Average Error: 52.7 → 1.5
Time: 4.8s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.25 \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a} \cdot \frac{{c}^{4}}{b}\right)\right) - \mathsf{fma}\left(c \cdot \frac{c}{{b}^{3}}, a, \frac{c}{b}\right) \]
(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))
   (* -0.25 (* (/ (* (/ (pow a 4.0) (pow b 6.0)) 20.0) a) (/ (pow c 4.0) b))))
  (fma (* c (/ c (pow b 3.0))) 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 fma(-2.0, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), (-0.25 * ((((pow(a, 4.0) / pow(b, 6.0)) * 20.0) / a) * (pow(c, 4.0) / b)))) - fma((c * (c / pow(b, 3.0))), 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(fma(-2.0, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), Float64(-0.25 * Float64(Float64(Float64(Float64((a ^ 4.0) / (b ^ 6.0)) * 20.0) / a) * Float64((c ^ 4.0) / b)))) - fma(Float64(c * Float64(c / (b ^ 3.0))), 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[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision] / a), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + 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}, -0.25 \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a} \cdot \frac{{c}^{4}}{b}\right)\right) - \mathsf{fma}\left(c \cdot \frac{c}{{b}^{3}}, a, \frac{c}{b}\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  2. Taylor expanded in c around 0 1.5

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{c}^{4} \cdot \left({\left(-2 \cdot \frac{{a}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

  3. Simplified1.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.25 \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a} \cdot \frac{{c}^{4}}{b}\right)\right) - \mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot c, a, \frac{c}{b}\right)} \]

  4. Final simplification1.5

    \[\leadsto \mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.25 \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a} \cdot \frac{{c}^{4}}{b}\right)\right) - \mathsf{fma}\left(c \cdot \frac{c}{{b}^{3}}, a, \frac{c}{b}\right) \]

Reproduce

herbie shell --seed 2022165 
(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)))