Cubic critical, narrow range

Average Error: 28.3 → 5.1

Time: 28.0s

Precision: binary64

Cost: 61252

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{\sqrt[3]{\left(c \cdot c\right) \cdot \left(a \cdot a\right)} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t_0}\right)}{\sqrt[3]{b \cdot b - t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, \left(c \cdot c\right) \cdot {a}^{3}, {a}^{3} \cdot \left(\left(c \cdot c\right) \cdot 1.125\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)}\\ \end{array} \]

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 (fma a (* c -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.027)
     (/
      (* (cbrt (* (* c c) (* a a))) (cbrt 9.0))
      (/ (* (* a -3.0) (+ b (sqrt t_0))) (cbrt (- (* b b) t_0))))
     (/
      1.0
      (fma
       -2.0
       (/ b c)
       (fma
        3.0
        (/
         (fma
          -0.5625
          (* (* c c) (pow a 3.0))
          (* (pow a 3.0) (* (* c c) 1.125)))
         (pow b 5.0))
        (fma 1.5 (/ a b) (* 3.0 (/ (* (* c (* a a)) 0.375) (pow b 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) {
	double t_0 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.027) {
		tmp = (cbrt(((c * c) * (a * a))) * cbrt(9.0)) / (((a * -3.0) * (b + sqrt(t_0))) / cbrt(((b * b) - t_0)));
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), fma(3.0, (fma(-0.5625, ((c * c) * pow(a, 3.0)), (pow(a, 3.0) * ((c * c) * 1.125))) / pow(b, 5.0)), fma(1.5, (a / b), (3.0 * (((c * (a * a)) * 0.375) / pow(b, 3.0))))));
	}
	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(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.027)
		tmp = Float64(Float64(cbrt(Float64(Float64(c * c) * Float64(a * a))) * cbrt(9.0)) / Float64(Float64(Float64(a * -3.0) * Float64(b + sqrt(t_0))) / cbrt(Float64(Float64(b * b) - t_0))));
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), fma(3.0, Float64(fma(-0.5625, Float64(Float64(c * c) * (a ^ 3.0)), Float64((a ^ 3.0) * Float64(Float64(c * c) * 1.125))) / (b ^ 5.0)), fma(1.5, Float64(a / b), Float64(3.0 * Float64(Float64(Float64(c * Float64(a * a)) * 0.375) / (b ^ 3.0)))))));
	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[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.027], N[(N[(N[Power[N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[9.0, 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(3.0 * N[(N[(-0.5625 * N[(N[(c * c), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 1.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision] + N[(3.0 * N[(N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -0.0269999999999999997

   1. Initial program 13.1

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

   2. Simplified 13.1

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

      [Start] 13.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 13.1	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 13.1	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 13.6	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 13.6	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 13.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 13.8	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 13.7	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 13.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 13.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 13.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 13.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 13.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 13.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 13.8	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Applied egg-rr 12.3

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

   4. Taylor expanded in b around 0 11.1

      \[\leadsto \frac{\color{blue}{{\left({c}^{2} \cdot \left(1 \cdot {a}^{2}\right)\right)}^{0.3333333333333333} \cdot \sqrt[3]{9}}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

   5. Simplified 10.9

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(c \cdot c\right) \cdot \left(a \cdot a\right)} \cdot \sqrt[3]{9}}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      Proof

      [Start] 11.1	\[ \frac{{\left({c}^{2} \cdot \left(1 \cdot {a}^{2}\right)\right)}^{0.3333333333333333} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      unpow1/3 [=>] 10.9	\[ \frac{\color{blue}{\sqrt[3]{{c}^{2} \cdot \left(1 \cdot {a}^{2}\right)}} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      unpow2 [=>] 10.9	\[ \frac{\sqrt[3]{\color{blue}{\left(c \cdot c\right)} \cdot \left(1 \cdot {a}^{2}\right)} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      *-lft-identity [=>] 10.9	\[ \frac{\sqrt[3]{\left(c \cdot c\right) \cdot \color{blue}{{a}^{2}}} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      unpow2 [=>] 10.9	\[ \frac{\sqrt[3]{\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

   if -0.0269999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 32.9

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

   2. Simplified 32.9

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

      [Start] 32.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 32.9	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 32.9	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 33.4	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 33.4	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 33.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 33.4	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 32.8	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 33.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 33.4	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Applied egg-rr 32.9

