Average Error: 28.4 → 5.9
Time: 4.5s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{3} \cdot \left(\left(a \cdot a\right) \cdot {b}^{-5}\right)\right)\right) \cdot -0.5625 + \left(\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -1.0546875 + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot -0.375 + \frac{c}{b} \cdot -0.5\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (+
  (* (expm1 (log1p (* (pow c 3.0) (* (* a a) (pow b -5.0))))) -0.5625)
  (+
   (* (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)) -1.0546875)
   (+ (* (/ (* a (pow c 2.0)) (pow b 3.0)) -0.375) (* (/ c b) -0.5)))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return (expm1(log1p((pow(c, 3.0) * ((a * a) * pow(b, -5.0))))) * -0.5625) + ((((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)) * -1.0546875) + ((((a * pow(c, 2.0)) / pow(b, 3.0)) * -0.375) + ((c / b) * -0.5)));
}

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}
↓
public static double code(double a, double b, double c) {
	return (Math.expm1(Math.log1p((Math.pow(c, 3.0) * ((a * a) * Math.pow(b, -5.0))))) * -0.5625) + ((((Math.pow(c, 4.0) * Math.pow(a, 3.0)) / Math.pow(b, 7.0)) * -1.0546875) + ((((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)) * -0.375) + ((c / b) * -0.5)));
}

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)
↓
def code(a, b, c):
	return (math.expm1(math.log1p((math.pow(c, 3.0) * ((a * a) * math.pow(b, -5.0))))) * -0.5625) + ((((math.pow(c, 4.0) * math.pow(a, 3.0)) / math.pow(b, 7.0)) * -1.0546875) + ((((a * math.pow(c, 2.0)) / math.pow(b, 3.0)) * -0.375) + ((c / b) * -0.5)))

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end
↓
function code(a, b, c)
	return Float64(Float64(expm1(log1p(Float64((c ^ 3.0) * Float64(Float64(a * a) * (b ^ -5.0))))) * -0.5625) + Float64(Float64(Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0)) * -1.0546875) + Float64(Float64(Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)) * -0.375) + Float64(Float64(c / b) * -0.5))))
end

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := N[(N[(N[(Exp[N[Log[1 + N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * -0.5625), $MachinePrecision] + N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision] + N[(N[(N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{3} \cdot \left(\left(a \cdot a\right) \cdot {b}^{-5}\right)\right)\right) \cdot -0.5625 + \left(\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -1.0546875 + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot -0.375 + \frac{c}{b} \cdot -0.5\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.4

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

  2. Simplified28.4

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

  3. Taylor expanded in b around inf 5.9

    \[\leadsto \color{blue}{-\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\right)} \]

  4. Applied egg-rr5.9

    \[\leadsto -\left(0.5625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{3} \cdot \left(\left(a \cdot a\right) \cdot {b}^{-5}\right)\right)\right)} + \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\right) \]

  5. Final simplification5.9

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left({c}^{3} \cdot \left(\left(a \cdot a\right) \cdot {b}^{-5}\right)\right)\right) \cdot -0.5625 + \left(\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -1.0546875 + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot -0.375 + \frac{c}{b} \cdot -0.5\right)\right) \]

Reproduce

herbie shell --seed 2022165 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))