math.log/2 on complex, real part

Percentage Accurate: 51.3% → 99.2%

Time: 20.0s

Alternatives: 5

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: 51.3% 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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ [re_m, im, base] = \mathsf{sort}([re_m, im, base])\\ \\ \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\_m\right)\right) \cdot \log base}{{\log base}^{2}} \end{array} \]

FPCore

re_m = (fabs.f64 re)
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
(FPCore (re_m im base)
 :precision binary64
 (/ (* (log (hypot (exp (log im)) re_m)) (log base)) (pow (log base) 2.0)))

C

re_m = fabs(re);
assert(re_m < im && im < base);
double code(double re_m, double im, double base) {
	return (log(hypot(exp(log(im)), re_m)) * log(base)) / pow(log(base), 2.0);
}

Java

re_m = Math.abs(re);
assert re_m < im && im < base;
public static double code(double re_m, double im, double base) {
	return (Math.log(Math.hypot(Math.exp(Math.log(im)), re_m)) * Math.log(base)) / Math.pow(Math.log(base), 2.0);
}

Python

re_m = math.fabs(re)
[re_m, im, base] = sort([re_m, im, base])
def code(re_m, im, base):
	return (math.log(math.hypot(math.exp(math.log(im)), re_m)) * math.log(base)) / math.pow(math.log(base), 2.0)

Julia

re_m = abs(re)
re_m, im, base = sort([re_m, im, base])
function code(re_m, im, base)
	return Float64(Float64(log(hypot(exp(log(im)), re_m)) * log(base)) / (log(base) ^ 2.0))
end

MATLAB

re_m = abs(re);
re_m, im, base = num2cell(sort([re_m, im, base])){:}
function tmp = code(re_m, im, base)
	tmp = (log(hypot(exp(log(im)), re_m)) * log(base)) / (log(base) ^ 2.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
code[re$95$m_, im_, base_] := N[(N[(N[Log[N[Sqrt[N[Exp[N[Log[im], $MachinePrecision]], $MachinePrecision] ^ 2 + re$95$m ^ 2], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] / N[Power[N[Log[base], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{e^{\log im \cdot 2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   4. exp-lft-sqr N/A

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

   5. accelerator-lowering-hypot.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(\color{blue}{e^{\log im}}, re\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   7. log-lowering-log.f64 46.1

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\color{blue}{\log im}}, re\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

4. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
5. Step-by-step derivation

   1. mul0-rgt 46.1

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

6. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
7. Step-by-step derivation

   1. metadata-eval 46.1

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

8. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}} + 0} \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}} + 0} \]

   3. log-lowering-log.f64 46.1

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{{\color{blue}{\log base}}^{2} + 0} \]

10. Applied egg-rr 46.1%

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}} + 0} \]

11. Final simplification 46.1%

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base}{{\log base}^{2}} \]
12. Add Preprocessing

Alternative 2: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{e^{\log im \cdot 2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   4. exp-lft-sqr N/A

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

   5. accelerator-lowering-hypot.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(\color{blue}{e^{\log im}}, re\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   7. log-lowering-log.f64 46.1

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\color{blue}{\log im}}, re\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

4. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
5. Step-by-step derivation

   1. mul0-rgt 46.1

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

6. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
7. Step-by-step derivation

   1. metadata-eval 46.1

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

8. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]
9. Step-by-step derivation

   1. rem-exp-log 99.2

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ [re_m, im, base] = \mathsf{sort}([re_m, im, base])\\ \\ \frac{\log base \cdot \log \left(\mathsf{fma}\left(re\_m, \frac{re\_m \cdot 0.5}{im}, im\right)\right)}{\log base \cdot \log base} \end{array} \]

FPCore

re_m = (fabs.f64 re)
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
(FPCore (re_m im base)
 :precision binary64
 (/
  (* (log base) (log (fma re_m (/ (* re_m 0.5) im) im)))
  (* (log base) (log base))))

C

re_m = fabs(re);
assert(re_m < im && im < base);
double code(double re_m, double im, double base) {
	return (log(base) * log(fma(re_m, ((re_m * 0.5) / im), im))) / (log(base) * log(base));
}

Julia

re_m = abs(re)
re_m, im, base = sort([re_m, im, base])
function code(re_m, im, base)
	return Float64(Float64(log(base) * log(fma(re_m, Float64(Float64(re_m * 0.5) / im), im))) / Float64(log(base) * log(base)))
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
code[re$95$m_, im_, base_] := N[(N[(N[Log[base], $MachinePrecision] * N[Log[N[(re$95$m * N[(N[(re$95$m * 0.5), $MachinePrecision] / im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{e^{\log im \cdot 2}} + re \cdot re}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   4. exp-lft-sqr N/A

