Average Error: 28.7 → 5.9
Time: 4.5s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\left(a \cdot \frac{a}{\frac{{b}^{5}}{{c}^{3}}}\right) \cdot -0.5625 - \mathsf{fma}\left(0.5, \frac{c}{b}, \mathsf{fma}\left(c \cdot c, \frac{0.375}{\frac{{b}^{3}}{a}}, {a}^{3} \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{1.0546875}}\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (* (* a (/ a (/ (pow b 5.0) (pow c 3.0)))) -0.5625)
  (fma
   0.5
   (/ c b)
   (fma
    (* c c)
    (/ 0.375 (/ (pow b 3.0) a))
    (* (pow a 3.0) (/ (pow c 4.0) (/ (pow b 7.0) 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 ((a * (a / (pow(b, 5.0) / pow(c, 3.0)))) * -0.5625) - fma(0.5, (c / b), fma((c * c), (0.375 / (pow(b, 3.0) / a)), (pow(a, 3.0) * (pow(c, 4.0) / (pow(b, 7.0) / 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(Float64(Float64(a * Float64(a / Float64((b ^ 5.0) / (c ^ 3.0)))) * -0.5625) - fma(0.5, Float64(c / b), fma(Float64(c * c), Float64(0.375 / Float64((b ^ 3.0) / a)), Float64((a ^ 3.0) * Float64((c ^ 4.0) / Float64((b ^ 7.0) / 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[(a * N[(a / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] - N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(0.375 / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.7

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

  2. Simplified28.6

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

  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. Simplified5.9

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

  5. Final simplification5.9

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

Reproduce

herbie shell --seed 2022156 
(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)))