Average Error: 28.4 → 5.8

Time: 10.3s

Precision: binary64

Cost: 34432

\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]

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

↓

\[\left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5\]

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

↓

\left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (+
  (-
   (-
    (* -2.0 (/ (* c (* (* a a) (* c c))) (pow b 5.0)))
    (/ (* a (* c c)) (pow b 3.0)))
   (/ c b))
  (* (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)) -5.0)))

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) {
	return (((-2.0 * ((c * ((a * a) * (c * c))) / pow(b, 5.0))) - ((a * (c * c)) / pow(b, 3.0))) - (c / b)) + (((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)) * -5.0);
}

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	7.0
Cost	14849

\[\begin{array}{l} \mathbf{if}\;b \leq 11.560297428247756:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array}\]

Alternative 2
Error	9.6
Cost	8577

\[\begin{array}{l} \mathbf{if}\;b \leq 13.36790805006234:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array}\]

Alternative 3
Error	9.8
Cost	7809

\[\begin{array}{l} \mathbf{if}\;b \leq 13.649810967366726:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array}\]

Alternative 4
Error	9.8
Cost	7681

\[\begin{array}{l} \mathbf{if}\;b \leq 13.36790805006234:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array}\]

Alternative 5
Error	11.9
Cost	7232

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

Alternative 6
Error	22.9
Cost	256

\[\frac{-c}{b}\]

Alternative 7
Error	55.6
Cost	64

\[-1\]

Alternative 8
Error	62.0
Cost	64

\[0\]

Alternative 9
Error	63.0
Cost	64

\[1\]

Derivation

Initial program 28.4
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Simplified28.4
\[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\]
Taylor expanded around inf 5.8
\[\leadsto \color{blue}{-\left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(\frac{c}{b} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} + 2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)}\]
Simplified5.8
\[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5}\]
Using strategy rm
Applied unpow3_binary64_4855.8
\[\leadsto \left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot c\right)}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5\]
Applied associate-*r*_binary64_3595.8
\[\leadsto \left(\left(-2 \cdot \frac{\color{blue}{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot c}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5\]
Simplified5.8
\[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5}\]
Final simplification5.8
\[\leadsto \left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5\]

Reproduce

herbie shell --seed 2021044 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))