Cubic critical, narrow range

Percentage Accurate: 54.9% → 92.3%

Time: 16.1s

Alternatives: 14

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (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]

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (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]

Alternative 1: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := t\_1 \cdot t\_2\\ \mathbf{if}\;b \leq 0.118:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, t\_0 \cdot t\_0, \left(t\_1 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \left(-t\_3\right)\right)}{\mathsf{fma}\left(b, \left(b \cdot b\right) \cdot t\_0, t\_3 \cdot t\_3 + t\_0 \cdot t\_3\right)}}{\left(a \cdot -3\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, t\_2, t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.5625}{{b}^{5}}, -0.375 \cdot \frac{c}{t\_0}\right), \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (fma c (* a -3.0) (* b b)))
        (t_2 (sqrt t_1))
        (t_3 (* t_1 t_2)))
   (if (<= b 0.118)
     (/
      (/
       (fma t_0 (* t_0 t_0) (* (* t_1 (* t_1 t_1)) (- t_3)))
       (fma b (* (* b b) t_0) (+ (* t_3 t_3) (* t_0 t_3))))
      (* (* a -3.0) (fma b b (fma b t_2 t_1))))
     (/
      1.0
      (fma
       a
       (fma
        a
        (*
         -3.0
         (fma a (/ (* (* c c) -0.5625) (pow b 5.0)) (* -0.375 (/ c t_0))))
        (/ 1.5 b))
       (* -2.0 (/ b c)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = fma(c, (a * -3.0), (b * b));
	double t_2 = sqrt(t_1);
	double t_3 = t_1 * t_2;
	double tmp;
	if (b <= 0.118) {
		tmp = (fma(t_0, (t_0 * t_0), ((t_1 * (t_1 * t_1)) * -t_3)) / fma(b, ((b * b) * t_0), ((t_3 * t_3) + (t_0 * t_3)))) / ((a * -3.0) * fma(b, b, fma(b, t_2, t_1)));
	} else {
		tmp = 1.0 / fma(a, fma(a, (-3.0 * fma(a, (((c * c) * -0.5625) / pow(b, 5.0)), (-0.375 * (c / t_0)))), (1.5 / b)), (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_2 = sqrt(t_1)
	t_3 = Float64(t_1 * t_2)
	tmp = 0.0
	if (b <= 0.118)
		tmp = Float64(Float64(fma(t_0, Float64(t_0 * t_0), Float64(Float64(t_1 * Float64(t_1 * t_1)) * Float64(-t_3))) / fma(b, Float64(Float64(b * b) * t_0), Float64(Float64(t_3 * t_3) + Float64(t_0 * t_3)))) / Float64(Float64(a * -3.0) * fma(b, b, fma(b, t_2, t_1))));
	else
		tmp = Float64(1.0 / fma(a, fma(a, Float64(-3.0 * fma(a, Float64(Float64(Float64(c * c) * -0.5625) / (b ^ 5.0)), Float64(-0.375 * Float64(c / t_0)))), Float64(1.5 / b)), Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, If[LessEqual[b, 0.118], N[(N[(N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * (-t$95$3)), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(t$95$3 * t$95$3), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b * b + N[(b * t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(a * N[(-3.0 * N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.11799999999999999

   1. Initial program 87.3%

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 88.7%

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

   8. Applied rewrites 90.0%

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

   if 0.11799999999999999 < b

   1. Initial program 52.5%

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-3 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -3 \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}} \]

