Cubic critical, narrow range

Average Error: 28.4 → 5.3

Time: 16.9s

Precision: binary64

Cost: 67396

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -4:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_0, -b\right) \cdot {\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \frac{\left(\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125\right) \cdot -0.16666666666666666}{a \cdot {b}^{7}}\right)\right)\right)\\ \end{array} \]

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 (pow (fma c (* a -3.0) (* b b)) 0.25)))
   (if (<= (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) b) (* 3.0 a)) -4.0)
     (* (fma t_0 t_0 (- b)) (pow (cbrt (/ 0.3333333333333333 a)) 3.0))
     (fma
      -0.5625
      (* (* a a) (/ (pow c 3.0) (pow b 5.0)))
      (fma
       -0.5
       (/ c b)
       (fma
        -0.375
        (* a (/ (* c c) (pow b 3.0)))
        (/
         (* (* (* (pow c 4.0) (pow a 4.0)) 6.328125) -0.16666666666666666)
         (* a (pow b 7.0)))))))))

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) {
	double t_0 = pow(fma(c, (a * -3.0), (b * b)), 0.25);
	double tmp;
	if (((sqrt(((b * b) + (c * (a * -3.0)))) - b) / (3.0 * a)) <= -4.0) {
		tmp = fma(t_0, t_0, -b) * pow(cbrt((0.3333333333333333 / a)), 3.0);
	} else {
		tmp = fma(-0.5625, ((a * a) * (pow(c, 3.0) / pow(b, 5.0))), fma(-0.5, (c / b), fma(-0.375, (a * ((c * c) / pow(b, 3.0))), ((((pow(c, 4.0) * pow(a, 4.0)) * 6.328125) * -0.16666666666666666) / (a * pow(b, 7.0))))));
	}
	return tmp;
}

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 = fma(c, Float64(a * -3.0), Float64(b * b)) ^ 0.25
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a)) <= -4.0)
		tmp = Float64(fma(t_0, t_0, Float64(-b)) * (cbrt(Float64(0.3333333333333333 / a)) ^ 3.0));
	else
		tmp = fma(-0.5625, Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0))), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a * Float64(Float64(c * c) / (b ^ 3.0))), Float64(Float64(Float64(Float64((c ^ 4.0) * (a ^ 4.0)) * 6.328125) * -0.16666666666666666) / Float64(a * (b ^ 7.0))))));
	end
	return tmp
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[Power[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -4.0], N[(N[(t$95$0 * t$95$0 + (-b)), $MachinePrecision] * N[Power[N[Power[N[(0.3333333333333333 / a), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4

   1. Initial program 9.7

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

   2. Simplified 9.6

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
      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)): 25 points increase in error, 10 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)))): 48 points increase in error, 42 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)))): 22 points increase in error, 23 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 9.7

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

   4. Applied egg-rr 10.6

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 30.8

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

   2. Taylor expanded in b around inf 4.6

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   3. Simplified 4.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \frac{\left(\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125\right) \cdot -0.16666666666666666}{a \cdot {b}^{7}}\right)\right)\right)} \]
      Proof
      (fma.f64 -9/16 (*.f64 (*.f64 a a) (/.f64 (pow.f64 c 3) (pow.f64 b 5))) (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (*.f64 a (/.f64 (*.f64 c c) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (/.f64 (pow.f64 c 3) (pow.f64 b 5))) (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (*.f64 a (/.f64 (*.f64 c c) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 c 3) (pow.f64 b 5)) (pow.f64 a 2))) (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (*.f64 a (/.f64 (*.f64 c c) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (*.f64 a (/.f64 (*.f64 c c) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (*.f64 a (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) a)) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 405/64) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite<= metadata-eval (+.f64 81/64 81/16))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 (Rewrite<= metadata-eval (*.f64 -9/8 -9/8)) 81/16)) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (*.f64 (pow.f64 c (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 a 4))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 c 2) (pow.f64 c 2))) (pow.f64 a 4))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (pow.f64 a (Rewrite<= metadata-eval (*.f64 2 2))))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 a 2) (pow.f64 a 2))))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (Rewrite=> unswap-sqr_binary64 (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (*.f64 (pow.f64 c 2) (pow.f64 a 2))))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))))) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 (*.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2)) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) -1/6) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) -1/6))))): 0 points increase in error, 0 points decrease in error
      (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= *-commutative_binary64 (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))))): 0 points increase in error, 0 points decrease in error
      (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<= fma-def_binary64 (+.f64 (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))))): 0 points increase in error, 0 points decrease in error
      (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 (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))) (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))))): 15 points increase in error, 6 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (+.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 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (+.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
3. Recombined 2 regimes into one program.

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -4:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25}, -b\right) \cdot {\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \frac{\left(\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125\right) \cdot -0.16666666666666666}{a \cdot {b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.3
Cost	60932

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -4:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_0, -b\right) \cdot {\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(c \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5625}{{b}^{5}} \cdot \left(\left(a \cdot a\right) \cdot {c}^{3}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	6.6
Cost	34244

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -0.2:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot {\left(\sqrt{\frac{0.3333333333333333}{a}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(1.5, \frac{a}{b}, \frac{1.125 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{3}}\right)\right)\right)}^{-1}\\ \end{array} \]

Alternative 3
Error	6.8
Cost	33924

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -0.2:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot {\left(\sqrt{\frac{0.3333333333333333}{a}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(a \cdot \left(c \cdot c\right)\right) \cdot \left(\frac{-0.375}{{b}^{3}} + \frac{c \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\\ \end{array} \]

Alternative 4
Error	6.8
Cost	28100

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -0.2:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(a \cdot \left(c \cdot c\right)\right) \cdot \left(\frac{-0.375}{{b}^{3}} + \frac{c \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\\ \end{array} \]

Alternative 5
Error	9.3
Cost	21188

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -0.15:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -0.15:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 7
Error	9.3
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -0.15:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 8
Error	9.3
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.15:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 9
Error	9.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.56:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 10
Error	11.7
Cost	7168

\[{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1} \]

Alternative 11
Error	23.0
Cost	320

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

Alternative 12
Error	23.0
Cost	320

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

Reproduce

herbie shell --seed 2022297 
(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)))