Average Error: 53.0 → 1.4
Time: 8.9s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[-2 \cdot \frac{a}{\frac{{b}^{5}}{a \cdot {c}^{3}}} - \mathsf{fma}\left(5, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (* -2.0 (/ a (/ (pow b 5.0) (* a (pow c 3.0)))))
  (fma
   5.0
   (* (/ (pow c 4.0) (pow b 7.0)) (pow a 3.0))
   (fma (/ c (/ (pow b 3.0) c)) a (/ c b)))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	return (-2.0 * (a / (pow(b, 5.0) / (a * pow(c, 3.0))))) - fma(5.0, ((pow(c, 4.0) / pow(b, 7.0)) * pow(a, 3.0)), fma((c / (pow(b, 3.0) / c)), a, (c / b)));
}

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end
↓
function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(a / Float64((b ^ 5.0) / Float64(a * (c ^ 3.0))))) - fma(5.0, Float64(Float64((c ^ 4.0) / (b ^ 7.0)) * (a ^ 3.0)), fma(Float64(c / Float64((b ^ 3.0) / c)), a, Float64(c / b))))
end

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := N[(N[(-2.0 * N[(a / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(5.0 * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-2 \cdot \frac{a}{\frac{{b}^{5}}{a \cdot {c}^{3}}} - \mathsf{fma}\left(5, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 53.0

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

  2. Simplified53.0

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}} \]

  3. Taylor expanded in b around inf 1.4

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

  4. Simplified1.4

    \[\leadsto \color{blue}{-2 \cdot \frac{a}{\frac{{b}^{5}}{a \cdot {c}^{3}}} - \mathsf{fma}\left(5, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right)} \]

  5. Final simplification1.4

    \[\leadsto -2 \cdot \frac{a}{\frac{{b}^{5}}{a \cdot {c}^{3}}} - \mathsf{fma}\left(5, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

Reproduce

herbie shell --seed 2022155 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))