Average Error: 33.8 → 9.1
Time: 5.5s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.6479930097769547 \cdot 10^{+85}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -2.058574872602062 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}\\ \mathbf{elif}\;b \leq 1.6833943237704546 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]
\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 -2.6479930097769547 \cdot 10^{+85}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq -2.058574872602062 \cdot 10^{-105}:\\
\;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}\\

\mathbf{elif}\;b \leq 1.6833943237704546 \cdot 10^{+136}:\\
\;\;\;\;\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 -2.6479930097769547e+85)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b -2.058574872602062e-105)
     (*
      -0.5
      (/ (/ (* 4.0 (* c a)) (- b (sqrt (- (* b b) (* 4.0 (* c a)))))) a))
     (if (<= b 1.6833943237704546e+136)
       (/ (- (- 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 <= -2.6479930097769547e+85) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= -2.058574872602062e-105) {
		tmp = -0.5 * (((4.0 * (c * a)) / (b - sqrt((b * b) - (4.0 * (c * a))))) / a);
	} else if (b <= 1.6833943237704546e+136) {
		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

Original33.8
Target21.0
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.64799300977695472e85

    1. Initial program 58.3

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

    2. Simplified58.3

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

    3. Taylor expanded around -inf 2.9

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

    if -2.64799300977695472e85 < b < -2.058574872602062e-105

    1. Initial program 40.8

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

    2. Simplified40.8

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

    4. Applied flip-+_binary64_175740.8

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

    5. Simplified15.4

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

    if -2.058574872602062e-105 < b < 1.6833943237704546e136

    1. Initial program 11.8

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

    if 1.6833943237704546e136 < b

    1. Initial program 56.4

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

    2. Simplified56.4

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

    3. Taylor expanded around inf 2.6

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

    4. Simplified2.6

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6479930097769547 \cdot 10^{+85}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -2.058574872602062 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}\\ \mathbf{elif}\;b \leq 1.6833943237704546 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))