Cubic critical, narrow range

Average Error: 28.7 → 0.7

Time: 22.3s

Precision: binary64

Cost: 27712

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

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

↓

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

C

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 sqrt(((a * (c * 3.0)) + ((b * b) - (b * b)))) / (((a * -3.0) * (b + sqrt(fma(a, (c * -3.0), (b * b))))) / sqrt((3.0 * (a * c))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	return Float64(sqrt(Float64(Float64(a * Float64(c * 3.0)) + Float64(Float64(b * b) - Float64(b * b)))) / Float64(Float64(Float64(a * -3.0) * Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b))))) / sqrt(Float64(3.0 * Float64(a * c)))))
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[Sqrt[N[(N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.7

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

2. Simplified 28.7

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
   Proof

   [Start] 28.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 28.7	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 28.7	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 29.2	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 29.2	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 29.3	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 29.3	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 28.7	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 28.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 28.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 28.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 28.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 28.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 29.3	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 29.3	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   associate-*r/ [=>] 29.3	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1 \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   associate-*l/ [<=] 29.2	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1}{3 \cdot a} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \]
   distribute-lft-out-- [=>] 28.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
   *-commutative [=>] 28.7	\[ \color{blue}{\left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a}} \]
   neg-mul-1 [=>] 28.7	\[ \left(\frac{\color{blue}{-1 \cdot b}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
   *-commutative [=>] 28.7	\[ \left(\frac{\color{blue}{b \cdot -1}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
   associate-/l* [=>] 28.7	\[ \left(\color{blue}{\frac{b}{\frac{-1}{-1}}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
   metadata-eval [=>] 28.7	\[ \left(\frac{b}{\color{blue}{1}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
   /-rgt-identity [=>] 28.7	\[ \left(\color{blue}{b} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
   cancel-sign-sub-inv [=>] 28.7	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}\right) \cdot \frac{-1}{3 \cdot a} \]
   +-commutative [=>] 28.7	\[ \left(b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}\right) \cdot \frac{-1}{3 \cdot a} \]
   *-commutative [=>] 28.7	\[ \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
   distribute-lft-neg-in [=>] 28.7	\[ \left(b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
   associate-*r* [=>] 28.7	\[ \left(b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
   *-commutative [=>] 28.7	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
   fma-def [=>] 28.7	\[ \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}\right) \cdot \frac{-1}{3 \cdot a} \]
   metadata-eval [=>] 28.7	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{-1}{3 \cdot a} \]
   associate-/r* [=>] 28.7	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{-1}{3}}{a}} \]
   metadata-eval [=>] 28.7	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{-0.3333333333333333}}{a} \]

3. Applied egg-rr 27.8

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]

4. Taylor expanded in b around 0 27.0

   \[\leadsto \frac{\sqrt{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt{\color{blue}{3 \cdot \left(c \cdot a\right)}}}} \]

5. Applied egg-rr 0.7

   \[\leadsto \frac{\sqrt{\color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt{3 \cdot \left(c \cdot a\right)}}} \]

6. Final simplification 0.7

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

Alternatives

Alternative 1
Error	8.8
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.023:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{b \cdot b - t_0}{a}}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	8.8
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.023:\\ \;\;\;\;\left(b \cdot b - t_0\right) \cdot \frac{\frac{-0.3333333333333333}{a}}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	27200

\[\begin{array}{l} t_0 := \sqrt{3 \cdot \left(a \cdot c\right)}\\ \frac{t_0}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{t_0}} \end{array} \]

Alternative 4
Error	9.0
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.023:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	9.0
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.023:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	9.0
Cost	14852

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.023:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	9.4
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -3\right) + b \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.5}{b} + -0.375 \cdot \left(c \cdot \frac{a}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 8
Error	9.4
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.5}{b} + -0.375 \cdot \left(c \cdot \frac{a}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 9
Error	11.7
Cost	7296

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

Alternative 10
Error	22.7
Cost	320

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

Alternative 11
Error	22.6
Cost	320

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

Reproduce

herbie shell --seed 2022349 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))