math.log10 on complex, real part

Percentage Accurate: 52.1% → 99.0%

Time: 6.6s

Alternatives: 5

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\log 10}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}\right)}^{-1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (pow (/ (log 10.0) (log (hypot im re))) -1.0))

C

double code(double re, double im) {
	return pow((log(10.0) / log(hypot(im, re))), -1.0);
}

Java

public static double code(double re, double im) {
	return Math.pow((Math.log(10.0) / Math.log(Math.hypot(im, re))), -1.0);
}

Python

def code(re, im):
	return math.pow((math.log(10.0) / math.log(math.hypot(im, re))), -1.0)

Julia

function code(re, im)
	return Float64(log(10.0) / log(hypot(im, re))) ^ -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = (log(10.0) / log(hypot(im, re))) ^ -1.0;
end

Wolfram

code[re_, im_] := N[Power[N[(N[Log[10.0], $MachinePrecision] / N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   4. lower-/.f64 54.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{\color{blue}{im \cdot im} + re \cdot re}\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}} \]

   10. lower-hypot.f64 99.2

       \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}} \]

5. Final simplification 99.2%

   \[\leadsto {\left(\frac{\log 10}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}\right)}^{-1} \]
6. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
5. Add Preprocessing

Alternative 3: 27.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\log 10}{\log im}\right)}^{-1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (pow (/ (log 10.0) (log im)) -1.0))

C

double code(double re, double im) {
	return pow((log(10.0) / log(im)), -1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (log(10.0d0) / log(im)) ** (-1.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.pow((Math.log(10.0) / Math.log(im)), -1.0);
}

Python

def code(re, im):
	return math.pow((math.log(10.0) / math.log(im)), -1.0)

Julia

function code(re, im)
	return Float64(log(10.0) / log(im)) ^ -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = (log(10.0) / log(im)) ^ -1.0;
end

Wolfram

code[re_, im_] := N[Power[N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   4. lower-/.f64 54.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{\color{blue}{im \cdot im} + re \cdot re}\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}} \]

   10. lower-hypot.f64 99.2

       \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}} \]

5. Taylor expanded in re around 0

   \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log im}}} \]
6. Step-by-step derivation

   1. lower-log.f64 29.4

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log im}}} \]

7. Applied rewrites 29.4%

   \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log im}}} \]

8. Final simplification 29.4%

   \[\leadsto {\left(\frac{\log 10}{\log im}\right)}^{-1} \]
9. Add Preprocessing

Alternative 4: 25.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right) \cdot 0.5}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* (fma (/ re im) (/ re im) (* (- (log im)) -2.0)) 0.5) (log 10.0)))

C

double code(double re, double im) {
	return (fma((re / im), (re / im), (-log(im) * -2.0)) * 0.5) / log(10.0);
}

Julia

function code(re, im)
	return Float64(Float64(fma(Float64(re / im), Float64(re / im), Float64(Float64(-log(im)) * -2.0)) * 0.5) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[((-N[Log[im], $MachinePrecision]) * -2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   3. pow1/2 N/A

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10} \]

   4. pow-to-exp N/A

      \[\leadsto \frac{\log \color{blue}{\left(e^{\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{2}}\right)}}{\log 10} \]

   5. rem-log-exp N/A

      \[\leadsto \frac{\color{blue}{\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{2}}}{\log 10} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{2}}}{\log 10} \]

   7. lower-log.f64 54.2

      \[\leadsto \frac{\color{blue}{\log \left(re \cdot re + im \cdot im\right)} \cdot 0.5}{\log 10} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(re \cdot re + im \cdot im\right)} \cdot \frac{1}{2}}{\log 10} \]

   9. +-commutative N/A

      \[\leadsto \frac{\log \color{blue}{\left(im \cdot im + re \cdot re\right)} \cdot \frac{1}{2}}{\log 10} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\color{blue}{im \cdot im} + re \cdot re\right) \cdot \frac{1}{2}}{\log 10} \]

   11. lower-fma.f64 54.2

       \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)} \cdot 0.5}{\log 10} \]

4. Applied rewrites 54.2%

   \[\leadsto \frac{\color{blue}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \cdot 0.5}}{\log 10} \]

5. Taylor expanded in im around inf

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

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \frac{1}{2}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   3. unpow2 N/A

      \[\leadsto \frac{\left(\frac{re \cdot re}{\color{blue}{im \cdot im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   4. times-frac N/A

      \[\leadsto \frac{\left(\color{blue}{\frac{re}{im} \cdot \frac{re}{im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \frac{1}{2}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\log \left(\frac{1}{im}\right) \cdot -2}\right) \cdot \frac{1}{2}}{\log 10} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\log \left(\frac{1}{im}\right) \cdot -2}\right) \cdot \frac{1}{2}}{\log 10} \]

   10. log-rec N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)} \cdot -2\right) \cdot \frac{1}{2}}{\log 10} \]

   11. lower-neg.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-\log im\right)} \cdot -2\right) \cdot \frac{1}{2}}{\log 10} \]

   12. lower-log.f64 27.1

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\color{blue}{\log im}\right) \cdot -2\right) \cdot 0.5}{\log 10} \]

7. Applied rewrites 27.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)} \cdot 0.5}{\log 10} \]
8. Add Preprocessing

Alternative 5: 27.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

double code(double re, double im) {
	return log(im) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(im) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(im) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(im) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
4. Step-by-step derivation

   1. lower-log.f64 29.4

      \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]

5. Applied rewrites 29.4%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]

6. Final simplification 29.4%

   \[\leadsto \frac{\log im}{\log 10} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))