Cubic critical, narrow range

Average Error: 44.32% → 0.44%

Time: 21.9s

Precision: binary64

Cost: 13568

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

Math

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

↓

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

FPCore

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

↓

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

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

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(Float64(-c) / Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))))
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[((-c) / N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.32

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

2. Simplified 44.32

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

   [Start] 44.32	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 44.32	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 44.32	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 44.32	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 44.32	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 44.32	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 44.32	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 44.32	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 45.11

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

4. Simplified 45.1

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

   [Start] 45.11	\[ \frac{\left(b \cdot a - a \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot -0.3333333333333333}{a \cdot a} \]
   times-frac [=>] 45.1	\[ \color{blue}{\frac{b \cdot a - a \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{-0.3333333333333333}{a}} \]
   *-commutative [=>] 45.1	\[ \frac{\color{blue}{a \cdot b} - a \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{-0.3333333333333333}{a} \]

5. Applied egg-rr 44.29

   \[\leadsto \color{blue}{\frac{\left({\left(a \cdot b\right)}^{2} - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) \cdot \left(a \cdot a\right)\right) \cdot -0.3333333333333333}{\left(a \cdot a\right) \cdot \left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}} \]

6. Taylor expanded in a around 0 0.92

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(c \cdot {a}^{3}\right)}}{\left(a \cdot a\right) \cdot \left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \]

7. Simplified 0.92

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

   [Start] 0.92	\[ \frac{-1 \cdot \left(c \cdot {a}^{3}\right)}{\left(a \cdot a\right) \cdot \left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \]
   mul-1-neg [=>] 0.92	\[ \frac{\color{blue}{-c \cdot {a}^{3}}}{\left(a \cdot a\right) \cdot \left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \]
   distribute-rgt-neg-in [=>] 0.92	\[ \frac{\color{blue}{c \cdot \left(-{a}^{3}\right)}}{\left(a \cdot a\right) \cdot \left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \]

8. Applied egg-rr 0.71

   \[\leadsto \color{blue}{\frac{c}{\frac{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-1}} \cdot {a}^{3}} \]

9. Simplified 0.44

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

   [Start] 0.71	\[ \frac{c}{\frac{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-1}} \cdot {a}^{3} \]
   associate-*l/ [=>] 0.7	\[ \color{blue}{\frac{c \cdot {a}^{3}}{\frac{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-1}}} \]
   associate-/r/ [=>] 0.7	\[ \color{blue}{\frac{c \cdot {a}^{3}}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot -1} \]
   *-commutative [=>] 0.7	\[ \frac{\color{blue}{{a}^{3} \cdot c}}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot -1 \]
   associate-*l/ [<=] 0.71	\[ \color{blue}{\left(\frac{{a}^{3}}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot c\right)} \cdot -1 \]
   *-commutative [<=] 0.71	\[ \color{blue}{-1 \cdot \left(\frac{{a}^{3}}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot c\right)} \]
   neg-mul-1 [<=] 0.71	\[ \color{blue}{-\frac{{a}^{3}}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot c} \]
   associate-*l/ [=>] 0.7	\[ -\color{blue}{\frac{{a}^{3} \cdot c}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   *-commutative [<=] 0.7	\[ -\frac{\color{blue}{c \cdot {a}^{3}}}{{a}^{3} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   associate-/r* [=>] 0.54	\[ -\color{blue}{\frac{\frac{c \cdot {a}^{3}}{{a}^{3}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-/l* [=>] 0.44	\[ -\frac{\color{blue}{\frac{c}{\frac{{a}^{3}}{{a}^{3}}}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   *-inverses [=>] 0.44	\[ -\frac{\frac{c}{\color{blue}{1}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   /-rgt-identity [=>] 0.44	\[ -\frac{\color{blue}{c}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]

10. Final simplification 0.44

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

Alternatives

Alternative 1
Error	10.49%
Cost	16004

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 3} \leq -0.395:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(c \cdot \left(a \cdot -3\right) - \left(t_0 + -3 \cdot \left(c \cdot a\right)\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(a \cdot 3\right) \cdot \left(\left(\frac{\left(c \cdot a\right) \cdot 0.375 + \left(c \cdot a\right) \cdot -0.75}{{b}^{3}} + 0.6666666666666666 \cdot \frac{b}{c \cdot a}\right) + 0.5 \cdot \frac{-1}{b}\right)}\\ \end{array} \]

Alternative 2
Error	15.27%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 8.8:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1.5 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 3
Error	18.15%
Cost	832

\[\frac{-1}{-1.5 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}} \]

Alternative 4
Error	87.94%
Cost	320

\[-0.1111111111111111 \cdot \frac{b}{a} \]

Alternative 5
Error	35.88%
Cost	320

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

Alternative 6
Error	35.82%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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)))