Cubic critical, medium range

Average Error: 43.7 → 0.6

Time: 13.9s

Precision: binary64

Cost: 14016

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

Math

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

↓

\[\begin{array}{l} t_0 := -3 \cdot \left(c \cdot a\right)\\ \frac{t_0}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}} \cdot \frac{0.3333333333333333}{a} \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -3.0 (* c a))))
   (* (/ t_0 (+ b (sqrt (fma b b t_0)))) (/ 0.3333333333333333 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) {
	double t_0 = -3.0 * (c * a);
	return (t_0 / (b + sqrt(fma(b, b, t_0)))) * (0.3333333333333333 / 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)
	t_0 = Float64(-3.0 * Float64(c * a))
	return Float64(Float64(t_0 / Float64(b + sqrt(fma(b, b, t_0)))) * Float64(0.3333333333333333 / 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_] := Block[{t$95$0 = N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 43.7

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

2. Simplified 43.6

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   Proof
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -3)))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 3)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 3)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c (*.f64 3 a)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 3 a) c))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) b) (/.f64 1/3 a)): 16 points increase in error, 6 points decrease in error
   (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) (neg.f64 b))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= metadata-eval (*.f64 -1 -1))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1)) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 -1 -1))) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> metadata-eval 1)) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= metadata-eval (/.f64 1 3)) a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 3) a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 3 a)))): 21 points increase in error, 29 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -1)) (*.f64 -1 (*.f64 3 a)))): 13 points increase in error, 14 points decrease in error
   (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -1)) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (neg.f64 -1) -1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (Rewrite=> metadata-eval 1) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval -1)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 44.3

   \[\leadsto \color{blue}{\left(\left({\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25} + \sqrt{b}\right) \cdot \left({\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25} - \sqrt{b}\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

4. Applied egg-rr 43.2

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

5. Taylor expanded in b around 0 0.6

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

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	14016

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

Alternative 2
Error	6.1
Cost	7424

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

Alternative 3
Error	57.4
Cost	320

\[-0.3333333333333333 \cdot \frac{b}{a} \]

Alternative 4
Error	12.3
Cost	320

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

Alternative 5
Error	12.2
Cost	320

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

Reproduce

herbie shell --seed 2022294 
(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)))