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

   5. accelerator-lowering-hypot.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(\color{blue}{e^{\log im}}, re\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   7. log-lowering-log.f64 46.1

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\color{blue}{\log im}}, re\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

4. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(e^{\log im}, re\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
5. Step-by-step derivation

   1. mul0-rgt 46.1

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

6. Applied egg-rr 46.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(e^{\log im}, re\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
7. Step-by-step derivation

   1. metadata-eval 46.1

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

8. Applied egg-rr 46.1%

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

9. Taylor expanded in re around 0

   \[\leadsto \frac{\log \color{blue}{\left(im + \frac{1}{2} \cdot \frac{{re}^{2}}{im}\right)} \cdot \log base + 0}{\log base \cdot \log base + 0} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{im} + im\right)} \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    2. *-commutative N/A

       \[\leadsto \frac{\log \left(\color{blue}{\frac{{re}^{2}}{im} \cdot \frac{1}{2}} + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    3. associate-*l/ N/A

       \[\leadsto \frac{\log \left(\color{blue}{\frac{{re}^{2} \cdot \frac{1}{2}}{im}} + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    4. associate-*r/ N/A

       \[\leadsto \frac{\log \left(\color{blue}{{re}^{2} \cdot \frac{\frac{1}{2}}{im}} + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\log \left({re}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{im} + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    6. associate-*r/ N/A

       \[\leadsto \frac{\log \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right)} + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    7. unpow2 N/A

       \[\leadsto \frac{\log \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right) + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    8. associate-*l* N/A

       \[\leadsto \frac{\log \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)} + im\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    9. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(re, re \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right), im\right)\right)} \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    10. associate-*r/ N/A

        \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{im}}, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    11. metadata-eval N/A

        \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \frac{\color{blue}{\frac{1}{2}}}{im}, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    12. associate-*r/ N/A

        \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, \color{blue}{\frac{re \cdot \frac{1}{2}}{im}}, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, \color{blue}{\frac{re \cdot \frac{1}{2}}{im}}, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

    14. *-lowering-*.f64 23.5

        \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, \frac{\color{blue}{re \cdot 0.5}}{im}, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + 0} \]

11. Simplified 23.5%

    \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(re, \frac{re \cdot 0.5}{im}, im\right)\right)} \cdot \log base + 0}{\log base \cdot \log base + 0} \]

12. Final simplification 23.5%

    \[\leadsto \frac{\log base \cdot \log \left(\mathsf{fma}\left(re, \frac{re \cdot 0.5}{im}, im\right)\right)}{\log base \cdot \log base} \]
13. Add Preprocessing

Alternative 4: 76.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

re_m = abs(re)
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
real(8) function code(re_m, im, base)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = log(im) / log(base)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

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

   2. log-lowering-log.f64 N/A

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

   3. log-lowering-log.f64 24.3

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

5. Simplified 24.3%

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

Alternative 5: 3.1% accurate, 562.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ [re_m, im, base] = \mathsf{sort}([re_m, im, base])\\ \\ 0 \end{array} \]

FPCore

re_m = (fabs.f64 re)
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
(FPCore (re_m im base) :precision binary64 0.0)

C

re_m = fabs(re);
assert(re_m < im && im < base);
double code(double re_m, double im, double base) {
	return 0.0;
}

Fortran

re_m = abs(re)
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
real(8) function code(re_m, im, base)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = 0.0d0
end function

Java

re_m = Math.abs(re);
assert re_m < im && im < base;
public static double code(double re_m, double im, double base) {
	return 0.0;
}

Python

re_m = math.fabs(re)
[re_m, im, base] = sort([re_m, im, base])
def code(re_m, im, base):
	return 0.0

Julia

re_m = abs(re)
re_m, im, base = sort([re_m, im, base])
function code(re_m, im, base)
	return 0.0
end

MATLAB

re_m = abs(re);
re_m, im, base = num2cell(sort([re_m, im, base])){:}
function tmp = code(re_m, im, base)
	tmp = 0.0;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m, im, and base should be sorted in increasing order before calling this function.
code[re$95$m_, im_, base_] := 0.0

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

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

   2. log-lowering-log.f64 N/A

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

   3. log-lowering-log.f64 24.3

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

5. Simplified 24.3%

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

7. Applied egg-rr 3.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))