Average Error: 20.2 → 19.0
Time: 4.9s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3316660457403754 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \end{array}\]
\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 -1.3316660457403754 \cdot 10^{+154}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot -2\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{a \cdot 2}\\

\end{array}\\

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

\end{array}
(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 -1.3316660457403754e+154)
   (if (>= b 0.0)
     (* (/ c (+ b (sqrt (- (* b b) (* c (* 4.0 a)))))) -2.0)
     (/ (- b) (* a 2.0)))
   (if (>= b 0.0)
     (*
      -2.0
      (/
       c
       (+
        b
        (*
         (sqrt (sqrt (- (* b b) (* c (* 4.0 a)))))
         (sqrt (sqrt (- (* b b) (* c (* 4.0 a)))))))))
     (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)))))
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 <= -1.3316660457403754e+154) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c / (b + sqrt((b * b) - (c * (4.0 * a))))) * -2.0;
		} else {
			tmp_1 = -b / (a * 2.0);
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = -2.0 * (c / (b + (sqrt(sqrt((b * b) - (c * (4.0 * a)))) * sqrt(sqrt((b * b) - (c * (4.0 * a)))))));
	} else {
		tmp = (sqrt((b * b) - (c * (4.0 * a))) - b) / (a * 2.0);
	}
	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 2 regimes

  2. if b < -1.3316660457403754e154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around 0 52.3

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

    if -1.3316660457403754e154 < b

    1. Initial program 14.8

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

    2. Simplified14.8

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

    4. Applied add-sqr-sqrt_binary6414.9

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3316660457403754 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(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))))