Average Error: 34.6 → 19.1
Time: 23.7s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.2020858296199258 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{1}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}\\ \mathbf{elif}\;b \leq 1.3233476909814753 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{a} \cdot -0.5 - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a}\\ \end{array}\]
\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 -1.2020858296199258 \cdot 10^{-268}:\\
\;\;\;\;2 \cdot \frac{1}{\frac{1}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}\\

\mathbf{elif}\;b \leq 1.3233476909814753 \cdot 10^{+154}:\\
\;\;\;\;\frac{b}{a} \cdot -0.5 - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{2 \cdot 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 -1.2020858296199258e-268)
   (* 2.0 (/ 1.0 (/ 1.0 (/ c (- (sqrt (- (* b b) (* 4.0 (* c a)))) b)))))
   (if (<= b 1.3233476909814753e+154)
     (- (* (/ b a) -0.5) (/ (sqrt (- (* b b) (* 4.0 (* c a)))) (* 2.0 a)))
     (/ (- 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 <= -1.2020858296199258e-268) {
		tmp = 2.0 * (1.0 / (1.0 / (c / (sqrt((b * b) - (4.0 * (c * a))) - b))));
	} else if (b <= 1.3233476909814753e+154) {
		tmp = ((b / a) * -0.5) - (sqrt((b * b) - (4.0 * (c * a))) / (2.0 * a));
	} else {
		tmp = -b / (2.0 * a);
	}
	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

Original34.6
Target21.6
Herbie19.1
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2020858296199258e-268

    1. Initial program 45.8

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

    3. Applied flip--_binary6445.8

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

    4. Simplified23.9

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

    5. Simplified23.9

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity_binary6423.9

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \color{blue}{1 \cdot b}}}{2 \cdot a}\]

    8. Applied *-un-lft-identity_binary6423.9

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - 1 \cdot b}}{2 \cdot a}\]

    9. Applied distribute-lft-out--_binary6423.9

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}}}{2 \cdot a}\]

    10. Applied times-frac_binary6423.9

      \[\leadsto \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]

    11. Applied times-frac_binary6423.9

      \[\leadsto \color{blue}{\frac{\frac{4}{1}}{2} \cdot \frac{\frac{a \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}}\]

    12. Simplified23.9

      \[\leadsto \color{blue}{2} \cdot \frac{\frac{a \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity_binary6423.9

      \[\leadsto 2 \cdot \frac{\frac{a \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \color{blue}{1 \cdot b}}}{a}\]

    15. Applied *-un-lft-identity_binary6423.9

      \[\leadsto 2 \cdot \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - 1 \cdot b}}{a}\]

    16. Applied distribute-lft-out--_binary6423.9

      \[\leadsto 2 \cdot \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}}}{a}\]

    17. Applied times-frac_binary6422.3

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{a}\]

    18. Simplified22.3

      \[\leadsto 2 \cdot \frac{\color{blue}{a} \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\]
    19. Using strategy rm

    20. Applied clear-num_binary6422.5

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{a}{a \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}\]

    21. Simplified18.8

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{1}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}\]

    if -1.2020858296199258e-268 < b < 1.3233476909814753e154

    1. Initial program 9.6

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

    3. Applied div-sub_binary649.6

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

    4. Simplified9.6

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.5} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    5. Simplified9.6

      \[\leadsto \frac{b}{a} \cdot -0.5 - \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\]

    if 1.3233476909814753e154 < b

    1. Initial program 64.0

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

    2. Taylor expanded around inf 52.2

      \[\leadsto \frac{\left(-b\right) - \color{blue}{0}}{2 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2020858296199258 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{1}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}\\ \mathbf{elif}\;b \leq 1.3233476909814753 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{a} \cdot -0.5 - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020285 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected #f

  :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)))