Average Error: 43.8 → 3.0
Time: 5.0s
Precision: binary64
\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), -0.5625, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right) + \left(\frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}\right) \cdot -1.0546875 \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (+
  (fma
   (* (/ (pow c 3.0) (pow b 5.0)) (* a a))
   -0.5625
   (fma -0.5 (/ c b) (/ (* (* c c) -0.375) (/ (pow b 3.0) a))))
  (* (* (/ (pow c 4.0) (pow b 7.0)) (pow a 3.0)) -1.0546875)))
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 fma(((pow(c, 3.0) / pow(b, 5.0)) * (a * a)), -0.5625, fma(-0.5, (c / b), (((c * c) * -0.375) / (pow(b, 3.0) / a)))) + (((pow(c, 4.0) / pow(b, 7.0)) * pow(a, 3.0)) * -1.0546875);
}

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(fma(Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)), -0.5625, fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * -0.375) / Float64((b ^ 3.0) / a)))) + Float64(Float64(Float64((c ^ 4.0) / (b ^ 7.0)) * (a ^ 3.0)) * -1.0546875))
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[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  3. Taylor expanded in b around inf 3.0

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

  4. Simplified3.0

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

  5. Final simplification3.0

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

Reproduce

herbie shell --seed 2022150 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))