Average Error: 28.5 → 5.0
Time: 5.2s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18.619244255333086:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -0.5625 - \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\
\mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18.619244255333086:\\
\;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -0.5625 - \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\\


\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<=
        (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
        -18.619244255333086)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (-
      (* (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)) -0.5625)
      (+
       (* 1.0546875 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)))
       (+ (* 0.375 (/ (* a (pow c 2.0)) (pow b 3.0))) (* 0.5 (/ c 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) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -18.619244255333086) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)) * -0.5625) - ((1.0546875 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0))) + ((0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (0.5 * (c / b))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -18.619244255333086

    1. Initial program 9.1

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

    2. Applied egg-rr8.2

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}}{3 \cdot a} \]

    if -18.619244255333086 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

    1. Initial program 30.2

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

    2. Taylor expanded in b around inf 4.8

      \[\leadsto \color{blue}{-\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18.619244255333086:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -0.5625 - \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))