Average Error: 52.6 → 1.5
Time: 4.4s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{7}}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125}{a}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{c \cdot -0.5}{b}\right)\right)\right) \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (fma
  (/ -0.16666666666666666 (pow b 7.0))
  (/ (* (* (pow c 4.0) (pow a 4.0)) 6.328125) a)
  (fma
   -0.375
   (* a (/ (* c c) (pow b 3.0)))
   (fma -0.5625 (* (/ (pow c 3.0) (pow b 5.0)) (* a a)) (/ (* c -0.5) b)))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return fma((-0.16666666666666666 / pow(b, 7.0)), (((pow(c, 4.0) * pow(a, 4.0)) * 6.328125) / a), fma(-0.375, (a * ((c * c) / pow(b, 3.0))), fma(-0.5625, ((pow(c, 3.0) / pow(b, 5.0)) * (a * a)), ((c * -0.5) / b))));
}

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end
↓
function code(a, b, c)
	return fma(Float64(-0.16666666666666666 / (b ^ 7.0)), Float64(Float64(Float64((c ^ 4.0) * (a ^ 4.0)) * 6.328125) / a), fma(-0.375, Float64(a * Float64(Float64(c * c) / (b ^ 3.0))), fma(-0.5625, Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)), Float64(Float64(c * -0.5) / b))))
end

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := N[(N[(-0.16666666666666666 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / a), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{7}}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125}{a}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{c \cdot -0.5}{b}\right)\right)\right)

Error

Derivation

  1. Initial program 52.6

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

  2. Taylor expanded in b around inf 1.5

    \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

  3. Simplified1.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{7}}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125}{a}, \mathsf{fma}\left(-0.375, \frac{c \cdot c}{{b}^{3}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{c \cdot -0.5}{b}\right)\right)\right)} \]

  4. Final simplification1.5

    \[\leadsto \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{7}}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125}{a}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{c \cdot -0.5}{b}\right)\right)\right) \]

Reproduce

herbie shell --seed 2022206 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))