Average Error: 33.6 → 6.7
Time: 5.3s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.4695905252840837 \cdot 10^{+123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.980743311430165 \cdot 10^{-252}:\\ \;\;\;\;\frac{c \cdot \sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 5.943737817038774 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -3.4695905252840837 \cdot 10^{+123}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq 5.980743311430165 \cdot 10^{-252}:\\
\;\;\;\;\frac{c \cdot \sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\mathbf{elif}\;b_2 \leq 5.943737817038774 \cdot 10^{+58}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a}\\

\end{array}
(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.4695905252840837e+123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 5.980743311430165e-252)
     (/ (* c (cbrt 1.0)) (- (sqrt (- (* b_2 b_2) (* c a))) b_2))
     (if (<= b_2 5.943737817038774e+58)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (* -2.0 (/ b_2 a))))))
double code(double a, double b_2, double c) {
	return (-b_2 - sqrt((b_2 * b_2) - (a * c))) / a;
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.4695905252840837e+123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5.980743311430165e-252) {
		tmp = (c * cbrt(1.0)) / (sqrt((b_2 * b_2) - (c * a)) - b_2);
	} else if (b_2 <= 5.943737817038774e+58) {
		tmp = (-b_2 - sqrt((b_2 * b_2) - (c * a))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.46959052528408367e123

    1. Initial program 60.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 2.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]

    if -3.46959052528408367e123 < b_2 < 5.980743311430165e-252

    1. Initial program 31.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num_binary6431.2

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]
    4. Using strategy rm

    5. Applied flip--_binary6431.2

      \[\leadsto \frac{1}{\frac{a}{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}\]

    6. Applied associate-/r/_binary6431.3

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}\]

    7. Applied add-cube-cbrt_binary6431.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\]

    8. Applied times-frac_binary6431.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{\sqrt[3]{1}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    9. Simplified14.7

      \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a}\right)} \cdot \frac{\sqrt[3]{1}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    10. Simplified14.7

      \[\leadsto \left(\left(a \cdot c\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a}\right) \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    11. Taylor expanded around 0 8.9

      \[\leadsto \color{blue}{c} \cdot \frac{\sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\]
    12. Using strategy rm

    13. Applied associate-*r/_binary648.8

      \[\leadsto \color{blue}{\frac{c \cdot \sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    if 5.980743311430165e-252 < b_2 < 5.9437378170387743e58

    1. Initial program 8.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    if 5.9437378170387743e58 < b_2

    1. Initial program 39.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num_binary6439.1

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    4. Taylor expanded around 0 5.3

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.4695905252840837 \cdot 10^{+123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.980743311430165 \cdot 10^{-252}:\\ \;\;\;\;\frac{c \cdot \sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 5.943737817038774 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))