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

   8. hypot-def 99.4%

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

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 2: 43.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{if}\;im \leq 1.02 \cdot 10^{-203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\log im \cdot \frac{1}{\log base}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ -1.0 re))) (log base))))
   (if (<= im 1.02e-203)
     t_0
     (if (<= im 7.2e-165)
       (/ 1.0 (/ (log base) (log im)))
       (if (<= im 2.55e-101)
         t_0
         (if (<= im 1.05e-61)
           (* (log im) (/ 1.0 (log base)))
           (if (<= im 2.7e-50) t_0 (/ (log im) (log base)))))))))

C

double code(double re, double im, double base) {
	double t_0 = -log((-1.0 / re)) / log(base);
	double tmp;
	if (im <= 1.02e-203) {
		tmp = t_0;
	} else if (im <= 7.2e-165) {
		tmp = 1.0 / (log(base) / log(im));
	} else if (im <= 2.55e-101) {
		tmp = t_0;
	} else if (im <= 1.05e-61) {
		tmp = log(im) * (1.0 / log(base));
	} else if (im <= 2.7e-50) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -log(((-1.0d0) / re)) / log(base)
    if (im <= 1.02d-203) then
        tmp = t_0
    else if (im <= 7.2d-165) then
        tmp = 1.0d0 / (log(base) / log(im))
    else if (im <= 2.55d-101) then
        tmp = t_0
    else if (im <= 1.05d-61) then
        tmp = log(im) * (1.0d0 / log(base))
    else if (im <= 2.7d-50) then
        tmp = t_0
    else
        tmp = log(im) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double t_0 = -Math.log((-1.0 / re)) / Math.log(base);
	double tmp;
	if (im <= 1.02e-203) {
		tmp = t_0;
	} else if (im <= 7.2e-165) {
		tmp = 1.0 / (Math.log(base) / Math.log(im));
	} else if (im <= 2.55e-101) {
		tmp = t_0;
	} else if (im <= 1.05e-61) {
		tmp = Math.log(im) * (1.0 / Math.log(base));
	} else if (im <= 2.7e-50) {
		tmp = t_0;
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	t_0 = -math.log((-1.0 / re)) / math.log(base)
	tmp = 0
	if im <= 1.02e-203:
		tmp = t_0
	elif im <= 7.2e-165:
		tmp = 1.0 / (math.log(base) / math.log(im))
	elif im <= 2.55e-101:
		tmp = t_0
	elif im <= 1.05e-61:
		tmp = math.log(im) * (1.0 / math.log(base))
	elif im <= 2.7e-50:
		tmp = t_0
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	t_0 = Float64(Float64(-log(Float64(-1.0 / re))) / log(base))
	tmp = 0.0
	if (im <= 1.02e-203)
		tmp = t_0;
	elseif (im <= 7.2e-165)
		tmp = Float64(1.0 / Float64(log(base) / log(im)));
	elseif (im <= 2.55e-101)
		tmp = t_0;
	elseif (im <= 1.05e-61)
		tmp = Float64(log(im) * Float64(1.0 / log(base)));
	elseif (im <= 2.7e-50)
		tmp = t_0;
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	t_0 = -log((-1.0 / re)) / log(base);
	tmp = 0.0;
	if (im <= 1.02e-203)
		tmp = t_0;
	elseif (im <= 7.2e-165)
		tmp = 1.0 / (log(base) / log(im));
	elseif (im <= 2.55e-101)
		tmp = t_0;
	elseif (im <= 1.05e-61)
		tmp = log(im) * (1.0 / log(base));
	elseif (im <= 2.7e-50)
		tmp = t_0;
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := Block[{t$95$0 = N[((-N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]) / N[Log[base], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.02e-203], t$95$0, If[LessEqual[im, 7.2e-165], N[(1.0 / N[(N[Log[base], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.55e-101], t$95$0, If[LessEqual[im, 1.05e-61], N[(N[Log[im], $MachinePrecision] * N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e-50], t$95$0, N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.02000000000000005e-203 or 7.19999999999999969e-165 < im < 2.5500000000000001e-101 or 1.05e-61 < im < 2.7e-50

   1. Initial program 51.2%

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

      1. mul0-rgt 51.2%

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

      2. +-rgt-identity 51.2%

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

      3. metadata-eval 51.2%

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

      4. +-rgt-identity 51.2%

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

      5. times-frac 51.3%

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

      6. *-inverses 51.3%

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

      7. *-rgt-identity 51.3%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around -inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. mul-1-neg 39.4%

