math.log/2 on complex, imaginary part

?

Percentage Accurate: 50.3% → 99.5%
Time: 9.3s
Precision: binary64
Cost: 13056

?

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
↓
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]
(FPCore (re im base)
 :precision binary64
 (/
  (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))
↓
(FPCore (re im base) :precision binary64 (/ (atan2 im re) (log base)))
double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}
↓
double code(double re, double im, double base) {
	return atan2(im, re) / log(base);
}

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function
↓
real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = atan2(im, re) / log(base)
end function

public static double code(double re, double im, double base) {
	return ((Math.atan2(im, re) * Math.log(base)) - (Math.log(Math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}
↓
public static double code(double re, double im, double base) {
	return Math.atan2(im, re) / Math.log(base);
}

def code(re, im, base):
	return ((math.atan2(im, re) * math.log(base)) - (math.log(math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))
↓
def code(re, im, base):
	return math.atan2(im, re) / math.log(base)

function code(re, im, base)
	return Float64(Float64(Float64(atan(im, re) * log(base)) - Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end
↓
function code(re, im, base)
	return Float64(atan(im, re) / log(base))
end

function tmp = code(re, im, base)
	tmp = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end
↓
function tmp = code(re, im, base)
	tmp = atan2(im, re) / log(base);
end

code[re_, im_, base_] := N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_, base_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
\frac{\tan^{-1}_* \frac{im}{re}}{\log base}

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.

Herbie found 2 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

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.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 54.6%

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

  2. Simplified99.5%

    \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
    Step-by-step derivation

    [Start]54.6%

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

    mul0-rgt [=>]99.3%

    \[ \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

    --rgt-identity [=>]99.3%

    \[ \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

    metadata-eval [=>]99.3%

    \[ \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

    +-rgt-identity [=>]99.3%

    \[ \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]

    times-frac [=>]99.5%

    \[ \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base} \cdot \frac{\log base}{\log base}} \]

    *-inverses [=>]99.5%

    \[ \frac{\tan^{-1}_* \frac{im}{re}}{\log base} \cdot \color{blue}{1} \]

    *-rgt-identity [=>]99.5%

    \[ \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]

  3. Final simplification99.5%

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]

Alternatives

Alternative 1
Accuracy99.5%
Cost13056
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]
Alternative 2
Accuracy6.9%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023178 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))