Cubic critical, medium range

Average Error: 43.6 → 0.6

Time: 18.0s

Precision: binary64

Cost: 14144

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (*
  (/ (/ (* 3.0 (* c a)) (+ b (sqrt (fma c (* a -3.0) (* b b))))) -3.0)
  (/ 1.0 a)))

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 (((3.0 * (c * a)) / (b + sqrt(fma(c, (a * -3.0), (b * b))))) / -3.0) * (1.0 / a);
}

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(Float64(Float64(3.0 * Float64(c * a)) / Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))) / -3.0) * Float64(1.0 / a))
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[(N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.6

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

2. Applied egg-rr 43.6

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

3. Simplified 43.6

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

   [Start] 43.6	\[ \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{-3} \cdot \frac{1}{a} \]
   fma-def [<=] 43.6	\[ \frac{b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{-3} \cdot \frac{1}{a} \]
   +-commutative [=>] 43.6	\[ \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}}{-3} \cdot \frac{1}{a} \]
   fma-def [=>] 43.6	\[ \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{-3} \cdot \frac{1}{a} \]

4. Applied egg-rr 43.0

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

5. Simplified 43.0

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

   [Start] 43.0	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{-3} \cdot \frac{1}{a} \]
   associate-*r/ [=>] 43.0	\[ \frac{\color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot 1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}}{-3} \cdot \frac{1}{a} \]
   *-rgt-identity [=>] 43.0	\[ \frac{\frac{\color{blue}{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{-3} \cdot \frac{1}{a} \]

6. Taylor expanded in b around 0 0.6

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

7. Final simplification 0.6

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

Alternatives

Alternative 1
Error	4.3
Cost	8448

\[\frac{\frac{-0.3333333333333333}{a}}{\left(0.6666666666666666 \cdot \frac{b}{c \cdot a} - \frac{\left(c \cdot a\right) \cdot -0.375 + \left(c \cdot a\right) \cdot 0.75}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}} \]

Alternative 2
Error	6.2
Cost	7232

\[\frac{\frac{-0.3333333333333333}{a}}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c \cdot a}, \frac{-0.5}{b}\right)} \]

Alternative 3
Error	6.2
Cost	1088

\[\frac{\frac{-0.3333333333333333}{a}}{0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{-1}{b}} \]

Alternative 4
Error	6.2
Cost	960

\[\frac{\frac{-0.3333333333333333}{a}}{\frac{b \cdot 0.6666666666666666}{c \cdot a} - \frac{0.5}{b}} \]

Alternative 5
Error	12.2
Cost	320

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

Reproduce

herbie shell --seed 2023027 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))