Average Error: 28.6 → 5.9
Time: 4.5s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, 0.75, \frac{c}{b}\right), -0.5, -1.0546875 \cdot \mathsf{fma}\left(\frac{a \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}}, 0.5333333333333333, \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (fma
  (fma (* (* c c) (/ a (pow b 3.0))) 0.75 (/ c b))
  -0.5
  (*
   -1.0546875
   (fma
    (/ (* a (* a (pow c 3.0))) (pow b 5.0))
    0.5333333333333333
    (/ (pow c 4.0) (/ (pow b 7.0) (pow a 3.0)))))))
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(fma(((c * c) * (a / pow(b, 3.0))), 0.75, (c / b)), -0.5, (-1.0546875 * fma(((a * (a * pow(c, 3.0))) / pow(b, 5.0)), 0.5333333333333333, (pow(c, 4.0) / (pow(b, 7.0) / pow(a, 3.0))))));
}

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 fma(fma(Float64(Float64(c * c) * Float64(a / (b ^ 3.0))), 0.75, Float64(c / b)), -0.5, Float64(-1.0546875 * fma(Float64(Float64(a * Float64(a * (c ^ 3.0))) / (b ^ 5.0)), 0.5333333333333333, Float64((c ^ 4.0) / Float64((b ^ 7.0) / (a ^ 3.0))))))
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[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.75 + N[(c / b), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(-1.0546875 * N[(N[(N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * 0.5333333333333333 + N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\mathsf{fma}\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, 0.75, \frac{c}{b}\right), -0.5, -1.0546875 \cdot \mathsf{fma}\left(\frac{a \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}}, 0.5333333333333333, \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.6

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

  2. Simplified28.6

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

  3. Taylor expanded in b around inf 6.1

    \[\leadsto \frac{\color{blue}{-\left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} + 1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)}}{3 \cdot a} \]

  4. Taylor expanded in c around 0 5.9

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

  5. Simplified5.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, 0.75, \frac{c}{b}\right), -0.5, -1.0546875 \cdot \mathsf{fma}\left(\frac{a \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}}, 0.5333333333333333, \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}\right)\right)} \]

  6. Final simplification5.9

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, 0.75, \frac{c}{b}\right), -0.5, -1.0546875 \cdot \mathsf{fma}\left(\frac{a \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}}, 0.5333333333333333, \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}\right)\right) \]

Reproduce

herbie shell --seed 2022155 
(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)))