math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.1%
Time: 6.8s
Alternatives: 2
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

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

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

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

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

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

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]

\begin{array}{l}

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

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 51.4% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

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

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

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

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

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

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]

\begin{array}{l}

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

Alternative 1: 99.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]
(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))
double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

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

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

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

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

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

\begin{array}{l}

\\
\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}
\end{array}
Derivation

  1. Initial program 49.2%

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

    1. +-commutative49.2%

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

    2. +-commutative49.2%

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

    3. sqr-neg49.2%

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

    4. sqr-neg49.2%

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

    5. sqr-neg49.2%

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

    6. sqr-neg49.2%

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

    7. hypot-def99.1%

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

  3. Simplified99.1%

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

  4. Final simplification99.1%

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

Alternative 2: 26.7% accurate, 1.5× speedup?

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]
(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))
double code(double re, double im) {
	return log(im) / log(10.0);
}

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

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

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

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

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

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

\begin{array}{l}

\\
\frac{\log im}{\log 10}
\end{array}
Derivation

  1. Initial program 49.2%

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

    1. +-commutative49.2%

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

    2. +-commutative49.2%

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

    3. sqr-neg49.2%

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

    4. sqr-neg49.2%

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

    5. sqr-neg49.2%

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

    6. sqr-neg49.2%

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

    7. hypot-def99.1%

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

  3. Simplified99.1%

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

  4. Taylor expanded in re around 0 27.9%

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

  5. Final simplification27.9%

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

Reproduce

?
herbie shell --seed 2023263 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))