Average Error: 34.4 → 10.5
Time: 15.4s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.0695451765040258 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.7543418175087657 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -1.0695451765040258 \cdot 10^{+148}:\\
\;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\

\mathbf{elif}\;b \leq 1.7543418175087657 \cdot 10^{-89}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-\frac{c}{b}\\


\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 -1.0695451765040258e+148)
   (/ (* b -2.0) (* 2.0 a))
   (if (<= b 1.7543418175087657e-89)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a))
     (- (/ c b)))))
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 <= -1.0695451765040258e+148) {
		tmp = (b * -2.0) / (2.0 * a);
	} else if (b <= 1.7543418175087657e-89) {
		tmp = (sqrt((b * b) - ((a * 4.0) * c)) - b) / (2.0 * a);
	} else {
		tmp = -(c / b);
	}
	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

Original34.4
Target21.4
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0695451765040258e148

    1. Initial program 62.0

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

    2. Taylor expanded in b around -inf 2.0

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a} \]

    if -1.0695451765040258e148 < b < 1.7543418175087657e-89

    1. Initial program 12.6

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

    if 1.7543418175087657e-89 < b

    1. Initial program 51.9

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

    2. Taylor expanded in b around inf 10.4

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

    3. Simplified10.4

      \[\leadsto \color{blue}{-\frac{c}{b}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.0695451765040258 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.7543418175087657 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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

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