   9. Applied rewrites 93.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.75, \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right)}{b \cdot b}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)}{c \cdot c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right), \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \frac{-9}{16} \cdot \color{blue}{\frac{{c}^{2}}{{b}^{5}}}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-3}{8}\right), \frac{\frac{3}{2}}{b}\right), -2 \cdot \frac{b}{c}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 93.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \frac{-0.5625 \cdot \left(c \cdot c\right)}{\color{blue}{{b}^{5}}}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right), \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.118:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right), \left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right) \cdot \left(-\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, \left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right), \left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) + \left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)\right)}}{\left(a \cdot -3\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.5625}{{b}^{5}}, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right), \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := \left(c \cdot a\right) \cdot -9\\ t_2 := \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(t\_1 \cdot t\_1\right)\right)\\ t_3 := \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), -0.5 \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \frac{b \cdot \left(\frac{0.5 \cdot \mathsf{fma}\left(0.25, t\_2 \cdot t\_2, \left(0.5 \cdot t\_1\right) \cdot t\_3\right)}{{b}^{6}} - \mathsf{fma}\left(0.5, t\_1, 0.5 \cdot \left(\frac{t\_3}{{b}^{4}} + \frac{t\_2}{b \cdot b}\right)\right)\right)}{\left(a \cdot -3\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{t\_0}, t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b)))
        (t_1 (* (* c a) -9.0))
        (t_2 (fma (* (* a a) (* c c)) 27.0 (* -0.25 (* t_1 t_1))))
        (t_3 (fma (* -27.0 (* a (* a a))) (* c (* c c)) (* -0.5 (* t_1 t_2)))))
   (/
    (*
     b
     (-
      (/ (* 0.5 (fma 0.25 (* t_2 t_2) (* (* 0.5 t_1) t_3))) (pow b 6.0))
      (fma 0.5 t_1 (* 0.5 (+ (/ t_3 (pow b 4.0)) (/ t_2 (* b b)))))))
    (* (* a -3.0) (fma b b (fma b (sqrt t_0) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = (c * a) * -9.0;
	double t_2 = fma(((a * a) * (c * c)), 27.0, (-0.25 * (t_1 * t_1)));
	double t_3 = fma((-27.0 * (a * (a * a))), (c * (c * c)), (-0.5 * (t_1 * t_2)));
	return (b * (((0.5 * fma(0.25, (t_2 * t_2), ((0.5 * t_1) * t_3))) / pow(b, 6.0)) - fma(0.5, t_1, (0.5 * ((t_3 / pow(b, 4.0)) + (t_2 / (b * b))))))) / ((a * -3.0) * fma(b, b, fma(b, sqrt(t_0), t_0)));
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(Float64(c * a) * -9.0)
	t_2 = fma(Float64(Float64(a * a) * Float64(c * c)), 27.0, Float64(-0.25 * Float64(t_1 * t_1)))
	t_3 = fma(Float64(-27.0 * Float64(a * Float64(a * a))), Float64(c * Float64(c * c)), Float64(-0.5 * Float64(t_1 * t_2)))
	return Float64(Float64(b * Float64(Float64(Float64(0.5 * fma(0.25, Float64(t_2 * t_2), Float64(Float64(0.5 * t_1) * t_3))) / (b ^ 6.0)) - fma(0.5, t_1, Float64(0.5 * Float64(Float64(t_3 / (b ^ 4.0)) + Float64(t_2 / Float64(b * b))))))) / Float64(Float64(a * -3.0) * fma(b, b, fma(b, sqrt(t_0), t_0))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * a), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(-0.25 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-27.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(b * N[(N[(N[(0.5 * N[(0.25 * N[(t$95$2 * t$95$2), $MachinePrecision] + N[(N[(0.5 * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * t$95$1 + N[(0.5 * N[(N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 54.9%

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

4. Applied rewrites 54.9%

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

6. Applied rewrites 55.8%

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

7. Taylor expanded in b around inf

   \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} - \left(\frac{1}{2} \cdot \left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\left(a \cdot -3\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)} \]

9. Applied rewrites 91.5%

   \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{0.5 \cdot \mathsf{fma}\left(0.25, \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right), \left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right) \cdot \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)\right)\right)\right)}{{b}^{6}} - \mathsf{fma}\left(0.5, \left(a \cdot c\right) \cdot -9, 0.5 \cdot \left(\frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{b \cdot b}\right)\right)\right)}}{\left(a \cdot -3\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)} \]

10. Final simplification 91.5%

    \[\leadsto \frac{b \cdot \left(\frac{0.5 \cdot \mathsf{fma}\left(0.25, \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right)\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right)\right), \left(0.5 \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right) \cdot \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), -0.5 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right)\right)\right)\right)\right)}{{b}^{6}} - \mathsf{fma}\left(0.5, \left(c \cdot a\right) \cdot -9, 0.5 \cdot \left(\frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), -0.5 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right)\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right)\right)}{b \cdot b}\right)\right)\right)}{\left(a \cdot -3\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))