Average Error: 44.3 → 2.9
Time: 6.5s
Precision: binary64
\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \left(a \cdot a\right) \cdot \left(\frac{{c}^{4} \cdot -1.0546875}{\frac{{b}^{7}}{a}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (fma
   -0.375
   (* a (/ (* c c) (pow b 3.0)))
   (*
    (* a a)
    (+
     (/ (* (pow c 4.0) -1.0546875) (/ (pow b 7.0) a))
     (* -0.5625 (/ (pow c 3.0) (pow b 5.0))))))))
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(-0.5, (c / b), fma(-0.375, (a * ((c * c) / pow(b, 3.0))), ((a * a) * (((pow(c, 4.0) * -1.0546875) / (pow(b, 7.0) / a)) + (-0.5625 * (pow(c, 3.0) / pow(b, 5.0)))))));
}

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(-0.5, Float64(c / b), fma(-0.375, Float64(a * Float64(Float64(c * c) / (b ^ 3.0))), Float64(Float64(a * a) * Float64(Float64(Float64((c ^ 4.0) * -1.0546875) / Float64((b ^ 7.0) / a)) + Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0)))))))
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[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * -1.0546875), $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $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(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \left(a \cdot a\right) \cdot \left(\frac{{c}^{4} \cdot -1.0546875}{\frac{{b}^{7}}{a}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)

Error

Derivation

  1. Initial program 44.3

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

  2. Simplified44.3

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

  3. Taylor expanded in a around 0 3.3

    \[\leadsto \color{blue}{\left(-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{a}^{4} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

  4. Taylor expanded in c around 0 2.9

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

  5. Simplified2.9

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

  6. Final simplification2.9

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

Reproduce

herbie shell --seed 2022210 
(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)))