math.log10 on complex, real part

Percentage Accurate: 51.0% → 99.1%

Time: 5.4s

Alternatives: 3

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: 51.0% 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: 99.1% 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 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 2: 25.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{im + re} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im}, 2, -1\right), re, im\right)}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/
  (log (* (sqrt (+ im re)) (sqrt (fma (fma (/ re im) 2.0 -1.0) re im))))
  (log 10.0)))

C

double code(double re, double im) {
	return log((sqrt((im + re)) * sqrt(fma(fma((re / im), 2.0, -1.0), re, im)))) / log(10.0);
}

Julia

function code(re, im)
	return Float64(log(Float64(sqrt(Float64(im + re)) * sqrt(fma(fma(Float64(re / im), 2.0, -1.0), re, im)))) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[Log[N[(N[Sqrt[N[(im + re), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * 2.0 + -1.0), $MachinePrecision] * re + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. flip-+ N/A

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

   4. difference-of-squares N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-squares N/A

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

   9. times-frac N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. flip-+ N/A

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

   13. sqrt-prod N/A

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

   14. pow1/2 N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 29.3%

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

5. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 25.1

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

7. Applied rewrites 25.1%

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

8. Final simplification 25.1%

   \[\leadsto \frac{\log \left(\sqrt{im + re} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im}, 2, -1\right), re, im\right)}\right)}{\log 10} \]
9. Add Preprocessing

Alternative 3: 26.9% 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 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 26.8

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

5. Applied rewrites 26.8%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))