Cubic critical, narrow range

Average Error: 28.9 → 0.3

Time: 19.8s

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 28.9

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

2. Simplified 28.9

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

   [Start] 28.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 28.9	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 28.9	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 28.9	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 28.9	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 28.9	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 28.9	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 28.9	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]
   neg-mul-1 [=>] 28.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \cdot \frac{-1}{3} \]
   neg-sub0 [=>] 28.9	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   associate-+l- [=>] 28.9	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   sub0-neg [=>] 28.9	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   neg-mul-1 [=>] 28.9	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   times-frac [=>] 28.9	\[ \color{blue}{\left(\frac{-1}{-1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\right)} \cdot \frac{-1}{3} \]
   metadata-eval [=>] 28.9	\[ \left(\color{blue}{1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\right) \cdot \frac{-1}{3} \]
   *-lft-identity [=>] 28.9	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}} \cdot \frac{-1}{3} \]
   cancel-sign-sub-inv [=>] 28.9	\[ \frac{b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}}{a} \cdot \frac{-1}{3} \]
   +-commutative [=>] 28.9	\[ \frac{b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}}{a} \cdot \frac{-1}{3} \]
   *-commutative [=>] 28.9	\[ \frac{b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   distribute-lft-neg-in [=>] 28.9	\[ \frac{b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   associate-*r* [=>] 28.9	\[ \frac{b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   *-commutative [=>] 28.9	\[ \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   fma-def [=>] 28.9	\[ \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}}{a} \cdot \frac{-1}{3} \]
   metadata-eval [=>] 28.9	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}}{a} \cdot \frac{-1}{3} \]
   metadata-eval [=>] 28.9	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \color{blue}{-0.3333333333333333} \]

3. Applied egg-rr 29.5

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

4. Taylor expanded in b around 0 0.6

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

5. Taylor expanded in c around 0 0.4

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

6. Simplified 0.4

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

   [Start] 0.4	\[ \frac{-1 \cdot \frac{c}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-*r/ [=>] 0.4	\[ \frac{\color{blue}{\frac{-1 \cdot c}{a}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   neg-mul-1 [<=] 0.4	\[ \frac{\frac{\color{blue}{-c}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]

7. Applied egg-rr 16.7

   \[\leadsto \color{blue}{\left(\frac{0}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} - e^{\mathsf{log1p}\left(\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right)}\right) + 1} \]

8. Simplified 0.3

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

   [Start] 16.7	\[ \left(\frac{0}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} - e^{\mathsf{log1p}\left(\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right)}\right) + 1 \]
   associate-+l- [=>] 16.7	\[ \color{blue}{\frac{0}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} - \left(e^{\mathsf{log1p}\left(\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right)} - 1\right)} \]
   div0 [=>] 16.7	\[ \color{blue}{0} - \left(e^{\mathsf{log1p}\left(\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right)} - 1\right) \]
   expm1-def [=>] 0.5	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right)\right)} \]
   expm1-log1p [=>] 0.5	\[ 0 - \color{blue}{\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}} \]
   neg-sub0 [<=] 0.5	\[ \color{blue}{-\frac{c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}} \]
   distribute-neg-frac [=>] 0.5	\[ \color{blue}{\frac{-c}{a \cdot \left(\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}} \]
   associate-*r* [=>] 0.4	\[ \frac{-c}{\color{blue}{\left(a \cdot \frac{1}{a}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   rgt-mult-inverse [=>] 0.3	\[ \frac{-c}{\color{blue}{1} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   *-lft-identity [=>] 0.3	\[ \frac{-c}{\color{blue}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	7680

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

Alternative 2
Error	9.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 5.5:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 3
Error	9.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 5.8:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right) \cdot -0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 4
Error	9.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 5.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 5
Error	11.4
Cost	832

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

Alternative 6
Error	22.6
Cost	320

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

Reproduce

herbie shell --seed 2022356 
(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)))