jeff quadratic root 2

Average Error: 20.0 → 6.8

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4137242978488287 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2377991560471444 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \left(\left(\sqrt[3]{c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \left(\sqrt[3]{c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \sqrt[3]{c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4137242978488287e+98)
   (if (>= b 0.0)
     (* -2.0 (/ c (+ b (- b (* 2.0 (* c (/ a b)))))))
     (/ (- (- (* 2.0 (* c (/ a b))) b) b) (* 2.0 a)))
   (if (<= b 4.2377991560471444e+139)
     (if (>= b 0.0)
       (*
        -2.0
        (/
         c
         (+
          b
          (*
           (fabs (cbrt (- (* b b) (* c (* a 4.0)))))
           (sqrt (cbrt (- (* b b) (* c (* a 4.0)))))))))
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
     (if (>= b 0.0)
       (*
        -2.0
        (/
         c
         (+
          b
          (-
           b
           (*
            2.0
            (*
             (*
              (cbrt (* c (/ (* (cbrt a) (cbrt a)) (* (cbrt b) (cbrt b)))))
              (*
               (cbrt (* c (/ (* (cbrt a) (cbrt a)) (* (cbrt b) (cbrt b)))))
               (cbrt (* c (/ (* (cbrt a) (cbrt a)) (* (cbrt b) (cbrt b)))))))
             (/ (cbrt a) (cbrt b))))))))
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))))

C

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

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4137242978488287e+98) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = -2.0 * (c / (b + (b - (2.0 * (c * (a / b))))));
		} else {
			tmp_1 = (((2.0 * (c * (a / b))) - b) - b) / (2.0 * a);
		}
		tmp = tmp_1;
	} else if (b <= 4.2377991560471444e+139) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (fabs(cbrt((b * b) - (c * (a * 4.0)))) * sqrt(cbrt((b * b) - (c * (a * 4.0)))))));
		} else {
			tmp_2 = (sqrt((b * b) - (c * (a * 4.0))) - b) / (2.0 * a);
		}
		tmp = tmp_2;
	} else if (b >= 0.0) {
		tmp = -2.0 * (c / (b + (b - (2.0 * ((cbrt(c * ((cbrt(a) * cbrt(a)) / (cbrt(b) * cbrt(b)))) * (cbrt(c * ((cbrt(a) * cbrt(a)) / (cbrt(b) * cbrt(b)))) * cbrt(c * ((cbrt(a) * cbrt(a)) / (cbrt(b) * cbrt(b)))))) * (cbrt(a) / cbrt(b)))))));
	} else {
		tmp = (sqrt((b * b) - (c * (a * 4.0))) - b) / (2.0 * a);
	}
	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 < -1.41372429784882871e98

   1. Initial program 46.7

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

   2. Simplified 46.7

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

   3. Taylor expanded around inf 46.7

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

   4. Simplified 46.7

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

   6. Applied *-un-lft-identity_binary64 46.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \frac{c \cdot a}{\color{blue}{1 \cdot b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\]

   7. Applied times-frac_binary64 46.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \color{blue}{\left(\frac{c}{1} \cdot \frac{a}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\]

   8. Simplified 46.7

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

   9. Taylor expanded around -inf 9.8

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

   10. Simplified 4.1

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

   if -1.41372429784882871e98 < b < 4.2377991560471444e139

   1. Initial program 8.8

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

   2. Simplified 8.8

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

   4. Applied add-cube-cbrt_binary64 9.1

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

   5. Applied sqrt-prod_binary64 9.1

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

   6. Simplified 9.1

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

   if 4.2377991560471444e139 < b

   1. Initial program 34.8

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

   2. Simplified 34.8

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

   3. Taylor expanded around inf 6.2

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

   4. Simplified 6.2

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

   6. Applied *-un-lft-identity_binary64 6.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \frac{c \cdot a}{\color{blue}{1 \cdot b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\]

   7. Applied times-frac_binary64 1.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \color{blue}{\left(\frac{c}{1} \cdot \frac{a}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\]

   8. Simplified 1.6

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

   10. Applied add-cube-cbrt_binary64 1.6

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

   11. Applied add-cube-cbrt_binary64 1.6

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

   12. Applied times-frac_binary64 1.6

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

   13. Applied associate-*r*_binary64 1.6

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

   15. Applied add-cube-cbrt_binary64 1.6

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4137242978488287 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2377991560471444 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b - 2 \cdot \left(\left(\sqrt[3]{c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \left(\sqrt[3]{c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \sqrt[3]{c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))