Cubic critical, narrow range

Percentage Accurate: 55.0% → 92.2%

Time: 19.9s

Alternatives: 19

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: 55.0% 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: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_2 := \sqrt{t\_1}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -30.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b \cdot b, -t\_1 \cdot t\_2\right)}{a \cdot \left(-3 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + t\_2, c \cdot \left(a \cdot -3\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-0.375}{t\_0}, c \cdot c, a \cdot \mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot t\_0}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(t\_0 \cdot t\_0\right)} \cdot -0.16666666666666666\right)\right), -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

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)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -30.5)
     (/
      (fma b (* b b) (- (* t_1 t_2)))
      (* a (* -3.0 (fma b b (fma b (+ b t_2) (* c (* a -3.0)))))))
     (fma
      a
      (fma
       (/ -0.375 t_0)
       (* c c)
       (*
        a
        (fma
         c
         (* (* c c) (/ -0.5625 (* (* b b) t_0)))
         (*
          (/ (* (* c (* c (* c c))) (* a 6.328125)) (* b (* t_0 t_0)))
          -0.16666666666666666))))
      (* -0.5 (/ c b))))))

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 tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -30.5) {
		tmp = fma(b, (b * b), -(t_1 * t_2)) / (a * (-3.0 * fma(b, b, fma(b, (b + t_2), (c * (a * -3.0))))));
	} else {
		tmp = fma(a, fma((-0.375 / t_0), (c * c), (a * fma(c, ((c * c) * (-0.5625 / ((b * b) * t_0))), ((((c * (c * (c * c))) * (a * 6.328125)) / (b * (t_0 * t_0))) * -0.16666666666666666)))), (-0.5 * (c / b)));
	}
	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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -30.5)
		tmp = Float64(fma(b, Float64(b * b), Float64(-Float64(t_1 * t_2))) / Float64(a * Float64(-3.0 * fma(b, b, fma(b, Float64(b + t_2), Float64(c * Float64(a * -3.0)))))));
	else
		tmp = fma(a, fma(Float64(-0.375 / t_0), Float64(c * c), Float64(a * fma(c, Float64(Float64(c * c) * Float64(-0.5625 / Float64(Float64(b * b) * t_0))), Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 6.328125)) / Float64(b * Float64(t_0 * t_0))) * -0.16666666666666666)))), Float64(-0.5 * Float64(c / b)));
	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]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -30.5], N[(N[(b * N[(b * b), $MachinePrecision] + (-N[(t$95$1 * t$95$2), $MachinePrecision])), $MachinePrecision] / N[(a * N[(-3.0 * N[(b * b + N[(b * N[(b + t$95$2), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-0.375 / t$95$0), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * N[(-0.5625 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -30.5

   1. Initial program 88.8%

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

   4. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-inv N/A

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

      7. lift--.f64 N/A

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

      8. flip3-- N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 89.7%

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

   8. Applied egg-rr 89.4%

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

   10. Applied egg-rr 90.0%

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

   if -30.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 53.2%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 94.2%

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

   7. Applied egg-rr 94.2%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-0.375}{b \cdot \left(b \cdot b\right)}, c \cdot c, a \cdot \mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(6.328125 \cdot a\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b} \cdot -0.16666666666666666\right)\right)}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -30.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b \cdot b, -\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}{a \cdot \left(-3 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, c \cdot \left(a \cdot -3\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-0.375}{b \cdot \left(b \cdot b\right)}, c \cdot c, a \cdot \mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot -0.16666666666666666\right)\right), -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot a\right)\\ t_1 := \left(a \cdot c\right) \cdot -9\\ t_2 := \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \left(t\_1 \cdot t\_1\right)\right)\\ t_3 := \mathsf{fma}\left(-27 \cdot t\_0, c \cdot \left(c \cdot c\right), -0.5 \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{t\_1}{a}, \mathsf{fma}\left(-0.5, \frac{t\_3}{a \cdot {b}^{4}} + \frac{t\_2}{a \cdot \left(b \cdot b\right)}, \frac{0.5 \cdot \mathsf{fma}\left(t\_1 \cdot 0.5, t\_3, 0.25 \cdot \left(t\_2 \cdot t\_2\right)\right)}{a \cdot {b}^{6}}\right)\right)}{\mathsf{fma}\left(2, b \cdot b, \mathsf{fma}\left(c, \mathsf{fma}\left(a, -4.5, c \cdot \mathsf{fma}\left(-1.125, \frac{a \cdot a}{b \cdot b}, \frac{-1.6875 \cdot \left(c \cdot t\_0\right)}{{b}^{4}}\right)\right), b \cdot b\right)\right)}}{-3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* a a)))
        (t_1 (* (* a c) -9.0))
        (t_2 (fma (* (* a a) (* c c)) 27.0 (* -0.25 (* t_1 t_1))))
        (t_3 (fma (* -27.0 t_0) (* c (* c c)) (* -0.5 (* t_1 t_2)))))
   (/
    (/
     (*
      b
      (fma
       -0.5
       (/ t_1 a)
       (fma
        -0.5
        (+ (/ t_3 (* a (pow b 4.0))) (/ t_2 (* a (* b b))))
        (/
         (* 0.5 (fma (* t_1 0.5) t_3 (* 0.25 (* t_2 t_2))))
         (* a (pow b 6.0))))))
     (fma
      2.0
      (* b b)
      (fma
       c
       (fma
        a
        -4.5
        (*
         c
         (fma
          -1.125
          (/ (* a a) (* b b))
          (/ (* -1.6875 (* c t_0)) (pow b 4.0)))))
       (* b b))))
    -3.0)))

