Average Error: 34.3 → 10.8
Time: 10.9s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9717588257544654 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.478133240814697 \cdot 10^{+28}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -2.9717588257544654 \cdot 10^{-49}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq 1.478133240814697 \cdot 10^{+28}:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{b + b}{a}\\

\end{array}
(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9717588257544654e-49)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 1.478133240814697e+28)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) a))
     (* -0.5 (/ (+ b b) a)))))
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 tmp;
	if (b <= -2.9717588257544654e-49) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 1.478133240814697e+28) {
		tmp = -0.5 * ((b + sqrt((b * b) - (4.0 * (c * a)))) / a);
	} else {
		tmp = -0.5 * ((b + b) / a);
	}
	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

Target

Original34.3
Target21.1
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.9717588257544654e-49

    1. Initial program 53.9

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

    2. Simplified53.9

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

    3. Taylor expanded around -inf 7.7

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)}\]

    if -2.9717588257544654e-49 < b < 1.478133240814697e28

    1. Initial program 15.6

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

    2. Simplified15.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

    if 1.478133240814697e28 < b

    1. Initial program 35.6

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

    2. Simplified35.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

    3. Taylor expanded around inf 6.9

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{b}}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9717588257544654 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.478133240814697 \cdot 10^{+28}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))