Average Error: 28.3 → 6.0
Time: 3.3s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c \cdot a}}, \left(a \cdot a\right) \cdot \left(\frac{-0.5625 \cdot {c}^{3}}{{b}^{5}} + a \cdot \frac{-1.0546875 \cdot {c}^{4}}{{b}^{7}}\right)\right)\right) \]
(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
   (/ c (/ (pow b 3.0) (* c a)))
   (*
    (* a a)
    (+
     (/ (* -0.5625 (pow c 3.0)) (pow b 5.0))
     (* a (/ (* -1.0546875 (pow c 4.0)) (pow b 7.0))))))))
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, (c / (pow(b, 3.0) / (c * a))), ((a * a) * (((-0.5625 * pow(c, 3.0)) / pow(b, 5.0)) + (a * ((-1.0546875 * pow(c, 4.0)) / pow(b, 7.0)))))));
}

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

Error

Derivation

  1. Initial program 28.3

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

  2. Simplified28.3

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

  3. Taylor expanded in c around 0 6.3

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

  4. Simplified6.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1.125, \frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{{c}^{4}}{b} \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-1.5, \frac{c}{b} \cdot a, \frac{-1.6875}{{b}^{5}} \cdot \left({c}^{3} \cdot {a}^{3}\right)\right)\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

  5. Taylor expanded in c around 0 6.0

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

  6. Simplified6.0

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

  7. Final simplification6.0

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

Reproduce

herbie shell --seed 2022212 
(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)))