Average Error: 52.2 → 1.6
Time: 3.8s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-1.0546875, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \frac{a \cdot \left(c \cdot c\right)}{\frac{{b}^{3}}{-0.375}}\right)\right)\right) \]
(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
   -1.0546875
   (* (/ (pow c 4.0) (pow b 7.0)) (pow a 3.0))
   (fma
    -0.5625
    (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
    (/ (* a (* c c)) (/ (pow b 3.0) -0.375))))))
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(-1.0546875, ((pow(c, 4.0) / pow(b, 7.0)) * pow(a, 3.0)), fma(-0.5625, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), ((a * (c * c)) / (pow(b, 3.0) / -0.375)))));
}

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

Error

Derivation

  1. Initial program 52.2

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

  2. Taylor expanded in a around 0 2.0

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

  3. Taylor expanded in c around 0 1.6

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

  4. Simplified1.6

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

  5. Final simplification1.6

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

Reproduce

herbie shell --seed 2022212 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))