Average Error: 33.9 → 6.4
Time: 17.3s
Precision: binary64
Cost: 8131
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.6307928858729175 \cdot 10^{+125}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \leq -6.943016465332528 \cdot 10^{-294}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 1.6550855570993975 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.6307928858729175 \cdot 10^{+125}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) - b_2}\\

\mathbf{elif}\;b_2 \leq -6.943016465332528 \cdot 10^{-294}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\mathbf{elif}\;b_2 \leq 1.6550855570993975 \cdot 10^{+91}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\

\end{array}
(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.6307928858729175e+125)
   (/ c (- (- b_2) b_2))
   (if (<= b_2 -6.943016465332528e-294)
     (/ c (- (sqrt (- (* b_2 b_2) (* c a))) b_2))
     (if (<= b_2 1.6550855570993975e+91)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))))
double code(double a, double b_2, double c) {
	return (-b_2 - sqrt((b_2 * b_2) - (a * c))) / a;
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.6307928858729175e+125) {
		tmp = c / (-b_2 - b_2);
	} else if (b_2 <= -6.943016465332528e-294) {
		tmp = c / (sqrt((b_2 * b_2) - (c * a)) - b_2);
	} else if (b_2 <= 1.6550855570993975e+91) {
		tmp = (-b_2 - sqrt((b_2 * b_2) - (c * a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error34.5
Cost60032
\[\frac{\sqrt[3]{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\sqrt[3]{a}}\]
Alternative 2
Error34.4
Cost40576
\[\left(\sqrt[3]{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\right) \cdot \frac{\sqrt[3]{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
Alternative 3
Error35.5
Cost40192
\[\frac{\left(-b_2\right) - \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}{a}\]
Alternative 4
Error53.1
Cost39872
\[\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} + \sqrt{-b_2}\right) \cdot \frac{\sqrt{-b_2} - \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
Alternative 5
Error53.3
Cost33728
\[\frac{\left(-b_2\right) - \frac{\sqrt{{b_2}^{6} - {\left(a \cdot c\right)}^{3}}}{\sqrt{{b_2}^{4} + a \cdot \left(c \cdot \left(b_2 \cdot b_2 + a \cdot c\right)\right)}}}{a}\]
Alternative 6
Error42.7
Cost27776
\[\frac{\frac{{\left(-b_2\right)}^{3} - {\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}{b_2 \cdot b_2 + \left(b_2 \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) - a \cdot c\right)}}{a}\]
Alternative 7
Error48.3
Cost27200
\[\sqrt{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \cdot \sqrt{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]
Alternative 8
Error43.4
Cost27072
\[\sqrt{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} \cdot \sqrt{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]
Alternative 9
Error35.2
Cost26880
\[\frac{\left(-b_2\right) - \left|\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a}\]
Alternative 10
Error34.8
Cost26880
\[\frac{\left(-b_2\right) - \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
Alternative 11
Error34.4
Cost26688
\[\frac{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}} \cdot \frac{-1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\]
Alternative 12
Error34.4
Cost26624
\[\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}\]
Alternative 13
Error48.7
Cost20160
\[\frac{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt{a}} \cdot \frac{-1}{\sqrt{a}}\]
Alternative 14
Error49.7
Cost20032
\[\sqrt[3]{{\left(\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)}^{3}}\]
Alternative 15
Error42.8
Cost20032
\[\frac{\left(-b_2\right) - \sqrt[3]{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}}{a}\]
Alternative 16
Error60.3
Cost19968
\[\frac{\log \left(e^{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}{a}\]
Alternative 17
Error60.9
Cost19968
\[\log \left(e^{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\right)\]
Alternative 18
Error49.3
Cost19968
\[e^{\log \left(\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)}\]
Alternative 19
Error61.9
Cost19968
\[\frac{e^{\log \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]
Alternative 20
Error37.2
Cost19968
\[\frac{\left(-b_2\right) - e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]
Alternative 21
Error61.5
Cost19968
\[\frac{\left(-b_2\right) - \log \left(e^{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}{a}\]
Alternative 22
Error45.5
Cost7424
\[-0.5 \cdot \frac{c}{b_2} - 0.125 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b_2}^{3}}\]
Alternative 23
Error31.9
Cost7360
\[\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]
Alternative 24
Error31.1
Cost7360
\[\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]
Alternative 25
Error33.9
Cost7296
\[\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]
Alternative 26
Error29.2
Cost7232
\[\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\]
Alternative 27
Error34.0
Cost7232
\[\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{-1}{a}\]
Alternative 28
Error33.9
Cost7168
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
Alternative 29
Error41.1
Cost7168
\[\frac{\frac{a \cdot c}{\sqrt{-a \cdot c} - b_2}}{a}\]
Alternative 30
Error29.0
Cost7104
\[\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\]
Alternative 31
Error45.7
Cost7040
\[\frac{\frac{a \cdot c}{\sqrt{-a \cdot c}}}{a}\]
Alternative 32
Error44.4
Cost6976
\[\frac{\left(-b_2\right) - \sqrt{-a \cdot c}}{a}\]
Alternative 33
Error40.4
Cost6912
\[\frac{c}{\sqrt{-a \cdot c} - b_2}\]
Alternative 34
Error43.7
Cost6848
\[\frac{-\sqrt{-a \cdot c}}{a}\]
Alternative 35
Error45.5
Cost6784
\[\frac{c}{\sqrt{-a \cdot c}}\]
Alternative 36
Error40.5
Cost832
\[\frac{c}{\left(0.5 \cdot \frac{a \cdot c}{b_2} - b_2\right) - b_2}\]
Alternative 37
Error46.8
Cost832
\[\frac{\left(0.5 \cdot \frac{a \cdot c}{b_2} - b_2\right) - b_2}{a}\]
Alternative 38
Error46.8
Cost832
\[\frac{b_2 \cdot -2 + 0.5 \cdot \frac{a \cdot c}{b_2}}{a}\]
Alternative 39
Error45.6
Cost704
\[-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\]
Alternative 40
Error43.3
Cost640
\[\frac{a \cdot \frac{c}{\left(-b_2\right) - b_2}}{a}\]
Alternative 41
Error45.2
Cost576
\[\frac{-0.5 \cdot \frac{a \cdot c}{b_2}}{a}\]
Alternative 42
Error43.3
Cost576
\[\frac{a \cdot \frac{c}{b_2 \cdot -2}}{a}\]
Alternative 43
Error39.9
Cost384
\[\frac{c}{\left(-b_2\right) - b_2}\]
Alternative 44
Error45.4
Cost384
\[\frac{\left(-b_2\right) - b_2}{a}\]
Alternative 45
Error39.9
Cost320
\[\frac{c}{b_2 \cdot -2}\]
Alternative 46
Error40.0
Cost320
\[-0.5 \cdot \frac{c}{b_2}\]
Alternative 47
Error45.4
Cost320
\[\frac{b_2 \cdot -2}{a}\]
Alternative 48
Error56.3
Cost320
\[\frac{b_2 - b_2}{a}\]
Alternative 49
Error61.6
Cost64
\[1\]
Alternative 50
Error56.3
Cost64
\[0\]
Alternative 51
Error61.6
Cost64
\[-1\]

