Average Error: 33.5 → 9.8

Time: 16.9s

Precision: binary64

Cost: 7746

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.423389220310686 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.891425723699142 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;b_2 \leq -3.423389220310686 \cdot 10^{+150}:\\
\;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 3.891425723699142 \cdot 10^{-38}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot -0.5}{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 -3.423389220310686e+150)
   (/ (- (- b_2) b_2) a)
   (if (<= b_2 3.891425723699142e-38)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) 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 <= -3.423389220310686e+150) {
		tmp = (-b_2 - b_2) / a;
	} else if (b_2 <= 3.891425723699142e-38) {
		tmp = (sqrt((b_2 * b_2) - (a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / 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	34.1
Cost	59840

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

Alternative 2
Error	34.3
Cost	46400

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

Alternative 3
Error	34.0
Cost	40640

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

Alternative 4
Error	34.0
Cost	40384

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

Alternative 5
Error	34.0
Cost	40384

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

Alternative 6
Error	35.2
Cost	40128

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

Alternative 7
Error	40.3
Cost	39232

\[\left(-0.5 \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{b_2}}\]

Alternative 8
Error	53.1
Cost	33664

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

Alternative 9
Error	40.9
Cost	32704

\[\left(-0.5 \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt{b_2}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt{b_2}}\]

Alternative 10
Error	42.4
Cost	27584

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

Alternative 11
Error	34.7
Cost	27328

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

Alternative 12
Error	53.0
Cost	27264

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

Alternative 13
Error	47.8
Cost	27072

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

Alternative 14
Error	33.9
Cost	26944

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

Alternative 15
Error	33.9
Cost	26944

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

Alternative 16
Error	33.9
Cost	26944

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

Alternative 17
Error	34.4
Cost	26816

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

Alternative 18
Error	34.8
Cost	26816

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

Alternative 19
Error	34.0
Cost	26688

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

Alternative 20
Error	57.6
Cost	20608

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

Alternative 21
Error	48.9
Cost	20160

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

Alternative 22
Error	42.5
Cost	20096

\[\frac{1}{\frac{a}{\sqrt[3]{{\left(\sqrt{b_2 \cdot b_2 - c \cdot a}\right)}^{3}} - b_2}}\]

Alternative 23
Error	48.8
Cost	20032

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

Alternative 24
Error	42.5
Cost	19968

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

Alternative 25
Error	49.9
Cost	19968

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

Alternative 26
Error	51.9
Cost	19968

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

Alternative 27
Error	61.6
Cost	19904

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

Alternative 28
Error	48.9
Cost	19904

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

Alternative 29
Error	36.8
Cost	19904

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

Alternative 30
Error	35.7
Cost	19904

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

Alternative 31
Error	60.5
Cost	19904

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

Alternative 32
Error	40.2
Cost	19776

\[\left(-0.5 \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \frac{\sqrt[3]{c}}{b_2}\]

Alternative 33
Error	40.2
Cost	19776

\[\frac{-0.5}{\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}} \cdot \frac{c}{\sqrt[3]{b_2}}\]

Alternative 34
Error	45.7
Cost	14016

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

Alternative 35
Error	49.6
Cost	13504

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

Alternative 36
Error	50.3
Cost	13184

\[\sqrt[3]{\frac{-0.125}{{\left(\frac{b_2}{c}\right)}^{3}}}\]

Alternative 37
Error	50.2
Cost	13120

\[e^{\log \left(-0.5 \cdot \frac{c}{b_2}\right)}\]

Alternative 38
Error	43.6
Cost	8000

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

Alternative 39
Error	43.6
Cost	8000

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

Alternative 40
Error	43.6
Cost	7872

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

Alternative 41
Error	43.6
Cost	7872

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

Alternative 42
Error	49.6
Cost	7808

\[\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}} - 0.125 \cdot \frac{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}{{b_2}^{3}}}{a}\]

Alternative 43
Error	33.6
Cost	7360

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

Alternative 44
Error	33.8
Cost	7232

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

Alternative 45
Error	33.6
Cost	7232

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

Alternative 46
Error	33.6
Cost	7232

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

Alternative 47
Error	33.5
Cost	7104

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

Alternative 48
Error	44.2
Cost	7040

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

Alternative 49
Error	43.5
Cost	6912

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

Alternative 50
Error	44.2
Cost	6912

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

Alternative 51
Error	43.5
Cost	6784

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

Alternative 52
Error	56.4
Cost	832

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

Alternative 53
Error	45.8
Cost	832

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

Alternative 54
Error	45.7
Cost	704

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

Alternative 55
Error	45.5
Cost	576

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

Alternative 56
Error	45.6
Cost	512

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

Alternative 57
Error	45.6
Cost	448

\[\frac{1}{\frac{a}{b_2 \cdot -2}}\]

Alternative 58
Error	45.5
Cost	384

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

Alternative 59
Error	39.8
Cost	320

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

Alternative 60
Error	45.5
Cost	320

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

Alternative 61
Error	56.3
Cost	320

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

Alternative 62
Error	39.8
Cost	320

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

Alternative 63
Error	61.6
Cost	64

\[1\]

Alternative 64
Error	56.3
Cost	64

\[0\]

Alternative 65
Error	61.6
Cost	64

\[-1\]

Error

Derivation

Split input into 3 regimes
if b_2 < -3.4233892203106858e150
1. Initial program 62.0
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified62.0
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
3. Taylor expanded around -inf 2.7
  \[\leadsto \frac{\color{blue}{-1 \cdot b_2} - b_2}{a}\]
4. Simplified2.7
  \[\leadsto \frac{\color{blue}{\left(-b_2\right)} - b_2}{a}\]
5. Simplified2.7
  \[\leadsto \color{blue}{\frac{\left(-b_2\right) - b_2}{a}}\]
if -3.4233892203106858e150 < b_2 < 3.891425723699142e-38
1. Initial program 12.7
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified12.7
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}}\]
if 3.891425723699142e-38 < b_2
1. Initial program 54.5
  \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Simplified54.5
  \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
3. Taylor expanded around inf 7.7
  \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
4. Using strategy rm
5. Applied associate-*r/_binary647.7
  \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}}\]
6. Simplified7.7
  \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}}\]
Recombined 3 regimes into one program.
Final simplification9.8
\[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.423389220310686 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.891425723699142 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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 < -3.4233892203106858e150`

`if -3.4233892203106858e150 < b_2 < 3.891425723699142e-38`

`if 3.891425723699142e-38 < b_2`

Reproduce