Average Error: 34.4 → 6.4

Time: 13.4s

Precision: binary64

Cost: 8131

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.575768253013841 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -5.584163476997717 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.51778818962992 \cdot 10^{+131}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.575768253013841 \cdot 10^{+151}:\\
\;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\

\mathbf{elif}\;b_2 \leq -5.584163476997717 \cdot 10^{-250}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq 4.51778818962992 \cdot 10^{+131}:\\
\;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\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 -1.575768253013841e+151)
   (/ (- (- b_2) b_2) a)
   (if (<= b_2 -5.584163476997717e-250)
     (- (/ (sqrt (- (* b_2 b_2) (* a c))) a) (/ b_2 a))
     (if (<= b_2 4.51778818962992e+131)
       (/ (- c) (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))))
       (* -0.5 (/ c b_2))))))

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 <= -1.575768253013841e+151) {
		tmp = (-b_2 - b_2) / a;
	} else if (b_2 <= -5.584163476997717e-250) {
		tmp = (sqrt((b_2 * b_2) - (a * c)) / a) - (b_2 / a);
	} else if (b_2 <= 4.51778818962992e+131) {
		tmp = -c / (b_2 + sqrt((b_2 * b_2) - (a * c)));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `a`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	39.1
Cost	73280

\[\frac{\frac{\sqrt{-a \cdot c}}{\sqrt[3]{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt{-a \cdot c}}{\sqrt[3]{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{\sqrt[3]{a}}\]

Alternative 2
Error	32.6
Cost	60160

\[\frac{\frac{a}{\sqrt[3]{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{-c}{\sqrt[3]{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{\sqrt[3]{a}}\]

Alternative 3
Error	35.0
Cost	59840

\[\frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a}}\]

Alternative 4
Error	35.1
Cost	46400

\[\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a}}\]

Alternative 5
Error	34.9
Cost	40640

\[\sqrt[3]{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \cdot \left(\sqrt[3]{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \cdot \sqrt[3]{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\right)\]

Alternative 6
Error	34.9
Cost	40384

\[\frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \left(\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)}{a}\]

Alternative 7
Error	34.9
Cost	40384

\[\left(\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right) \cdot \frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

Alternative 8
Error	36.1
Cost	40128

\[\frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right) - b_2}{a}\]

Alternative 9
Error	53.3
Cost	33664

\[\frac{\frac{\sqrt{{b_2}^{6} - {\left(a \cdot c\right)}^{3}}}{\sqrt{{b_2}^{4} + a \cdot \left(c \cdot \left(a \cdot c + b_2 \cdot b_2\right)\right)}} - b_2}{a}\]

Alternative 10
Error	46.5
Cost	27584

\[\frac{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3} - {b_2}^{3}}{a \cdot \left(b_2 \cdot \left(b_2 + \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) - a \cdot c\right)}\]

Alternative 11
Error	35.7
Cost	27328

\[\frac{\sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)} - b_2}{a}\]

Alternative 12
Error	37.8
Cost	27264

\[\frac{a}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{\frac{-c}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

Alternative 13
Error	53.3
Cost	27264

\[\frac{\sqrt{\frac{{b_2}^{6} - {\left(a \cdot c\right)}^{3}}{{b_2}^{4} + a \cdot \left(c \cdot \left(a \cdot c + b_2 \cdot b_2\right)\right)}} - b_2}{a}\]

Alternative 14
Error	48.1
Cost	27072

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

Alternative 15
Error	34.7
Cost	26944

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

Alternative 16
Error	34.7
Cost	26944

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

Alternative 17
Error	34.7
Cost	26944

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

Alternative 18
Error	35.7
Cost	26816

\[\frac{\left|\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a}\]

Alternative 19
Error	35.2
Cost	26816

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

Alternative 20
Error	34.9
Cost	26688

\[\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\sqrt[3]{a}}\]

Alternative 21
Error	34.9
Cost	26560

\[\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}\]

Alternative 22
Error	47.3
Cost	21184

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

Alternative 23
Error	38.6
Cost	20288

\[\frac{\frac{-a \cdot c}{b_2 + \sqrt[3]{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}}}{a}\]

Alternative 24
Error	49.1
Cost	20160

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

Alternative 25
Error	52.1
Cost	19968

\[\frac{\sqrt{\sqrt[3]{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{3}}} - b_2}{a}\]

Alternative 26
Error	42.8
Cost	19968

\[\frac{\sqrt[3]{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}} - b_2}{a}\]

Alternative 27
Error	37.7
Cost	19904

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

Alternative 28
Error	61.5
Cost	19904

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

Alternative 29
Error	49.1
Cost	19904

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

Alternative 30
Error	36.4
Cost	19904

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

Alternative 31
Error	60.3
Cost	19904

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

Alternative 32
Error	35.3
Cost	7488

\[\left(a \cdot c\right) \cdot \frac{\frac{-1}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

