jeff quadratic root 1

Average Error: 19.6 → 17.6

Time: 12.0s

Precision: binary64

Math

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.130306482326166 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{0} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3570002931602082 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{{\left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{4}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
   (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.130306482326166e+159)
   (if (>= b 0.0)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))) a))
     (/ (* c 2.0) (- (sqrt 0.0) b)))
   (if (<= b 1.3570002931602082e+154)
     (if (>= b 0.0)
       (* -0.5 (/ (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))) a))
       (/
        (* c 2.0)
        (-
         (*
          (fabs (cbrt (- (* b b) (* (* 4.0 a) c))))
          (sqrt
           (*
            (cbrt (pow (sqrt (cbrt (- (* b b) (* (* 4.0 a) c)))) 4.0))
            (cbrt (cbrt (- (* b b) (* (* 4.0 a) c)))))))
         b)))
     (if (>= b 0.0)
       (* -0.5 (/ (+ b (sqrt 0.0)) a))
       (/ (* c 2.0) (- (sqrt (- (* b b) (* (* 4.0 a) c))) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + sqrt((b * b) - ((4.0 * a) * c)));
	}
	return tmp;
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.130306482326166e+159) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = -0.5 * ((b + sqrt((b * b) - ((4.0 * a) * c))) / a);
		} else {
			tmp_1 = (c * 2.0) / (sqrt(0.0) - b);
		}
		tmp = tmp_1;
	} else if (b <= 1.3570002931602082e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((b + sqrt((b * b) - ((4.0 * a) * c))) / a);
		} else {
			tmp_2 = (c * 2.0) / ((fabs(cbrt((b * b) - ((4.0 * a) * c))) * sqrt(cbrt(pow(sqrt(cbrt((b * b) - ((4.0 * a) * c))), 4.0)) * cbrt(cbrt((b * b) - ((4.0 * a) * c))))) - b);
		}
		tmp = tmp_2;
	} else if (b >= 0.0) {
		tmp = -0.5 * ((b + sqrt(0.0)) / a);
	} else {
		tmp = (c * 2.0) / (sqrt((b * b) - ((4.0 * a) * c)) - b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -6.13030648232616621e159

   1. Initial program 38.2

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Simplified 38.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

   3. Taylor expanded around inf 32.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{\sqrt{0} - b}\\ \end{array}\]

   if -6.13030648232616621e159 < b < 1.35700029316020823e154

   1. Initial program 8.5

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Simplified 8.5

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt_binary64 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} \cdot 2}{\sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]

   5. Applied sqrt-prod_binary64 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]

   6. Simplified 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} \cdot 2}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt_binary64 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - b}\\ \end{array}\]

   9. Applied cbrt-prod_binary64 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - b}\\ \end{array}\]

   10. Simplified 8.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{{\left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{4}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - b}\\ \end{array}\]

   if 1.35700029316020823e154 < b

   1. Initial program 64.0

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

   3. Taylor expanded around inf 52.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\sqrt{0}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.130306482326166 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{0} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3570002931602082 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{{\left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{4}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))