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

   6. Simplified 39.4%

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

   if 1.02000000000000005e-203 < im < 7.19999999999999969e-165

   1. Initial program 56.6%

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

      1. mul0-rgt 56.6%

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

      2. +-rgt-identity 56.6%

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

      3. metadata-eval 56.6%

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

      4. +-rgt-identity 56.6%

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

      5. times-frac 56.6%

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

      6. *-inverses 56.6%

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

      7. *-rgt-identity 56.6%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 25.4%

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

      1. clear-num 25.4%

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

      2. inv-pow 25.4%

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

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\frac{\log base}{\log im}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 25.4%

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

   8. Simplified 25.4%

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

   if 2.5500000000000001e-101 < im < 1.05e-61

   1. Initial program 83.3%

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

      1. mul0-rgt 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. metadata-eval 83.3%

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

      4. +-rgt-identity 83.3%

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

      5. times-frac 83.5%

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

      6. *-inverses 83.5%

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

      7. *-rgt-identity 83.5%

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

      8. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in re around 0 48.6%

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

      1. clear-num 48.7%

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

      2. associate-/r/ 48.8%

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

   6. Applied egg-rr 48.8%

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

   if 2.7e-50 < im

   1. Initial program 46.7%

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

      1. mul0-rgt 46.7%

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

      2. +-rgt-identity 46.7%

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

      3. metadata-eval 46.7%

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

      4. +-rgt-identity 46.7%

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

      5. times-frac 46.6%

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

      6. *-inverses 46.6%

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

      7. *-rgt-identity 46.6%

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

      8. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in re around 0 79.0%

      \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-101}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\log im \cdot \frac{1}{\log base}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 3: 43.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{re}\right)\\ t_1 := \frac{-t_0}{\log base}\\ \mathbf{if}\;im \leq 1.02 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-102}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\log base}\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{-61}:\\ \;\;\;\;\log im \cdot \frac{1}{\log base}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (let* ((t_0 (log (/ -1.0 re))) (t_1 (/ (- t_0) (log base))))
   (if (<= im 1.02e-203)
     t_1
     (if (<= im 1.7e-168)
       (/ 1.0 (/ (log base) (log im)))
       (if (<= im 2.15e-102)
         (* t_0 (/ -1.0 (log base)))
         (if (<= im 2.35e-61)
           (* (log im) (/ 1.0 (log base)))
           (if (<= im 3.2e-50) t_1 (/ (log im) (log base)))))))))

C

double code(double re, double im, double base) {
	double t_0 = log((-1.0 / re));
	double t_1 = -t_0 / log(base);
	double tmp;
	if (im <= 1.02e-203) {
		tmp = t_1;
	} else if (im <= 1.7e-168) {
		tmp = 1.0 / (log(base) / log(im));
	} else if (im <= 2.15e-102) {
		tmp = t_0 * (-1.0 / log(base));
	} else if (im <= 2.35e-61) {
		tmp = log(im) * (1.0 / log(base));
	} else if (im <= 3.2e-50) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(((-1.0d0) / re))
    t_1 = -t_0 / log(base)
    if (im <= 1.02d-203) then
        tmp = t_1
    else if (im <= 1.7d-168) then
        tmp = 1.0d0 / (log(base) / log(im))
    else if (im <= 2.15d-102) then
        tmp = t_0 * ((-1.0d0) / log(base))
    else if (im <= 2.35d-61) then
        tmp = log(im) * (1.0d0 / log(base))
    else if (im <= 3.2d-50) then
        tmp = t_1
    else
        tmp = log(im) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double t_0 = Math.log((-1.0 / re));
	double t_1 = -t_0 / Math.log(base);
	double tmp;
	if (im <= 1.02e-203) {
		tmp = t_1;
	} else if (im <= 1.7e-168) {
		tmp = 1.0 / (Math.log(base) / Math.log(im));
	} else if (im <= 2.15e-102) {
		tmp = t_0 * (-1.0 / Math.log(base));
	} else if (im <= 2.35e-61) {
		tmp = Math.log(im) * (1.0 / Math.log(base));
	} else if (im <= 3.2e-50) {
		tmp = t_1;
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	t_0 = math.log((-1.0 / re))
	t_1 = -t_0 / math.log(base)
	tmp = 0
	if im <= 1.02e-203:
		tmp = t_1
	elif im <= 1.7e-168:
		tmp = 1.0 / (math.log(base) / math.log(im))
	elif im <= 2.15e-102:
		tmp = t_0 * (-1.0 / math.log(base))
	elif im <= 2.35e-61:
		tmp = math.log(im) * (1.0 / math.log(base))
	elif im <= 3.2e-50:
		tmp = t_1
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	t_0 = log(Float64(-1.0 / re))
	t_1 = Float64(Float64(-t_0) / log(base))
	tmp = 0.0
	if (im <= 1.02e-203)
		tmp = t_1;
	elseif (im <= 1.7e-168)
		tmp = Float64(1.0 / Float64(log(base) / log(im)));
	elseif (im <= 2.15e-102)
		tmp = Float64(t_0 * Float64(-1.0 / log(base)));
	elseif (im <= 2.35e-61)
		tmp = Float64(log(im) * Float64(1.0 / log(base)));
	elseif (im <= 3.2e-50)
		tmp = t_1;
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	t_0 = log((-1.0 / re));
	t_1 = -t_0 / log(base);
	tmp = 0.0;
	if (im <= 1.02e-203)
		tmp = t_1;
	elseif (im <= 1.7e-168)
		tmp = 1.0 / (log(base) / log(im));
	elseif (im <= 2.15e-102)
		tmp = t_0 * (-1.0 / log(base));
	elseif (im <= 2.35e-61)
		tmp = log(im) * (1.0 / log(base));
	elseif (im <= 3.2e-50)
		tmp = t_1;
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := Block[{t$95$0 = N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-t$95$0) / N[Log[base], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.02e-203], t$95$1, If[LessEqual[im, 1.7e-168], N[(1.0 / N[(N[Log[base], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.15e-102], N[(t$95$0 * N[(-1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.35e-61], N[(N[Log[im], $MachinePrecision] * N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.2e-50], t$95$1, N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 1.02000000000000005e-203 or 2.3499999999999998e-61 < im < 3.2e-50

