Cubic critical, narrow range

Average Error: 28.8 → 5.8

Time: 25.7s

Precision: binary64

Cost: 47680

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

Math

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

↓

\[\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (fma
  -0.16666666666666666
  (* (/ (pow a 3.0) b) (* (* (* c c) (* (* c c) (pow b -6.0))) 6.328125))
  (fma
   -0.5625
   (* (* (* c c) (* c (pow b -5.0))) (* a a))
   (fma -0.5 (/ c b) (/ (* (* c c) -0.375) (/ (pow b 3.0) 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 fma(-0.16666666666666666, ((pow(a, 3.0) / b) * (((c * c) * ((c * c) * pow(b, -6.0))) * 6.328125)), fma(-0.5625, (((c * c) * (c * pow(b, -5.0))) * (a * a)), fma(-0.5, (c / b), (((c * c) * -0.375) / (pow(b, 3.0) / 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 fma(-0.16666666666666666, Float64(Float64((a ^ 3.0) / b) * Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 6.328125)), fma(-0.5625, Float64(Float64(Float64(c * c) * Float64(c * (b ^ -5.0))) * Float64(a * a)), fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * -0.375) / Float64((b ^ 3.0) / 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[(-0.16666666666666666 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.8

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

2. Simplified 28.8

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   Proof
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -3)))) b) (*.f64 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 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 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 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 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 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 3 a)): 19 points increase in error, 10 points decrease in error
   (/.f64 (-.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) 1)) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) (Rewrite<= metadata-eval (/.f64 -1 -1))) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) -1) -1)) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= /-rgt-identity_binary64 (/.f64 b 1))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 b (Rewrite<= metadata-eval (/.f64 -1 -1)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 b -1) -1))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 b)) -1)) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 b)) -1)) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 b)) -1)) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 (neg.f64 b)))) -1) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> remove-double-neg_binary64 b)) -1) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (neg.f64 (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 (/.f64 (Rewrite<= sub-neg_binary64 (-.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
   (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 -1 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 -1 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 -1 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (neg.f64 -1) -1) (/.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 (/.f64 (Rewrite=> metadata-eval 1) -1) (/.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=> metadata-eval -1) (/.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
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.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. Taylor expanded in a around 0 5.8

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

4. Simplified 5.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right)} \]
   Proof
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (Rewrite<= metadata-eval (+.f64 81/16 81/64)))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (+.f64 81/16 (Rewrite<= metadata-eval (*.f64 -9/8 -9/8))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -9/8 -9/8) (/.f64 (pow.f64 c 4) (pow.f64 b 6)))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -9/8 -9/8) (/.f64 (pow.f64 c (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 b 6))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -9/8 -9/8) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 c 2) (pow.f64 c 2))) (pow.f64 b 6))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -9/8 -9/8) (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (pow.f64 b (Rewrite<= metadata-eval (*.f64 2 3))))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -9/8 -9/8) (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 b 3) (pow.f64 b 3))))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -9/8 -9/8) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) (/.f64 (pow.f64 c 2) (pow.f64 b 3))))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2)))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 a 3) (/.f64 b (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))))) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (pow.f64 a 2)))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (*.f64 c c) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) -3/8) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/2 (/.f64 c b) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 c 2) (/.f64 (pow.f64 b 3) a)) -3/8))))): 1 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/2 (/.f64 c b) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) -3/8)))): 0 points increase in error, 1 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/2 (/.f64 c b) (Rewrite<= *-commutative_binary64 (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 5.8

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right)} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

6. Applied egg-rr 5.8

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

7. Final simplification 5.8

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

Alternatives

Alternative 1
Error	11.0
Cost	35980

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ t_1 := \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \mathbf{if}\;t_0 \leq -0.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.00085:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	11.0
Cost	35980

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.00085:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -1.125}{\frac{{b}^{3}}{a \cdot a}}\right)}{a \cdot 3}\\ \mathbf{elif}\;t_0 \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 3
Error	11.0
Cost	35980

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.3:\\ \;\;\;\;\sqrt[3]{{\left(\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a} \cdot -0.3333333333333333\right)}^{3}}\\ \mathbf{elif}\;t_0 \leq -0.00085:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, \frac{\left(c \cdot c\right) \cdot -1.125}{\frac{{b}^{3}}{a \cdot a}}\right)}{a \cdot 3}\\ \mathbf{elif}\;t_0 \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Error	11.1
Cost	35916

\[\begin{array}{l} t_0 := -0.3333333333333333 \cdot \frac{1.5 \cdot \frac{a \cdot c}{b} + 1.125 \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right)}{a}\\ t_1 := c \cdot \left(a \cdot -3\right)\\ t_2 := \frac{\sqrt{b \cdot b + t_1} - b}{a \cdot 3}\\ \mathbf{if}\;t_2 \leq -0.3:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -0.00085:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_1\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	11.1
Cost	35916

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ t_1 := -0.3333333333333333 \cdot \frac{1.5 \cdot \frac{a \cdot c}{b} + 1.125 \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right)}{a}\\ \mathbf{if}\;t_0 \leq -0.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.00085:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.7
Cost	33536

\[\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{{b}^{3}}\right)\right) \]

Alternative 7
Error	11.2
Cost	29644

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ t_1 := -0.3333333333333333 \cdot \frac{1.5 \cdot \frac{a \cdot c}{b} + 1.125 \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right)}{a}\\ \mathbf{if}\;t_0 \leq -0.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.00085:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	7.7
Cost	27328

\[\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b} \cdot \frac{-0.375}{b}\right)\right) \]

Alternative 9
Error	11.8
Cost	1728

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

Alternative 10
Error	22.5
Cost	320

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

Reproduce

herbie shell --seed 2022321 
(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)))