Average Error: 34.4 → 10.5
Time: 17.0s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.0695451765040258 \cdot 10^{+148}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.7543418175087657 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\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 -1.0695451765040258 \cdot 10^{+148}:\\
\;\;\;\;\frac{-b}{a}\\

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

\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) a)
   (if (<= b 1.7543418175087657e-89)
     (/ (- (sqrt (- (pow b 2.0) (* 4.0 (* a c)))) b) (* a 2.0))
     (- (/ 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 / a;
	} else if (b <= 1.7543418175087657e-89) {
		tmp = (sqrt(pow(b, 2.0) - (4.0 * (a * c))) - b) / (a * 2.0);
	} 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
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0695451765040258e148

    1. Initial program 62.0

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

    2. Taylor expanded in b around -inf 2.0

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

    3. Simplified2.0

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

    if -1.0695451765040258e148 < b < 1.7543418175087657e-89

    1. Initial program 12.6

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

    2. Taylor expanded in b around 0 12.6

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

    if 1.7543418175087657e-89 < b

    1. Initial program 51.9

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

    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

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

Reproduce

herbie shell --seed 2021215 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))