Average Error: 34.5 → 6.7
Time: 5.1s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.9209858444379851 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq 7.420373164236022 \cdot 10^{-279}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 1.304608330782827 \cdot 10^{+64}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.9209858444379851 \cdot 10^{+146}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\mathbf{elif}\;b \leq 7.420373164236022 \cdot 10^{-279}:\\
\;\;\;\;\frac{4}{2} \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\

\mathbf{elif}\;b \leq 1.304608330782827 \cdot 10^{+64}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \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 -1.9209858444379851e+146)
   (* (/ c b) -1.0)
   (if (<= b 7.420373164236022e-279)
     (* (/ 4.0 2.0) (/ c (- (sqrt (- (* b b) (* 4.0 (* c a)))) b)))
     (if (<= b 1.304608330782827e+64)
       (- (/ (- b) (* 2.0 a)) (/ (sqrt (- (* b b) (* 4.0 (* c a)))) (* 2.0 a)))
       (* -1.0 (/ b a))))))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= -1.9209858444379851e+146)) {
		tmp = ((double) ((c / b) * -1.0));
	} else {
		double tmp_1;
		if ((b <= 7.420373164236022e-279)) {
			tmp_1 = ((double) ((4.0 / 2.0) * (c / ((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (c * a)))))))) - b)))));
		} else {
			double tmp_2;
			if ((b <= 1.304608330782827e+64)) {
				tmp_2 = ((double) ((((double) -(b)) / ((double) (2.0 * a))) - (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (c * a)))))))) / ((double) (2.0 * a)))));
			} else {
				tmp_2 = ((double) (-1.0 * (b / a)));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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

Original34.5
Target21.2
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.9209858444379851e146

    1. Initial program 63.3

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

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    if -1.9209858444379851e146 < b < 7.4203731642360218e-279

    1. Initial program 33.1

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

    3. Applied flip--_binary6433.1

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

    4. Simplified15.2

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

    5. Simplified15.2

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

    7. Applied *-un-lft-identity_binary6415.2

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

    8. Applied times-frac_binary6415.2

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

    9. Applied times-frac_binary6415.2

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

    10. Simplified15.2

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

    11. Simplified8.0

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

    if 7.4203731642360218e-279 < b < 1.304608330782827e64

    1. Initial program 9.2

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

    3. Applied div-sub_binary649.2

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

    4. Simplified9.2

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

    5. Simplified9.2

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

    if 1.304608330782827e64 < b

    1. Initial program 41.4

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

    3. Applied flip--_binary6461.7

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

    4. Simplified60.9

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

    5. Simplified60.9

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

    6. Taylor expanded around 0 5.6

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9209858444379851 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \leq 7.420373164236022 \cdot 10^{-279}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 1.304608330782827 \cdot 10^{+64}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))