Average Error: 52.4 → 1.6
Time: 18.6s
Precision: binary64
Cost: 34944
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\mathsf{fma}\left(-2, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right) \]
(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
   (* (* (* c c) (* c (pow b -5.0))) (* a a))
   (*
    (/ (* (pow a 3.0) -0.25) b)
    (* (* (* c c) (* (* c c) (pow b -6.0))) 20.0)))
  (fma (* (/ c (* b b)) (/ c b)) 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, (((c * c) * (c * pow(b, -5.0))) * (a * a)), (((pow(a, 3.0) * -0.25) / b) * (((c * c) * ((c * c) * pow(b, -6.0))) * 20.0))) - fma(((c / (b * b)) * (c / b)), 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(c * c) * Float64(c * (b ^ -5.0))) * Float64(a * a)), Float64(Float64(Float64((a ^ 3.0) * -0.25) / b) * Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 20.0))) - fma(Float64(Float64(c / Float64(b * b)) * Float64(c / b)), 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[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * -0.25), $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / b), $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, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right)

Error

Derivation

  1. Initial program 52.4

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

  2. Taylor expanded in a around 0 1.6

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

  3. Simplified1.6

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

  4. Applied egg-rr1.6

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

  5. Applied egg-rr1.6

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

  6. Applied egg-rr1.6

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

  7. Final simplification1.6

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

Alternatives

Alternative 1
Error2.1
Cost20736
\[-2 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \left(\frac{c}{b} + a \cdot \left(\left(c \cdot c\right) \cdot {b}^{-3}\right)\right) \]
Alternative 2
Error2.2
Cost20672
\[-2 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{c}{b} \cdot \mathsf{fma}\left(\frac{c}{b}, \frac{a}{b}, 1\right) \]
Alternative 3
Error3.2
Cost1024
\[\frac{c}{b} \cdot \left(c \cdot \frac{\frac{-a}{b}}{b}\right) - \frac{c}{b} \]
Alternative 4
Error3.2
Cost832
\[\frac{c}{b} \cdot \left(-1 - c \cdot \frac{a}{b \cdot b}\right) \]
Alternative 5
Error6.3
Cost256
\[\frac{-c}{b} \]

Error

Reproduce

herbie shell --seed 2022241 
(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)))