Average Error: 33.7 → 13.0
Time: 4.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.48011838473109798 \cdot 10^{-73}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.7852092117837367 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{2}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}, \sqrt[3]{\frac{c}{b}}, -\frac{b}{a}\right)\\ \end{array}\]
\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 -8.48011838473109798 \cdot 10^{-73}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 4.7852092117837367 \cdot 10^{-95}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{2}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \mathsf{fma}\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}, \sqrt[3]{\frac{c}{b}}, -\frac{b}{a}\right)\\

\end{array}
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 <= -8.480118384731098e-73)) {
		temp = (-1.0 * (c / b));
	} else {
		double temp_1;
		if ((b <= 4.7852092117837367e-95)) {
			temp_1 = (((fma(b, b, -((b * b) - (4.0 * (a * c)))) / a) / 2.0) / (-b + sqrt(((b * b) - (4.0 * (a * c))))));
		} else {
			temp_1 = (1.0 * fma((cbrt((c / b)) * cbrt((c / b))), cbrt((c / b)), -(b / a)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.7
Target20.6
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.480118384731098e-73

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 8.5

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

    if -8.480118384731098e-73 < b < 4.7852092117837367e-95

    1. Initial program 17.4

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

    3. Applied div-inv17.5

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

    5. Applied flip--20.5

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

    6. Applied associate-*l/20.5

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

    7. Simplified20.4

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

    if 4.7852092117837367e-95 < b

    1. Initial program 25.5

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

    2. Taylor expanded around inf 11.4

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

    3. Simplified11.4

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

    5. Applied add-cube-cbrt11.4

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

    6. Applied fma-neg11.4

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}, \sqrt[3]{\frac{c}{b}}, -\frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.48011838473109798 \cdot 10^{-73}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.7852092117837367 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{2}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}, \sqrt[3]{\frac{c}{b}}, -\frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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