Average Error: 32.0 → 0.3
Time: 7.3s
Precision: binary64
Cost: 25856
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\right)\right) \]
(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
 (log1p (expm1 (/ (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 log1p(expm1((atan2(im, re) / log(base))));
}

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.log1p(Math.expm1((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.log1p(math.expm1((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 log1p(expm1(Float64(atan(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[Log[1 + N[(Exp[N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]] - 1), $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}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
    Proof
    (/.f64 (atan2.f64 im re) (log.f64 base)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (atan2.f64 im re) (log.f64 base)) 1)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (atan2.f64 im re) (log.f64 base)) (Rewrite<= *-inverses_binary64 (/.f64 (log.f64 base) (log.f64 base)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (atan2.f64 im re) (log.f64 base)) (*.f64 (log.f64 base) (log.f64 base)))): 60 points increase in error, 26 points decrease in error
    (/.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (*.f64 (atan2.f64 im re) (log.f64 base)) 0)) (*.f64 (log.f64 base) (log.f64 base))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (*.f64 (atan2.f64 im re) (log.f64 base)) 0) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (log.f64 base) (log.f64 base)) 0))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (*.f64 (atan2.f64 im re) (log.f64 base)) 0) (+.f64 (*.f64 (log.f64 base) (log.f64 base)) (Rewrite<= metadata-eval (*.f64 0 0)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (*.f64 (atan2.f64 im re) (log.f64 base)) (Rewrite<= mul0-rgt_binary64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) 0))) (+.f64 (*.f64 (log.f64 base) (log.f64 base)) (*.f64 0 0))): 119 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\right)\right)} \]

  4. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\right)\right) \]

Alternatives

Alternative 1
Error0.3
Cost13056
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]
Alternative 2
Error55.2
Cost6528
\[\tan^{-1}_* \frac{im}{re} \]

Error

Reproduce

herbie shell --seed 2022329 
(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))))