Average Error: 33.7 → 6.7
Time: 9.0s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.6404428421693406 \cdot 10^{+54}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -1.069698752870928 \cdot 10^{-294}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 2.4425538060656697 \cdot 10^{+108}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \end{array} \]
\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 -2.6404428421693406 \cdot 10^{+54}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq -1.069698752870928 \cdot 10^{-294}:\\
\;\;\;\;-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 2.4425538060656697 \cdot 10^{+108}:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{b \cdot 2}{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 -2.6404428421693406e+54)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b -1.069698752870928e-294)
     (* -0.5 (* 4.0 (/ c (- b (sqrt (- (* b b) (* 4.0 (* c a))))))))
     (if (<= b 2.4425538060656697e+108)
       (* -0.5 (/ (+ b (sqrt (+ (* b b) (* c (* a -4.0))))) a))
       (* -0.5 (/ (* b 2.0) 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 <= -2.6404428421693406e+54) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= -1.069698752870928e-294) {
		tmp = -0.5 * (4.0 * (c / (b - sqrt((b * b) - (4.0 * (c * a))))));
	} else if (b <= 2.4425538060656697e+108) {
		tmp = -0.5 * ((b + sqrt((b * b) + (c * (a * -4.0)))) / a);
	} else {
		tmp = -0.5 * ((b * 2.0) / 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

Original33.7
Target20.3
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 < -2.6404428421693406e54

    1. Initial program 57.6

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

    2. Simplified57.6

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

    3. Taylor expanded around -inf 3.5

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)} \]

    if -2.6404428421693406e54 < b < -1.06969875287092807e-294

    1. Initial program 29.4

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

    2. Simplified29.4

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

    4. Applied flip-+_binary6429.4

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

    5. Simplified16.9

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

    6. Simplified16.9

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

    8. Applied *-un-lft-identity_binary6416.9

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

    9. Applied *-un-lft-identity_binary6416.9

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

    10. Applied times-frac_binary6416.9

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

    11. Applied times-frac_binary6416.9

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

    12. Simplified16.9

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

    13. Simplified9.1

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

    if -1.06969875287092807e-294 < b < 2.4425538060656697e108

    1. Initial program 8.8

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

    2. Simplified8.8

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

    4. Applied sub-neg_binary648.8

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

    5. Simplified8.8

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

    if 2.4425538060656697e108 < b

    1. Initial program 48.3

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

    2. Simplified48.3

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

    3. Taylor expanded around inf 3.8

      \[\leadsto -0.5 \cdot \frac{\color{blue}{2 \cdot b}}{a} \]

    4. Simplified3.8

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b \cdot 2}}{a} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6404428421693406 \cdot 10^{+54}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -1.069698752870928 \cdot 10^{-294}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 2.4425538060656697 \cdot 10^{+108}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))