Error

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.6307928858729175e125

    1. Initial program 61.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--_binary6461.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified33.8

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified33.8

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity_binary6433.8

      \[\leadsto \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

    8. Applied *-un-lft-identity_binary6433.8

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{1 \cdot a}\]

    9. Applied times-frac_binary6433.8

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}}\]

    10. Simplified33.8

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

    11. Simplified32.3

      \[\leadsto 1 \cdot \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    12. Taylor expanded around -inf 1.7

      \[\leadsto 1 \cdot \frac{c}{\color{blue}{-1 \cdot b_2} - b_2}\]

    13. Simplified1.7

      \[\leadsto 1 \cdot \frac{c}{\color{blue}{\left(-b_2\right)} - b_2}\]

    14. Simplified1.7

      \[\leadsto \color{blue}{\frac{c}{\left(-b_2\right) - b_2}}\]

    if -1.6307928858729175e125 < b_2 < -6.94301646533252763e-294

    1. Initial program 34.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--_binary6434.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified15.7

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified15.7

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

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

      \[\leadsto \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

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

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{1 \cdot a}\]

    9. Applied times-frac_binary6415.7

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}}\]

    10. Simplified15.7

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

    11. Simplified8.3

      \[\leadsto 1 \cdot \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    12. Simplified8.3

      \[\leadsto \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    if -6.94301646533252763e-294 < b_2 < 1.6550855570993975e91

    1. Initial program 8.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified8.7

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    if 1.6550855570993975e91 < b_2

    1. Initial program 45.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.2

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    3. Simplified4.2

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}}\]

    4. Simplified4.2

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.6307928858729175 \cdot 10^{+125}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \leq -6.943016465332528 \cdot 10^{-294}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 1.6550855570993975 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))