Average Error: 52.9 → 1.5
Time: 4.1s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -5 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot {c}^{4}\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{a}}, c, \frac{c}{b}\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (fma
   -2.0
   (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
   (* -5.0 (* (/ (pow a 3.0) (pow b 7.0)) (pow c 4.0))))
  (fma (/ c (/ (pow b 3.0) a)) c (/ 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(-2.0, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), (-5.0 * ((pow(a, 3.0) / pow(b, 7.0)) * pow(c, 4.0)))) - fma((c / (pow(b, 3.0) / a)), c, (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(-2.0, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), Float64(-5.0 * Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * (c ^ 4.0)))) - fma(Float64(c / Float64((b ^ 3.0) / a)), c, 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[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * c + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 52.9

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

  2. Simplified52.9

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

  3. Taylor expanded in b around inf 1.8

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

  4. Simplified1.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{{b}^{7}}, \mathsf{fma}\left(-4, \frac{{a}^{3}}{{b}^{5}} \cdot {c}^{3}, -2 \cdot \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a \cdot a, \frac{c}{b} \cdot a\right)\right)\right)} \cdot \frac{0.5}{a} \]

  5. Taylor expanded in c around 0 1.5

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

  6. Simplified1.5

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

  7. Final simplification1.5

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

Reproduce

herbie shell --seed 2022204 
(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)))