Average Error: 43.7 → 2.9
Time: 3.8s
Precision: binary64
\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\mathsf{fma}\left(\frac{{c}^{4}}{a}, \frac{-5 \cdot {a}^{4}}{{b}^{7}}, a \cdot \left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a}} - \frac{c \cdot c}{{b}^{3}}\right)\right) - \frac{c}{b} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (fma
   (/ (pow c 4.0) a)
   (/ (* -5.0 (pow a 4.0)) (pow b 7.0))
   (*
    a
    (- (/ (* -2.0 (pow c 3.0)) (/ (pow b 5.0) a)) (/ (* c c) (pow b 3.0)))))
  (/ 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 fma((pow(c, 4.0) / a), ((-5.0 * pow(a, 4.0)) / pow(b, 7.0)), (a * (((-2.0 * pow(c, 3.0)) / (pow(b, 5.0) / a)) - ((c * c) / pow(b, 3.0))))) - (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(fma(Float64((c ^ 4.0) / a), Float64(Float64(-5.0 * (a ^ 4.0)) / (b ^ 7.0)), Float64(a * Float64(Float64(Float64(-2.0 * (c ^ 3.0)) / Float64((b ^ 5.0) / a)) - Float64(Float64(c * c) / (b ^ 3.0))))) - 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[(N[(N[Power[c, 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(N[(-5.0 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

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

  3. Taylor expanded in b around inf 3.2

    \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}} + \left(-2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + -2 \cdot \frac{c \cdot a}{b}\right)\right)\right)} \cdot \frac{0.5}{a} \]

  4. Taylor expanded in c around 0 2.9

    \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{c}^{4} \cdot \left(16 \cdot {a}^{4} + 4 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

  5. Simplified2.9

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

  6. Final simplification2.9

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

Reproduce

herbie shell --seed 2022213 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))