Average Error: 31.1 → 0.3
Time: 9.4s
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}

Error

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Your Program's Arguments

Results

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Derivation

  1. Initial program 31.1

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

  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)))): 74 points increase in error, 25 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))): 131 points increase in error, 0 points decrease in error

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2022291 
(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))))