Average Error: 52.4 → 1.6
Time: 5.1s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.5625, \mathsf{fma}\left(-1.0546875 \cdot \frac{{c}^{4}}{{b}^{7}}, {a}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a}{\frac{{b}^{3}}{c \cdot c}} \cdot -0.375\right)\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (fma
  (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
  -0.5625
  (fma
   (* -1.0546875 (/ (pow c 4.0) (pow b 7.0)))
   (pow a 3.0)
   (fma -0.5 (/ c b) (* (/ a (/ (pow b 3.0) (* c c))) -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((((a * a) / pow(b, 5.0)) * pow(c, 3.0)), -0.5625, fma((-1.0546875 * (pow(c, 4.0) / pow(b, 7.0))), pow(a, 3.0), fma(-0.5, (c / b), ((a / (pow(b, 3.0) / (c * c))) * -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(Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), -0.5625, fma(Float64(-1.0546875 * Float64((c ^ 4.0) / (b ^ 7.0))), (a ^ 3.0), fma(-0.5, Float64(c / b), Float64(Float64(a / Float64((b ^ 3.0) / Float64(c * c))) * -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[(N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(-1.0546875 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $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(\frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.5625, \mathsf{fma}\left(-1.0546875 \cdot \frac{{c}^{4}}{{b}^{7}}, {a}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a}{\frac{{b}^{3}}{c \cdot c}} \cdot -0.375\right)\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.4

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

  2. Simplified52.4

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

  3. Taylor expanded in b around inf 2.0

    \[\leadsto \color{blue}{\left(-\left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} + 1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

  4. Simplified1.9

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-1.5, \frac{c}{b} \cdot a, \frac{-1.125}{\frac{{b}^{3}}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}\right) - \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{{b}^{5}} \cdot {a}^{3}, 3.1640625 \cdot \left(\frac{{c}^{4}}{{b}^{7}} \cdot {a}^{4}\right)\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

  5. Taylor expanded in c around 0 1.6

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

  6. Simplified1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2022151 
(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)))