Average Error: 20.6 → 6.9
Time: 4.9s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.0667631505325645 \cdot 10^{+124}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.4627280993291758 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}\\ \end{array}\]
\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 -5.0667631505325645 \cdot 10^{+124}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\

\end{array}\\

\mathbf{elif}\;b \leq 2.4627280993291758 \cdot 10^{+95}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\end{array}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}\\

\end{array}
(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 -5.0667631505325645e+124)
   (if (>= b 0.0)
     (/ (/ (* 4.0 (* a c)) (- (sqrt (- (* b b) (* 4.0 (* a c)))) b)) (* a 2.0))
     (/ (* c 2.0) (+ (* 2.0 (* c (/ a b))) (* b -2.0))))
   (if (<= b 2.4627280993291758e+95)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
       (* c (/ 2.0 (- (sqrt (- (* b b) (* 4.0 (* a c)))) b))))
     (if (>= b 0.0)
       (/ (+ (* 2.0 (* c (/ a b))) (* b -2.0)) (* a 2.0))
       (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* 4.0 a)))) b))))))
double code(double a, double b, double c) {
	double tmp;
	if ((b >= 0.0)) {
		tmp = (((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
	} else {
		tmp = (((double) (2.0 * c)) / ((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))));
	}
	return tmp;
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= -5.0667631505325645e+124)) {
		double tmp_1;
		if ((b >= 0.0)) {
			tmp_1 = ((((double) (4.0 * ((double) (a * c)))) / ((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))) - b))) / ((double) (a * 2.0)));
		} else {
			tmp_1 = (((double) (c * 2.0)) / ((double) (((double) (2.0 * ((double) (c * (a / b))))) + ((double) (b * -2.0)))));
		}
		tmp = tmp_1;
	} else {
		double tmp_2;
		if ((b <= 2.4627280993291758e+95)) {
			double tmp_3;
			if ((b >= 0.0)) {
				tmp_3 = (((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))))) / ((double) (a * 2.0)));
			} else {
				tmp_3 = ((double) (c * (2.0 / ((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))) - b)))));
			}
			tmp_2 = tmp_3;
		} else {
			double tmp_4;
			if ((b >= 0.0)) {
				tmp_4 = (((double) (((double) (2.0 * ((double) (c * (a / b))))) + ((double) (b * -2.0)))) / ((double) (a * 2.0)));
			} else {
				tmp_4 = (((double) (c * 2.0)) / ((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (4.0 * a)))))))) - b)));
			}
			tmp_2 = tmp_4;
		}
		tmp = tmp_2;
	}
	return tmp;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -5.066763150532564e124

    1. Initial program Error: 33.4 bits

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

    2. Taylor expanded around -inf Error: 7.1 bits

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

    3. SimplifiedError: 2.0 bits

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

    5. Applied flip--Error: 2.0 bits

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

    6. SimplifiedError: 2.0 bits

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

    7. SimplifiedError: 2.0 bits

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

    if -5.066763150532564e124 < b < 2.46272809932917583e95

    1. Initial program Error: 9.2 bits

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

    2. SimplifiedError: 9.3 bits

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

    if 2.46272809932917583e95 < b

    1. Initial program Error: 47.2 bits

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

    2. Taylor expanded around inf Error: 8.9 bits

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

    3. SimplifiedError: 4.0 bits

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

  4. Final simplificationError: 6.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.0667631505325645 \cdot 10^{+124}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.4627280993291758 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))))))