Average Error: 33.9 → 6.8
Time: 6.7s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.193186085288575 \cdot 10^{+108}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 6.924914397486699 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 2.404688834790521 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -5.193186085288575 \cdot 10^{+108}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \leq 6.924914397486699 \cdot 10^{-140}:\\
\;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\

\mathbf{elif}\;b \leq 2.404688834790521 \cdot 10^{+51}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{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 -5.193186085288575e+108)
   (- (/ c b))
   (if (<= b 6.924914397486699e-140)
     (* 2.0 (/ c (- (sqrt (- (* b b) (* 4.0 (* c a)))) b)))
     (if (<= b 2.404688834790521e+51)
       (/
        (-
         (- b)
         (*
          (cbrt (sqrt (- (* b b) (* 4.0 (* c a)))))
          (*
           (cbrt (sqrt (- (* b b) (* 4.0 (* c a)))))
           (cbrt (sqrt (- (* b b) (* 4.0 (* c a))))))))
        (* 2.0 a))
       (/ (- b) 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 <= -5.193186085288575e+108) {
		tmp = -(c / b);
	} else if (b <= 6.924914397486699e-140) {
		tmp = 2.0 * (c / (sqrt((b * b) - (4.0 * (c * a))) - b));
	} else if (b <= 2.404688834790521e+51) {
		tmp = (-b - (cbrt(sqrt((b * b) - (4.0 * (c * a)))) * (cbrt(sqrt((b * b) - (4.0 * (c * a)))) * cbrt(sqrt((b * b) - (4.0 * (c * a))))))) / (2.0 * a);
	} else {
		tmp = -b / 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

Original33.9
Target20.8
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.1931860852885754e108

    1. Initial program 60.6

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

    2. Taylor expanded around -inf 2.4

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

    3. Simplified2.4

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

    if -5.1931860852885754e108 < b < 6.924914397486699e-140

    1. Initial program 27.5

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

    3. Applied flip--_binary64_39828.1

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

    4. Simplified16.1

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

    5. Simplified16.1

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

    7. Applied *-un-lft-identity_binary64_42316.1

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

    8. Applied times-frac_binary64_42916.1

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

    9. Applied times-frac_binary64_42916.1

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

    10. Simplified16.1

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

    11. Simplified9.7

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

    if 6.924914397486699e-140 < b < 2.40468883479052089e51

    1. Initial program 6.2

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

    3. Applied add-cube-cbrt_binary64_4587.0

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

    if 2.40468883479052089e51 < b

    1. Initial program 39.1

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

    3. Applied flip--_binary64_39862.2

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

    4. Simplified61.5

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

    5. Simplified61.5

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

    6. Taylor expanded around 0 5.2

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

    7. Simplified5.2

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.193186085288575 \cdot 10^{+108}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 6.924914397486699 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 2.404688834790521 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(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)))