Average Error: 34.6 → 7.0
Time: 9.9s
Precision: binary64
Cost: 8323
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.303984552860991 \cdot 10^{+140}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.201370751245736 \cdot 10^{-277}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 5.513576328640909 \cdot 10^{+99}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.303984552860991 \cdot 10^{+140}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \leq 3.201370751245736 \cdot 10^{-277}:\\
\;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\

\mathbf{elif}\;b \leq 5.513576328640909 \cdot 10^{+99}:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \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 -1.303984552860991e+140)
   (- (/ c b))
   (if (<= b 3.201370751245736e-277)
     (* -0.5 (* 4.0 (/ c (- b (sqrt (- (* b b) (* 4.0 (* c a))))))))
     (if (<= b 5.513576328640909e+99)
       (* -0.5 (/ (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) a))
       (- (/ c b) (/ 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 <= -1.303984552860991e+140) {
		tmp = -(c / b);
	} else if (b <= 3.201370751245736e-277) {
		tmp = -0.5 * (4.0 * (c / (b - sqrt((b * b) - (4.0 * (c * a))))));
	} else if (b <= 5.513576328640909e+99) {
		tmp = -0.5 * ((b + sqrt((b * b) + (a * (c * -4.0)))) / a);
	} else {
		tmp = (c / b) - (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

Original34.6
Target21.3
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Alternatives

Alternative 1
Error11.0
Cost8002
\[\begin{array}{l} \mathbf{if}\;b \leq -3.935594578192137 \cdot 10^{-124}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.0794489170197573 \cdot 10^{+97}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
Alternative 2
Error14.1
Cost7746
\[\begin{array}{l} \mathbf{if}\;b \leq -5.7015362524443945 \cdot 10^{-124}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.192658630200912 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
Alternative 3
Error14.3
Cost7746
\[\begin{array}{l} \mathbf{if}\;b \leq -5.380455948034893 \cdot 10^{-124}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 2.0197238409036384 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\sqrt{a \cdot \left(c \cdot -4\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
Alternative 4
Error14.3
Cost7618
\[\begin{array}{l} \mathbf{if}\;b \leq -5.7015362524443945 \cdot 10^{-124}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.156489435550409 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
Alternative 5
Error23.2
Cost577
\[\begin{array}{l} \mathbf{if}\;b \leq -3.1124603004849597 \cdot 10^{-207}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
Alternative 6
Error39.8
Cost256
\[-\frac{c}{b}\]
Alternative 7
Error56.2
Cost64
\[0\]
Alternative 8
Error61.6
Cost64
\[1\]

Error

Derivation

  1. Split input into 4 regimes

  2. if b < -1.30398455286099105e140

    1. Initial program 62.7

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

    2. Simplified62.7

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

    3. Taylor expanded around -inf 1.2

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

    4. Simplified1.2

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

    5. Simplified1.2

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

    if -1.30398455286099105e140 < b < 3.2013707512457359e-277

    1. Initial program 33.6

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

    2. Simplified33.6

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

    4. Applied flip-+_binary64_39333.7

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

    5. Simplified16.9

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

    7. Applied *-un-lft-identity_binary64_41916.9

      \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\color{blue}{1 \cdot a}}\]

    8. Applied *-un-lft-identity_binary64_41916.9

      \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{1 \cdot a}\]

    9. Applied times-frac_binary64_42516.9

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{1 \cdot a}\]

    10. Applied times-frac_binary64_42516.9

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\frac{4}{1}}{1} \cdot \frac{\frac{a \cdot c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\right)}\]

    11. Simplified16.9

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

    12. Simplified9.3

      \[\leadsto -0.5 \cdot \left(4 \cdot \color{blue}{\left(1 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}\right)\]

    13. Simplified9.3

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

    if 3.2013707512457359e-277 < b < 5.5135763286409085e99

    1. Initial program 9.5

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

    2. Simplified9.4

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

    4. Applied sub-neg_binary64_4129.4

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

    5. Simplified9.5

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

    6. Simplified9.5

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

    if 5.5135763286409085e99 < b

    1. Initial program 47.8

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

    2. Simplified47.9

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

    3. Taylor expanded around inf 4.2

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

    4. Simplified4.2

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.303984552860991 \cdot 10^{+140}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.201370751245736 \cdot 10^{-277}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 5.513576328640909 \cdot 10^{+99}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected #f

  :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)))