Average Error: 34.3 → 8.7
Time: 8.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.426533654513455 \cdot 10^{+95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.545269959537344 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.579920310062727 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{0.5}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.426533654513455 \cdot 10^{+95}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 5.545269959537344 \cdot 10^{-138}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 5.579920310062727 \cdot 10^{+36}:\\
\;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{0.5}{\sqrt[3]{a}}\\

\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 -2.426533654513455e+95)
   (- (/ c b) (/ b a))
   (if (<= b 5.545269959537344e-138)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (if (<= b 5.579920310062727e+36)
       (*
        (/
         (/ (* a (* c -4.0)) (+ b (sqrt (- (* b b) (* 4.0 (* c a))))))
         (* (cbrt a) (cbrt a)))
        (/ 0.5 (cbrt 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 <= -2.426533654513455e+95) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.545269959537344e-138) {
		tmp = (sqrt((b * b) - (4.0 * (c * a))) - b) / (a * 2.0);
	} else if (b <= 5.579920310062727e+36) {
		tmp = (((a * (c * -4.0)) / (b + sqrt((b * b) - (4.0 * (c * a))))) / (cbrt(a) * cbrt(a))) * (0.5 / cbrt(a));
	} 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

Original34.3
Target21.4
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.42653365451345503e95

    1. Initial program 46.8

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

    2. Simplified46.8

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

    3. Taylor expanded around -inf 3.9

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

    if -2.42653365451345503e95 < b < 5.54526995953734389e-138

    1. Initial program 11.1

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

    if 5.54526995953734389e-138 < b < 5.57992031006272696e36

    1. Initial program 37.5

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

    2. Simplified37.5

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

    4. Applied clear-num_binary6437.6

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

    5. Simplified37.6

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

    7. Applied div-inv_binary6437.6

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

    8. Applied add-cube-cbrt_binary6437.9

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{2}}}\]

    9. Applied times-frac_binary6437.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{\sqrt[3]{a}}{\frac{1}{2}}}}\]

    10. Applied add-cube-cbrt_binary6437.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{\sqrt[3]{a}}{\frac{1}{2}}}\]

    11. Applied times-frac_binary6437.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt[3]{a}}{\frac{1}{2}}}}\]

    12. Simplified37.9

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt[3]{a}}{\frac{1}{2}}}\]

    13. Simplified37.9

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

    15. Applied flip--_binary6437.9

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

    16. Simplified18.2

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

    17. Simplified18.2

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

    if 5.57992031006272696e36 < b

    1. Initial program 56.9

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

    2. Simplified56.9

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

    3. Taylor expanded around inf 3.7

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

    4. Simplified3.7

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.426533654513455 \cdot 10^{+95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.545269959537344 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.579920310062727 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{0.5}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))