Quadratic roots, full range

Average Error: 33.9 → 8.7

Time: 8.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.738191357919323 \cdot 10^{+116}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq 3.504400908400674 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(t_0 - b\right) \cdot 0.5}{a}\\ \mathbf{elif}\;b \leq 2.980663629993403 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \frac{0.5}{a}}{b + t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.738191357919323e+116)
   (/ (- b) a)
   (let* ((t_0 (sqrt (fma a (* c -4.0) (* b b)))))
     (if (<= b 3.504400908400674e-178)
       (/ (* (- t_0 b) 0.5) a)
       (if (<= b 2.980663629993403e+79)
         (/ (* (* c (* a -4.0)) (/ 0.5 a)) (+ b t_0))
         (- (/ c b)))))))

C

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) {
	double tmp;
	if (b <= -3.738191357919323e+116) {
		tmp = -b / a;
	} else {
		double t_0 = sqrt(fma(a, (c * -4.0), (b * b)));
		double tmp_1;
		if (b <= 3.504400908400674e-178) {
			tmp_1 = ((t_0 - b) * 0.5) / a;
		} else if (b <= 2.980663629993403e+79) {
			tmp_1 = ((c * (a * -4.0)) * (0.5 / a)) / (b + t_0);
		} else {
			tmp_1 = -(c / b);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -3.7381913579193229e116

   1. Initial program 51.5

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

   2. Simplified 51.5

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

   3. Taylor expanded in b around -inf 4.2

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]

   4. Simplified 4.2

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -3.7381913579193229e116 < b < 3.50440090840067388e-178

   1. Initial program 10.7

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

   2. Simplified 10.8

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

   3. Applied associate-*r/_binary64 10.7

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}} \]

   4. Applied *-un-lft-identity_binary64 10.7

      \[\leadsto \frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - \color{blue}{1 \cdot b}\right) \cdot 0.5}{a} \]

   5. Applied cancel-sign-sub-inv_binary64 10.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + \left(-1\right) \cdot b\right)} \cdot 0.5}{a} \]

   if 3.50440090840067388e-178 < b < 2.9806636299934028e79

   1. Initial program 35.4

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

   2. Simplified 35.4

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

   3. Applied flip--_binary64 35.4

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}} \cdot \frac{0.5}{a} \]

   4. Applied associate-*l/_binary64 35.5

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}} \]

   5. Simplified 15.3

      \[\leadsto \frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + 0\right) \cdot \frac{0.5}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b} \]

   if 2.9806636299934028e79 < b

   1. Initial program 58.3

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

   2. Simplified 58.3

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

   3. Taylor expanded in a around 0 3.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   4. Simplified 3.0

      \[\leadsto \color{blue}{-\frac{c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.738191357919323 \cdot 10^{+116}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.504400908400674 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{elif}\;b \leq 2.980663629993403 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))