Average Error: 34.1 → 10.4
Time: 10.3s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.952904602593089 \cdot 10^{-31}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2.428003017385294 \cdot 10^{+117}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{-a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \end{array} \]
\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 -5.952904602593089 \cdot 10^{-31}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq 2.428003017385294 \cdot 10^{+117}:\\
\;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{-a} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{b \cdot 2}{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 -5.952904602593089e-31)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 2.428003017385294e+117)
     (* (/ (+ b (sqrt (fma a (* c -4.0) (* b b)))) (- a)) 0.5)
     (* -0.5 (/ (* b 2.0) 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 <= -5.952904602593089e-31) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 2.428003017385294e+117) {
		tmp = ((b + sqrt(fma(a, (c * -4.0), (b * b)))) / -a) * 0.5;
	} else {
		tmp = -0.5 * ((b * 2.0) / a);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.1
Target20.9
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.9529046025930888e-31

    1. Initial program 54.6

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

    2. Simplified54.6

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

    3. Taylor expanded around -inf 7.2

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

    if -5.9529046025930888e-31 < b < 2.42800301738529409e117

    1. Initial program 14.5

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

    2. Simplified14.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
    3. Using strategy rm

    4. Applied frac-2neg_binary6414.5

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

    5. Applied distribute-frac-neg_binary6414.5

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

    if 2.42800301738529409e117 < b

    1. Initial program 51.0

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

    2. Simplified51.0

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

    3. Taylor expanded around inf 3.9

      \[\leadsto -0.5 \cdot \frac{\color{blue}{2 \cdot b}}{a} \]

    4. Simplified3.9

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b \cdot 2}}{a} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.952904602593089 \cdot 10^{-31}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2.428003017385294 \cdot 10^{+117}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{-a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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)))