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

   4. Taylor expanded in b around inf 3.4

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + \left(3 \cdot \frac{-0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right) + \left(0.2222222222222222 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{c}^{2} \cdot a} + -0.75 \cdot \left(c \cdot \left(a \cdot \left(0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \left(1.5 \cdot \frac{a}{b} + 3 \cdot \frac{0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}}\right)\right)}} \]

   5. Simplified 3.4

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \mathsf{fma}\left(0.2222222222222222, \frac{{\left(-1.125 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot \left(c \cdot c\right)}, -0.75 \cdot \left(c \cdot \left(a \cdot \left(\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375\right)\right)\right)\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)}} \]
      Proof

      [Start] 3.4	\[ \frac{1}{-2 \cdot \frac{b}{c} + \left(3 \cdot \frac{-0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right) + \left(0.2222222222222222 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{c}^{2} \cdot a} + -0.75 \cdot \left(c \cdot \left(a \cdot \left(0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \left(1.5 \cdot \frac{a}{b} + 3 \cdot \frac{0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}}\right)\right)} \]
      fma-def [=>] 3.4	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, 3 \cdot \frac{-0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right) + \left(0.2222222222222222 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{c}^{2} \cdot a} + -0.75 \cdot \left(c \cdot \left(a \cdot \left(0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \left(1.5 \cdot \frac{a}{b} + 3 \cdot \frac{0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}}\right)\right)}} \]
      fma-def [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \color{blue}{\mathsf{fma}\left(3, \frac{-0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right) + \left(0.2222222222222222 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{c}^{2} \cdot a} + -0.75 \cdot \left(c \cdot \left(a \cdot \left(0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 1.5 \cdot \frac{a}{b} + 3 \cdot \frac{0.75 \cdot \left(c \cdot {a}^{2}\right) + -0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}}\right)}\right)} \]

   6. Taylor expanded in c around 0 3.4

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{{c}^{2} \cdot \left(0.2222222222222222 \cdot \frac{1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}}{a} + -0.28125 \cdot {a}^{3}\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]

   7. Simplified 3.4

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{\left(c \cdot c\right) \cdot \mathsf{fma}\left(0.2222222222222222, \frac{{a}^{4} \cdot 6.328125}{a}, {a}^{3} \cdot -0.28125\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      Proof

      [Start] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), {c}^{2} \cdot \left(0.2222222222222222 \cdot \frac{1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}}{a} + -0.28125 \cdot {a}^{3}\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      unpow2 [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{\left(c \cdot c\right)} \cdot \left(0.2222222222222222 \cdot \frac{1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}}{a} + -0.28125 \cdot {a}^{3}\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      fma-def [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \left(c \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(0.2222222222222222, \frac{1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}}{a}, -0.28125 \cdot {a}^{3}\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      distribute-rgt-out [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \left(c \cdot c\right) \cdot \mathsf{fma}\left(0.2222222222222222, \frac{\color{blue}{{a}^{4} \cdot \left(1.265625 + 5.0625\right)}}{a}, -0.28125 \cdot {a}^{3}\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      metadata-eval [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \left(c \cdot c\right) \cdot \mathsf{fma}\left(0.2222222222222222, \frac{{a}^{4} \cdot \color{blue}{6.328125}}{a}, -0.28125 \cdot {a}^{3}\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      *-commutative [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \left(c \cdot c\right) \cdot \mathsf{fma}\left(0.2222222222222222, \frac{{a}^{4} \cdot 6.328125}{a}, \color{blue}{{a}^{3} \cdot -0.28125}\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]

   8. Taylor expanded in c around 0 3.4

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{{c}^{2} \cdot \left(-0.28125 \cdot {a}^{3} + 1.40625 \cdot {a}^{3}\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]

   9. Simplified 3.4

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{{a}^{3} \cdot \left(1.125 \cdot \left(c \cdot c\right)\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      Proof

