Average Error: 31.7 → 0.4
Time: 11.2s
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}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]
    Proof
    (/.f64 (log.f64 (hypot.f64 re im)) (log.f64 base)): 0 points increase in error, 0 points decrease in error
    (/.f64 (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))))) (log.f64 base)): 130 points increase in error, 0 points decrease in error
    (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) 1)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (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 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (*.f64 (log.f64 base) (log.f64 base)))): 32 points increase in error, 12 points decrease in error
    (/.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) 0)) (*.f64 (log.f64 base) (log.f64 base))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (Rewrite<= mul0-rgt_binary64 (*.f64 (atan2.f64 im re) 0))) (*.f64 (log.f64 base) (log.f64 base))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (*.f64 (atan2.f64 im re) 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 (log.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) (log.f64 base)) (*.f64 (atan2.f64 im re) 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

  3. Final simplification0.4

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

Alternatives

Alternative 1
Error35.7
Cost13316
\[\begin{array}{l} \mathbf{if}\;im \leq 2.962683588379831 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \end{array} \]
Alternative 2
Error46.7
Cost12992
\[\frac{\log im}{\log base} \]

Error

Reproduce

herbie shell --seed 2022291 
(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))))