Cubic critical, narrow range

Average Error: 28.6 → 5.2

Time: 8.9s

Precision: binary64

\[1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -10.14256529719701:\\ \;\;\;\;\begin{array}{l} t_0 := b \cdot b - 3 \cdot \left(a \cdot c\right)\\ \frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\right) - 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<=
      (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
      -10.14256529719701)
   (let* ((t_0 (- (* b b) (* 3.0 (* a c)))))
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a)))
   (-
    (-
     (- (* -0.5 (/ c b)) (* 0.375 (/ (* c (* a c)) (pow b 3.0))))
     (* 1.0546875 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0))))
    (* 0.5625 (/ (* (* a a) (pow c 3.0)) (pow b 5.0))))))

C

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 tmp;
	if (((sqrt((b * b) - ((3.0 * a) * c)) - b) / (3.0 * a)) <= -10.14256529719701) {
		double t_0_1 = (b * b) - (3.0 * (a * c));
		tmp = ((t_0_1 - (b * b)) / (b + sqrt(t_0_1))) / (3.0 * a);
	} else {
		tmp = (((-0.5 * (c / b)) - (0.375 * ((c * (a * c)) / pow(b, 3.0)))) - (1.0546875 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)))) - (0.5625 * (((a * a) * pow(c, 3.0)) / pow(b, 5.0)));
	}
	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)) < -10.142565297197009

   1. Initial program 9.4

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

   2. Simplified 9.4

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}} \]
   3. Using strategy rm

   4. Applied flip--_binary64 9.5

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

   5. Simplified 8.6

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

   6. Simplified 8.7

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

   if -10.142565297197009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 30.5

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

   2. Simplified 30.5

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

   3. Taylor expanded around inf 4.9

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

   4. Simplified 4.9

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -10.14256529719701:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\right) - 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-8 a 94906265.62425156) (< 1.0536712127723509e-8 b 94906265.62425156) (< 1.0536712127723509e-8 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))