Average Error: 34.2 → 7.4
Time: 6.1s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.0779796988596927 \cdot 10^{+137}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4533084893508145 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 8.216375787380612 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}} \cdot \left(-2 \cdot \frac{c}{\sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -2.0779796988596927 \cdot 10^{+137}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 1.4533084893508145 \cdot 10^{-308}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 8.216375787380612 \cdot 10^{+73}:\\
\;\;\;\;\frac{1}{\sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}} \cdot \left(-2 \cdot \frac{c}{\sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\right)\\

\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 -2.0779796988596927e+137)
   (- (/ c b) (/ b a))
   (if (<= b 1.4533084893508145e-308)
     (/
      (-
       (*
        (sqrt (sqrt (- (* b b) (* c (* a 4.0)))))
        (sqrt (sqrt (- (* b b) (* c (* a 4.0))))))
       b)
      (* a 2.0))
     (if (<= b 8.216375787380612e+73)
       (*
        (/
         1.0
         (*
          (cbrt (+ b (sqrt (- (* b b) (* c (* a 4.0))))))
          (cbrt (+ b (sqrt (- (* b b) (* c (* a 4.0))))))))
        (* -2.0 (/ c (cbrt (+ b (sqrt (- (* b b) (* c (* a 4.0)))))))))
       (- (/ 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 <= -2.0779796988596927e+137) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4533084893508145e-308) {
		tmp = ((sqrt(sqrt((b * b) - (c * (a * 4.0)))) * sqrt(sqrt((b * b) - (c * (a * 4.0))))) - b) / (a * 2.0);
	} else if (b <= 8.216375787380612e+73) {
		tmp = (1.0 / (cbrt(b + sqrt((b * b) - (c * (a * 4.0)))) * cbrt(b + sqrt((b * b) - (c * (a * 4.0)))))) * (-2.0 * (c / cbrt(b + sqrt((b * b) - (c * (a * 4.0))))));
	} 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

Derivation

  1. Split input into 4 regimes

  2. if b < -2.07797969885969273e137

    1. Initial program 56.5

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

    2. Simplified56.5

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

    3. Taylor expanded around -inf 2.9

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

    if -2.07797969885969273e137 < b < 1.4533084893508145e-308

    1. Initial program 9.8

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

    2. Simplified9.8

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

    4. Applied add-sqr-sqrt_binary64_43510.1

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

    if 1.4533084893508145e-308 < b < 8.21637578738061166e73

    1. Initial program 31.0

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

    2. Simplified31.0

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

    4. Applied flip--_binary64_38931.0

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

    5. Simplified16.8

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

    6. Simplified16.8

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

    8. Applied add-cube-cbrt_binary64_44617.5

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

    9. Applied times-frac_binary64_42015.4

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

    10. Applied times-frac_binary64_42011.3

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

    11. Simplified10.9

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

    12. Simplified10.7

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

    if 8.21637578738061166e73 < b

    1. Initial program 57.8

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

    2. Simplified57.8

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

    3. Taylor expanded around inf 2.6

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

    4. Simplified2.6

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.0779796988596927 \cdot 10^{+137}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4533084893508145 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 8.216375787380612 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}} \cdot \left(-2 \cdot \frac{c}{\sqrt[3]{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020301 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))