Average Error: 34.0 → 11.2
Time: 12.3s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.5179383292385342 \cdot 10^{-74}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 0.018137467673081126:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]
\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 -1.5179383292385342 \cdot 10^{-74}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq 0.018137467673081126:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\


\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 -1.5179383292385342e-74)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 0.018137467673081126)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 a))
     (* -0.5 (* 2.0 (- (/ b 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 <= -1.5179383292385342e-74) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 0.018137467673081126) {
		tmp = (-b - sqrt((b * b) - (4.0 * (c * a)))) / (2.0 * a);
	} else {
		tmp = -0.5 * (2.0 * ((b / a) - (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

Original34.0
Target21.1
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5179383292385342e-74

    1. Initial program 53.1

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

    2. Simplified53.2

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

    3. Taylor expanded in b around -inf 9.8

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

    if -1.5179383292385342e-74 < b < 0.0181374676730811264

    1. Initial program 15.1

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

    if 0.0181374676730811264 < b

    1. Initial program 32.0

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

    2. Simplified32.1

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

    3. Taylor expanded in b around inf 7.6

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

    4. Simplified7.6

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5179383292385342 \cdot 10^{-74}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 0.018137467673081126:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022066 
(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)))