   1. Initial program 51.5%

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

      1. mul0-rgt 51.5%

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

      2. +-rgt-identity 51.5%

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

      3. metadata-eval 51.5%

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

      4. +-rgt-identity 51.5%

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

      5. times-frac 51.6%

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

      6. *-inverses 51.6%

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

      7. *-rgt-identity 51.6%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around -inf 39.5%

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

      1. associate-*r/ 39.5%

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

      2. mul-1-neg 39.5%

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

   6. Simplified 39.5%

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

   if 1.02000000000000005e-203 < im < 1.70000000000000011e-168

   1. Initial program 56.6%

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

      1. mul0-rgt 56.6%

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

      2. +-rgt-identity 56.6%

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

      3. metadata-eval 56.6%

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

      4. +-rgt-identity 56.6%

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

      5. times-frac 56.6%

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

      6. *-inverses 56.6%

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

      7. *-rgt-identity 56.6%

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

      8. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 25.4%

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

      1. clear-num 25.4%

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

      2. inv-pow 25.4%

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

   6. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{{\left(\frac{\log base}{\log im}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 25.4%

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

   8. Simplified 25.4%

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

   if 1.70000000000000011e-168 < im < 2.1499999999999999e-102

   1. Initial program 46.0%

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

      1. mul0-rgt 46.0%

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

      2. +-rgt-identity 46.0%

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

      3. metadata-eval 46.0%

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

      4. +-rgt-identity 46.0%

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

      5. times-frac 46.0%

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

      6. *-inverses 46.0%

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

      7. *-rgt-identity 46.0%

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

      8. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. add-exp-log 54.6%

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

      2. log-rec 54.6%

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

   7. Applied egg-rr 54.6%

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

   8. Taylor expanded in re around -inf 15.8%

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

      1. mul-1-neg 15.8%

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

      2. *-commutative 15.8%

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

      3. distribute-rgt-neg-in 15.8%

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

      4. exp-neg 15.8%

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

      5. rem-exp-log 38.0%

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

   10. Simplified 38.0%

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

   if 2.1499999999999999e-102 < im < 2.3499999999999998e-61

   1. Initial program 83.3%

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

      1. mul0-rgt 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. metadata-eval 83.3%

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

      4. +-rgt-identity 83.3%

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

      5. times-frac 83.5%

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

      6. *-inverses 83.5%

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

      7. *-rgt-identity 83.5%

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

      8. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in re around 0 48.6%

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

      1. clear-num 48.7%

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

      2. associate-/r/ 48.8%

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

   6. Applied egg-rr 48.8%

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

   if 3.2e-50 < im

   1. Initial program 46.7%

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

      1. mul0-rgt 46.7%

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

      2. +-rgt-identity 46.7%

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

      3. metadata-eval 46.7%

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

      4. +-rgt-identity 46.7%

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

      5. times-frac 46.6%

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

      6. *-inverses 46.6%

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

      7. *-rgt-identity 46.6%

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

      8. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in re around 0 79.0%

      \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log im}}\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-102}:\\ \;\;\;\;\log \left(\frac{-1}{re}\right) \cdot \frac{-1}{\log base}\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{-61}:\\ \;\;\;\;\log im \cdot \frac{1}{\log base}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 4: 4.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double re, double im, double base) {
	return log(base) * 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(base) * log(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. mul0-rgt 51.6%

