jeff quadratic root 2

Average Error: 19.9 → 7.3

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9.653956009291089 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot \left(2 \cdot \frac{a}{b}\right) - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.877750719758907 \cdot 10^{+44}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{2 \cdot a}\\ \end{array}\]

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 -9.653956009291089e+148)
   (if (>= b 0.0)
     (* 2.0 (/ c (- (- (* 2.0 (* c (/ a b))) b) b)))
     (/ (- (- (* c (* 2.0 (/ a b))) b) b) (* 2.0 a)))
   (if (<= b 4.877750719758907e+44)
     (if (>= b 0.0)
       (*
        2.0
        (/
         c
         (-
          (- b)
          (*
           (sqrt (sqrt (- (* b b) (* c (* a 4.0)))))
           (sqrt (sqrt (- (* b b) (* c (* a 4.0)))))))))
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
     (if (>= b 0.0)
       (* 2.0 (/ c (- (- (* 2.0 (* c (/ a b))) b) b)))
       (/
        (-
         (*
          (fabs (cbrt (- (* b b) (* c (* a 4.0)))))
          (sqrt (cbrt (- (* 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 = (((double) (2.0 * c)) / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))));
	} else {
		tmp = (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
	}
	return tmp;
}

↓

double code(double a, double b, double c) {
	double tmp;
	if ((b <= -9.653956009291089e+148)) {
		double tmp_1;
		if ((b >= 0.0)) {
			tmp_1 = ((double) (2.0 * (c / ((double) (((double) (((double) (2.0 * ((double) (c * (a / b))))) - b)) - b)))));
		} else {
			tmp_1 = (((double) (((double) (((double) (c * ((double) (2.0 * (a / b))))) - b)) - b)) / ((double) (2.0 * a)));
		}
		tmp = tmp_1;
	} else {
		double tmp_2;
		if ((b <= 4.877750719758907e+44)) {
			double tmp_3;
			if ((b >= 0.0)) {
				tmp_3 = ((double) (2.0 * (c / ((double) (((double) -(b)) - ((double) (((double) sqrt(((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))))) * ((double) sqrt(((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))))))))))));
			} else {
				tmp_3 = (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))) - b)) / ((double) (2.0 * a)));
			}
			tmp_2 = tmp_3;
		} else {
			double tmp_4;
			if ((b >= 0.0)) {
				tmp_4 = ((double) (2.0 * (c / ((double) (((double) (((double) (2.0 * ((double) (c * (a / b))))) - b)) - b)))));
			} else {
				tmp_4 = (((double) (((double) (((double) fabs(((double) cbrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))))) * ((double) sqrt(((double) cbrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))))))) - b)) / ((double) (2.0 * a)));
			}
			tmp_2 = tmp_4;
		}
		tmp = tmp_2;
	}
	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 < -9.6539560092910892e148

   1. Initial program Error: 61.8 bits

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

   2. Simplified Error: 61.8 bits

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

   3. Taylor expanded around inf Error: 61.8 bits

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

   4. Simplified Error: 61.8 bits

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

   5. Taylor expanded around -inf Error: 10.0 bits

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

   6. Simplified Error: 2.6 bits

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

   if -9.6539560092910892e148 < b < 4.87775071975890684e44

   1. Initial program Error: 9.7 bits

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

   2. Simplified Error: 9.7 bits

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

   4. Applied add-sqr-sqrt Error: 9.7 bits

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

   5. Applied sqrt-prod Error: 9.8 bits

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

   if 4.87775071975890684e44 < b

   1. Initial program Error: 24.4 bits

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

   2. Simplified Error: 24.4 bits

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

   3. Taylor expanded around inf Error: 6.9 bits

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

   4. Simplified Error: 3.9 bits

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

   6. Applied add-cube-cbrt Error: 3.9 bits

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

   7. Applied sqrt-prod Error: 3.9 bits

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

   8. Simplified Error: 3.9 bits

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

4. Final simplification Error: 7.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.653956009291089 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot \left(2 \cdot \frac{a}{b}\right) - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.877750719758907 \cdot 10^{+44}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))