quadp (p42, positive)

Average Error: 33.8 → 9.7

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.5568198144016078 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{elif}\;b \le 40732783.803636283:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

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 temp;
	if ((b <= -7.700313305414632e+138)) {
		temp = ((1.0 / 2.0) * ((2.0 * (c / b)) - (2.0 * (b / a))));
	} else {
		double temp_1;
		if ((b <= 1.5568198144016078e-163)) {
			temp_1 = ((1.0 / 2.0) * ((-b + sqrt(((b * b) - (4.0 * (a * c))))) / a));
		} else {
			double temp_2;
			if ((b <= 40732783.80363628)) {
				temp_2 = ((1.0 / 2.0) * ((4.0 * (a * c)) / (a * (-b - sqrt(((b * b) - (4.0 * (a * c))))))));
			} else {
				temp_2 = ((1.0 / 2.0) * (-2.0 * (c / b)));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	21.1
Herbie	9.7

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -7.700313305414632e+138

   1. Initial program 57.3

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

   3. Applied clear-num 57.4

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 57.4

      \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}\]

   6. Applied times-frac 57.4

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

   7. Applied add-cube-cbrt 57.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   8. Applied times-frac 57.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

   9. Simplified 57.4

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   10. Simplified 57.3

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

   11. Taylor expanded around -inf 3.0

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)}\]

   if -7.700313305414632e+138 < b < 1.5568198144016078e-163

   1. Initial program 10.6

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

   3. Applied clear-num 10.7

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 10.7

      \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}\]

   6. Applied times-frac 10.7

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

   7. Applied add-cube-cbrt 10.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   8. Applied times-frac 10.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

   9. Simplified 10.7

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   10. Simplified 10.6

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

   if 1.5568198144016078e-163 < b < 40732783.80363628

   1. Initial program 32.3

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

   3. Applied flip-+ 32.3

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

   4. Simplified 17.3

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 17.3

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

   7. Applied *-un-lft-identity 17.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]

   8. Applied times-frac 17.3

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

   9. Applied times-frac 17.3

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}}\]

   10. Simplified 17.3

       \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\]

   11. Simplified 22.5

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]

   if 40732783.80363628 < b

   1. Initial program 55.9

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

   3. Applied clear-num 55.9

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 55.9

      \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}\]

   6. Applied times-frac 55.9

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

   7. Applied add-cube-cbrt 55.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   8. Applied times-frac 55.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

   9. Simplified 55.9

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   10. Simplified 55.9

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

   11. Taylor expanded around inf 6.0

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \frac{c}{b}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.5568198144016078 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{elif}\;b \le 40732783.803636283:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))