Cubic critical, medium range

Average Error: 68.48% → 0.63%

Time: 20.1s

Precision: binary64

Cost: 14528

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

Math

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

↓

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

FPCore

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

↓

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

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

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

Derivation

1. Initial program 68.48

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

2. Applied egg-rr 68.43

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

3. Simplified 68.48

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

   [Start] 68.43	\[ \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{-3} \cdot \frac{1}{a} \]
   fma-def [<=] 68.48	\[ \frac{b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{-3} \cdot \frac{1}{a} \]
   +-commutative [=>] 68.48	\[ \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}}{-3} \cdot \frac{1}{a} \]
   fma-def [=>] 68.48	\[ \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{-3} \cdot \frac{1}{a} \]

4. Applied egg-rr 67.56

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

5. Simplified 67.55

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

   [Start] 67.56	\[ \frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\frac{a \cdot -3}{\frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}} \]
   associate-/r/ [=>] 67.55	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot -3} \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]
   associate-*l/ [=>] 67.56	\[ \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{a \cdot -3}} \]
   associate-*r/ [=>] 67.55	\[ \frac{\color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot 1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}}{a \cdot -3} \]
   *-rgt-identity [=>] 67.55	\[ \frac{\frac{\color{blue}{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{a \cdot -3} \]

6. Applied egg-rr 0.63

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

7. Final simplification 0.63

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

Alternatives

Alternative 1
Error	0.85%
Cost	14016

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

Alternative 2
Error	9.41%
Cost	13696

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

Alternative 3
Error	9.76%
Cost	7680

\[\left(\frac{c}{b} \cdot 1.5\right) \cdot -0.3333333333333333 + -0.3333333333333333 \cdot \left(\left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} \cdot 1.125\right)\right) \]

Alternative 4
Error	9.77%
Cost	7616

\[-0.3333333333333333 \cdot \mathsf{fma}\left(\frac{c \cdot c}{b \cdot \frac{b \cdot b}{a}}, 1.125, \frac{c}{b} \cdot 1.5\right) \]

Alternative 5
Error	18.8%
Cost	320

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

Reproduce

herbie shell --seed 2023089 
(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)))