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

   2. +-rgt-identity 51.6%

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

   3. metadata-eval 51.6%

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

   4. +-rgt-identity 51.6%

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

   5. times-frac 51.7%

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

   6. *-inverses 51.7%

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

   7. *-rgt-identity 51.7%

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

   8. hypot-def 99.4%

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

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 27.9%

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

   1. log1p-expm1-u 27.6%

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

6. Applied egg-rr 27.6%

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

   1. log1p-expm1-u 27.9%

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

   2. div-inv 27.9%

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

   3. add-exp-log 14.4%

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

   4. neg-log 14.4%

      \[\leadsto \log im \cdot e^{\color{blue}{-\log \log base}} \]

   5. add-sqr-sqrt 0.0%

      \[\leadsto \log im \cdot e^{\color{blue}{\sqrt{-\log \log base} \cdot \sqrt{-\log \log base}}} \]

   6. sqrt-unprod 2.6%

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

   7. sqr-neg 2.6%

      \[\leadsto \log im \cdot e^{\sqrt{\color{blue}{\log \log base \cdot \log \log base}}} \]

   8. sqrt-unprod 2.6%

      \[\leadsto \log im \cdot e^{\color{blue}{\sqrt{\log \log base} \cdot \sqrt{\log \log base}}} \]

   9. add-sqr-sqrt 2.6%

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

   10. add-exp-log 5.0%

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

8. Applied egg-rr 5.0%

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

9. Final simplification 5.0%

   \[\leadsto \log base \cdot \log im \]

Alternative 5: 26.2% 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 51.6%

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

   1. mul0-rgt 51.6%

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

   2. +-rgt-identity 51.6%

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

   3. metadata-eval 51.6%

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

   4. +-rgt-identity 51.6%

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

   5. times-frac 51.7%

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

   6. *-inverses 51.7%

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

   7. *-rgt-identity 51.7%

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

   8. hypot-def 99.4%

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

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 27.9%

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

5. Final simplification 27.9%

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

Alternative 6: 3.9% 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 51.6%

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

   1. mul0-rgt 51.6%

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

   2. +-rgt-identity 51.6%

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

   3. metadata-eval 51.6%

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

   4. +-rgt-identity 51.6%

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

   5. times-frac 51.7%

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

   6. *-inverses 51.7%

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

   7. *-rgt-identity 51.7%

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

   8. hypot-def 99.4%

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

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 27.9%

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

   1. clear-num 27.9%

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

   2. associate-/r/ 27.9%

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

6. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{\frac{1}{\log base} \cdot \log im} \]
7. Step-by-step derivation

   1. inv-pow 27.9%

      \[\leadsto \color{blue}{{\log base}^{-1}} \cdot \log im \]

   2. metadata-eval 27.9%

      \[\leadsto {\log base}^{\color{blue}{\left(1 - 2\right)}} \cdot \log im \]

   3. pow-div 27.9%

      \[\leadsto \color{blue}{\frac{{\log base}^{1}}{{\log base}^{2}}} \cdot \log im \]

   4. pow1 27.9%

      \[\leadsto \frac{\color{blue}{\log base}}{{\log base}^{2}} \cdot \log im \]

   5. *-commutative 27.9%

      \[\leadsto \color{blue}{\log im \cdot \frac{\log base}{{\log base}^{2}}} \]

   6. associate-*r/ 28.0%

      \[\leadsto \color{blue}{\frac{\log im \cdot \log base}{{\log base}^{2}}} \]

   7. frac-2neg 28.0%

      \[\leadsto \color{blue}{\frac{-\log im \cdot \log base}{-{\log base}^{2}}} \]

   8. *-commutative 28.0%

      \[\leadsto \frac{-\color{blue}{\log base \cdot \log im}}{-{\log base}^{2}} \]

   9. add-sqr-sqrt 28.0%

      \[\leadsto \frac{-\log base \cdot \log im}{-\color{blue}{\sqrt{{\log base}^{2}} \cdot \sqrt{{\log base}^{2}}}} \]

   10. sqrt-unprod 28.0%

       \[\leadsto \frac{-\log base \cdot \log im}{-\color{blue}{\sqrt{{\log base}^{2} \cdot {\log base}^{2}}}} \]

   11. unpow2 28.0%

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

   12. associate-*r* 27.9%

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

8. Applied egg-rr 4.3%

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

   1. distribute-lft-neg-in 4.3%

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

   2. *-commutative 4.3%

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

   3. associate-/l* 4.3%

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

   4. *-inverses 4.3%

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

   5. /-rgt-identity 4.3%

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

10. Simplified 4.3%

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

11. Final simplification 4.3%

    \[\leadsto \log im \]

Reproduce

herbie shell --seed 2023174 
(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))))