Average Error: 28.7 → 5.7

Time: 11.3s

Precision: binary64

Cost: 34688

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

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

↓

\[\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}

↓

\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (-
  (-
   (* -0.5 (/ c b))
   (+
    (* 0.375 (/ (* a (* c c)) (pow b 3.0)))
    (* 0.5625 (/ (* (* a a) (pow c 3.0)) (pow b 5.0)))))
  (* 1.0546875 (/ (* (* a a) (* a (pow c 4.0))) (pow b 7.0)))))

double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	return ((-0.5 * (c / b)) - ((0.375 * ((a * (c * c)) / pow(b, 3.0))) + (0.5625 * (((a * a) * pow(c, 3.0)) / pow(b, 5.0))))) - (1.0546875 * (((a * a) * (a * pow(c, 4.0))) / pow(b, 7.0)));
}

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	7.7
Cost	20992

\[-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\]

Alternative 2
Error	9.5
Cost	8577

\[\begin{array}{l} \mathbf{if}\;b \leq 107.89175933158471:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array}\]

Alternative 3
Error	9.8
Cost	7745

\[\begin{array}{l} \mathbf{if}\;b \leq 99.99455083264182:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array}\]

Alternative 4
Error	16.9
Cost	7681

\[\begin{array}{l} \mathbf{if}\;b \leq 112.68706462491534:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 5
Error	16.9
Cost	7681

\[\begin{array}{l} \mathbf{if}\;b \leq 112.68706462491534:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 6
Error	16.9
Cost	7681

\[\begin{array}{l} \mathbf{if}\;b \leq 112.68706462491534:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 7
Error	22.6
Cost	320

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

Alternative 8
Error	55.7
Cost	64

\[-1\]

Alternative 9
Error	62.0
Cost	64

\[0\]

Alternative 10
Error	63.0
Cost	64

\[1\]

Derivation

Initial program 28.7
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
Simplified28.7
\[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]
Taylor expanded around inf 5.7
\[\leadsto \color{blue}{-\left(1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(0.5 \cdot \frac{c}{b} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)}\]
Simplified5.7
\[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}}\]
Using strategy rm
Applied unpow3_binary64_15085.7
\[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot {c}^{4}}{{b}^{7}}\]
Applied associate-*l*_binary64_13835.7
\[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot {c}^{4}\right)}}{{b}^{7}}\]
Simplified5.7
\[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}}\]
Final simplification5.7
\[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))