Alternative 33
Error	32.6
Cost	7424

\[\frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

Alternative 34
Error	35.4
Cost	7424

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

Alternative 35
Error	29.8
Cost	7424

\[\frac{1}{\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-c}}\]

Alternative 36
Error	31.7
Cost	7424

\[\frac{a \cdot \frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

Alternative 37
Error	29.8
Cost	7232

\[\frac{-1}{\frac{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\]

Alternative 38
Error	34.6
Cost	7232

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

Alternative 39
Error	34.4
Cost	7232

\[\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\]

Alternative 40
Error	41.7
Cost	7232

\[\frac{\frac{-a \cdot c}{\sqrt{-a \cdot c} + b_2}}{a}\]

Alternative 41
Error	34.4
Cost	7232

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

Alternative 42
Error	29.6
Cost	7232

\[c \cdot \frac{-1}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

Alternative 43
Error	44.4
Cost	7232

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

Alternative 44
Error	29.6
Cost	7168

\[\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

Alternative 45
Error	46.3
Cost	7104

\[\frac{\frac{-a \cdot c}{\sqrt{-a \cdot c}}}{a}\]

Alternative 46
Error	34.4
Cost	7104

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

Alternative 47
Error	48.8
Cost	7104

\[\frac{-a \cdot c}{a \cdot \sqrt{-a \cdot c}}\]

Alternative 48
Error	44.9
Cost	6912

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

Alternative 49
Error	46.1
Cost	6848

\[\frac{-c}{\sqrt{-a \cdot c}}\]

Alternative 50
Error	44.3
Cost	6784

\[\frac{\sqrt{-a \cdot c}}{a}\]

Alternative 51
Error	44.6
Cost	1152

\[\frac{\frac{-a \cdot c}{b_2 + \left(b_2 - 0.5 \cdot \frac{a \cdot c}{b_2}\right)}}{a}\]

Alternative 52
Error	56.1
Cost	832

\[\frac{\left(b_2 + \frac{c}{\frac{b_2}{a}} \cdot -0.5\right) - b_2}{a}\]

Alternative 53
Error	45.2
Cost	576

\[\frac{\frac{c}{\frac{b_2}{a}} \cdot -0.5}{a}\]

Alternative 54
Error	45.3
Cost	384

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

Alternative 55
Error	45.3
Cost	320

\[\frac{b_2 \cdot -2}{a}\]

Alternative 56
Error	56.0
Cost	320

\[\frac{b_2 - b_2}{a}\]

Alternative 57
Error	39.4
Cost	320

\[-0.5 \cdot \frac{c}{b_2}\]

Alternative 58
Error	61.6
Cost	64

\[1\]

Alternative 59
Error	56.0
Cost	64

\[0\]

Alternative 60
Error	61.6
Cost	64

\[-1\]

Derivation

Split input into 4 regimes
if b_2 < -1.5757682530138411e151
1. Initial program 62.7
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified62.7
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
3. Taylor expanded around -inf 2.9
  \[\leadsto \frac{\color{blue}{-1 \cdot b_2} - b_2}{a}\]
4. Simplified2.9
  \[\leadsto \frac{\color{blue}{\left(-b_2\right)} - b_2}{a}\]
5. Simplified2.9
  \[\leadsto \color{blue}{\frac{\left(-b_2\right) - b_2}{a}}\]
if -1.5757682530138411e151 < b_2 < -5.5841634769977169e-250
1. Initial program 7.9
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified7.9
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
3. Using strategy rm
4. Applied div-sub_binary647.9
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}}\]
5. Simplified7.9
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}}\]
if -5.5841634769977169e-250 < b_2 < 4.5177881896299201e131
1. Initial program 32.2
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified32.2
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
3. Using strategy rm
4. Applied flip--_binary6432.3
  \[\leadsto \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}{a}\]
5. Simplified16.0
  \[\leadsto \frac{\frac{\color{blue}{-a \cdot c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}{a}\]
6. Simplified16.0
  \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
7. Using strategy rm
8. Applied distribute-frac-neg_binary6416.0
  \[\leadsto \frac{\color{blue}{-\frac{a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
9. Applied distribute-frac-neg_binary6416.0
  \[\leadsto \color{blue}{-\frac{\frac{a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}}\]
10. Simplified8.7
  \[\leadsto -\color{blue}{\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]
11. Simplified8.7
  \[\leadsto \color{blue}{\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]
if 4.5177881896299201e131 < b_2
1. Initial program 61.8
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified61.8
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
3. Taylor expanded around inf 2.1
  \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
4. Simplified2.1
  \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
Recombined 4 regimes into one program.
Final simplification6.4
\[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.575768253013841 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -5.584163476997717 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.51778818962992 \cdot 10^{+131}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

Error

Try it out

Alternatives

Error

Derivation

`if b_2 < -1.5757682530138411e151`

`if -1.5757682530138411e151 < b_2 < -5.5841634769977169e-250`

`if -5.5841634769977169e-250 < b_2 < 4.5177881896299201e131`

`if 4.5177881896299201e131 < b_2`

Reproduce