      [Start] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), {c}^{2} \cdot \left(-0.28125 \cdot {a}^{3} + 1.40625 \cdot {a}^{3}\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      *-commutative [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{\left(-0.28125 \cdot {a}^{3} + 1.40625 \cdot {a}^{3}\right) \cdot {c}^{2}}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      distribute-rgt-out [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{\left({a}^{3} \cdot \left(-0.28125 + 1.40625\right)\right)} \cdot {c}^{2}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      metadata-eval [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \left({a}^{3} \cdot \color{blue}{1.125}\right) \cdot {c}^{2}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      unpow2 [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \left({a}^{3} \cdot 1.125\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]
      associate-*l* [=>] 3.4	\[ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, {a}^{3} \cdot \left(c \cdot c\right), \color{blue}{{a}^{3} \cdot \left(1.125 \cdot \left(c \cdot c\right)\right)}\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{\sqrt[3]{\left(c \cdot c\right) \cdot \left(a \cdot a\right)} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\sqrt[3]{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{\mathsf{fma}\left(-0.5625, \left(c \cdot c\right) \cdot {a}^{3}, {a}^{3} \cdot \left(\left(c \cdot c\right) \cdot 1.125\right)\right)}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{a}{b}, 3 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot 0.375}{{b}^{3}}\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.2
Cost	48068

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{\sqrt[3]{9} \cdot \sqrt[3]{c \cdot \left(c \cdot \left(a \cdot a\right)\right)}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t_0}\right)}{\sqrt[3]{b \cdot b - t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \left(0.375 \cdot \frac{c}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}\\ \end{array} \]

Alternative 2
Error	6.2
Cost	48068

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{\sqrt[3]{\left(c \cdot c\right) \cdot \left(a \cdot a\right)} \cdot \sqrt[3]{9}}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t_0}\right)}{\sqrt[3]{b \cdot b - t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \left(0.375 \cdot \frac{c}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}\\ \end{array} \]

Alternative 3
Error	6.5
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{b \cdot b - t_0}{a}}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \left(0.375 \cdot \frac{c}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}\\ \end{array} \]

Alternative 4
Error	6.5
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{b \cdot b - t_0}{b + \sqrt{t_0}} \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \left(0.375 \cdot \frac{c}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}\\ \end{array} \]

Alternative 5
Error	6.7
Cost	28036

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;-0.3333333333333333 \cdot {\left(\frac{a}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \left(0.375 \cdot \frac{c}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}\\ \end{array} \]

Alternative 6
Error	6.9
Cost	27524

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;-0.3333333333333333 \cdot {\left(\frac{a}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(a \cdot -3\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, \frac{b}{a \cdot c}, \frac{-0.5}{b}\right) + a \cdot \frac{c \cdot -0.375}{{b}^{3}}\right)}\\ \end{array} \]

Alternative 7
Error	6.9
Cost	21764

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{-0.3333333333333333}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(a \cdot -3\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, \frac{b}{a \cdot c}, \frac{-0.5}{b}\right) + a \cdot \frac{c \cdot -0.375}{{b}^{3}}\right)}\\ \end{array} \]

Alternative 8
Error	6.9
Cost	21188

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{-0.3333333333333333}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(3 \cdot a\right) \cdot \left(-0.5 \cdot \frac{-1}{b} + \left(\frac{\left(a \cdot c\right) \cdot -0.375 + \left(a \cdot c\right) \cdot 0.75}{{b}^{3}} + \frac{b}{a \cdot c} \cdot -0.6666666666666666\right)\right)}\\ \end{array} \]

Alternative 9
Error	6.9
Cost	16004

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(3 \cdot a\right) \cdot \left(-0.5 \cdot \frac{-1}{b} + \left(\frac{\left(a \cdot c\right) \cdot -0.375 + \left(a \cdot c\right) \cdot 0.75}{{b}^{3}} + \frac{b}{a \cdot c} \cdot -0.6666666666666666\right)\right)}\\ \end{array} \]

Alternative 10
Error	9.1
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.027:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 11
Error	10.1
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 200:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 12
Error	11.5
Cost	832

\[\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 13
Error	22.9
Cost	320

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

Alternative 14
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023056 
(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)))