math.log/2 on complex, real part

?

Percentage Accurate: 50.5% → 99.4%
Time: 17.2s
Precision: binary64
Cost: 19456

?

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

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

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

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

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

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

\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}
↓
\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}

Local Percentage Accuracy?

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.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 55.9%

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

  2. Simplified99.5%

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]
    Step-by-step derivation

    [Start]55.9

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

    mul0-rgt [=>]55.9

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

    +-rgt-identity [=>]55.9

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

    metadata-eval [=>]55.9

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

    +-rgt-identity [=>]55.9

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

    times-frac [=>]56.0

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

    *-inverses [=>]56.0

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

    *-rgt-identity [=>]56.0

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

    hypot-def [=>]99.5

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

  3. Final simplification99.5%

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

Alternatives

Alternative 1
Accuracy43.5%
Cost13645
\[\begin{array}{l} t_0 := \log \left(\frac{-1}{re}\right)\\ \mathbf{if}\;re \leq -9.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-t_0}{\log base}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-59} \lor \neg \left(re \leq -1.45 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\log base}\\ \end{array} \]
Alternative 2
Accuracy43.5%
Cost13581
\[\begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{-11} \lor \neg \left(re \leq -1.9 \cdot 10^{-59}\right) \land re \leq -1.4 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]
Alternative 3
Accuracy32.7%
Cost13124
\[\begin{array}{l} \mathbf{if}\;im \leq 3.3 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]
Alternative 4
Accuracy13.9%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.1111111111111111\\ \end{array} \]
Alternative 5
Accuracy14.1%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666\\ \end{array} \]
Alternative 6
Accuracy14.3%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]
Alternative 7
Accuracy14.5%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333\\ \end{array} \]
Alternative 8
Accuracy14.8%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1.1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 9
Accuracy15.0%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666\\ \end{array} \]
Alternative 10
Accuracy15.1%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 11
Accuracy15.1%
Cost196
\[\begin{array}{l} \mathbf{if}\;base \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 12
Accuracy9.3%
Cost64
\[-6 \]
Alternative 13
Accuracy9.6%
Cost64
\[-4 \]
Alternative 14
Accuracy9.8%
Cost64
\[-3 \]
Alternative 15
Accuracy10.1%
Cost64
\[-2 \]
Alternative 16
Accuracy10.4%
Cost64
\[-1 \]

Error

Reproduce?

herbie shell --seed 2023159 
(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))))