C

double code(double a, double b, double c) {
	double t_0 = a * (a * a);
	double t_1 = (a * c) * -9.0;
	double t_2 = fma(((a * a) * (c * c)), 27.0, (-0.25 * (t_1 * t_1)));
	double t_3 = fma((-27.0 * t_0), (c * (c * c)), (-0.5 * (t_1 * t_2)));
	return ((b * fma(-0.5, (t_1 / a), fma(-0.5, ((t_3 / (a * pow(b, 4.0))) + (t_2 / (a * (b * b)))), ((0.5 * fma((t_1 * 0.5), t_3, (0.25 * (t_2 * t_2)))) / (a * pow(b, 6.0)))))) / fma(2.0, (b * b), fma(c, fma(a, -4.5, (c * fma(-1.125, ((a * a) / (b * b)), ((-1.6875 * (c * t_0)) / pow(b, 4.0))))), (b * b)))) / -3.0;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(a * a))
	t_1 = Float64(Float64(a * c) * -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 * t_0), Float64(c * Float64(c * c)), Float64(-0.5 * Float64(t_1 * t_2)))
	return Float64(Float64(Float64(b * fma(-0.5, Float64(t_1 / a), fma(-0.5, Float64(Float64(t_3 / Float64(a * (b ^ 4.0))) + Float64(t_2 / Float64(a * Float64(b * b)))), Float64(Float64(0.5 * fma(Float64(t_1 * 0.5), t_3, Float64(0.25 * Float64(t_2 * t_2)))) / Float64(a * (b ^ 6.0)))))) / fma(2.0, Float64(b * b), fma(c, fma(a, -4.5, Float64(c * fma(-1.125, Float64(Float64(a * a) / Float64(b * b)), Float64(Float64(-1.6875 * Float64(c * t_0)) / (b ^ 4.0))))), Float64(b * b)))) / -3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * c), $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 * t$95$0), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(-0.5 * N[(t$95$1 / a), $MachinePrecision] + N[(-0.5 * N[(N[(t$95$3 / N[(a * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(N[(t$95$1 * 0.5), $MachinePrecision] * t$95$3 + N[(0.25 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.5 + N[(c * N[(-1.125 * N[(N[(a * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.6875 * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]]]]]

Derivation

1. Initial program 56.7%

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

4. Applied egg-rr 56.7%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-inv N/A

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

   7. lift--.f64 N/A

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

   8. flip3-- N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in b around inf

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

9. Simplified 92.5%

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

10. Taylor expanded in c around 0

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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), \frac{1}{4} \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \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, \frac{-1}{4} \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)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{2 \cdot {b}^{2} + \left(c \cdot \left(-3 \cdot a + \left(\frac{-3}{2} \cdot a + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) + {b}^{2}\right)}}}{-3} \]
11. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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), \frac{1}{4} \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \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, \frac{-1}{4} \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)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{\mathsf{fma}\left(2, {b}^{2}, c \cdot \left(-3 \cdot a + \left(\frac{-3}{2} \cdot a + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) + {b}^{2}\right)}}}{-3} \]

    2. unpow2 N/A

       \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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), \frac{1}{4} \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \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, \frac{-1}{4} \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)}{a \cdot {b}^{6}}\right)\right)}{\mathsf{fma}\left(2, \color{blue}{b \cdot b}, c \cdot \left(-3 \cdot a + \left(\frac{-3}{2} \cdot a + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) + {b}^{2}\right)}}{-3} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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), \frac{1}{4} \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \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, \frac{-1}{4} \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)}{a \cdot {b}^{6}}\right)\right)}{\mathsf{fma}\left(2, \color{blue}{b \cdot b}, c \cdot \left(-3 \cdot a + \left(\frac{-3}{2} \cdot a + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) + {b}^{2}\right)}}{-3} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{\frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right), c \cdot \left(c \cdot c\right), \frac{-1}{2} \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, \frac{-1}{4} \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), \frac{1}{4} \cdot \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \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, \frac{-1}{4} \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)}{a \cdot {b}^{6}}\right)\right)}{\mathsf{fma}\left(2, b \cdot b, \color{blue}{\mathsf{fma}\left(c, -3 \cdot a + \left(\frac{-3}{2} \cdot a + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right), {b}^{2}\right)}\right)}}{-3} \]

12. Simplified 92.5%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(-0.5, \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)}{a \cdot {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)}{a \cdot \left(b \cdot b\right)}, \frac{0.5 \cdot \mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \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), 0.25 \cdot \left(\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)\right)\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{\mathsf{fma}\left(2, b \cdot b, \mathsf{fma}\left(c, \mathsf{fma}\left(a, -4.5, c \cdot \mathsf{fma}\left(-1.125, \frac{a \cdot a}{b \cdot b}, \frac{-1.6875 \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot c\right)}{{b}^{4}}\right)\right), b \cdot b\right)\right)}}}{-3} \]

13. Final simplification 92.5%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\left(a \cdot c\right) \cdot -9}{a}, \mathsf{fma}\left(-0.5, \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)}{a \cdot {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)}{a \cdot \left(b \cdot b\right)}, \frac{0.5 \cdot \mathsf{fma}\left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot 0.5, \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), 0.25 \cdot \left(\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)\right)\right)}{a \cdot {b}^{6}}\right)\right)}{\mathsf{fma}\left(2, b \cdot b, \mathsf{fma}\left(c, \mathsf{fma}\left(a, -4.5, c \cdot \mathsf{fma}\left(-1.125, \frac{a \cdot a}{b \cdot b}, \frac{-1.6875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{4}}\right)\right), b \cdot b\right)\right)}}{-3} \]
14. Add Preprocessing

Alternative 3: 91.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(Float64(a * c) * -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(Float64(b * fma(-0.5, Float64(t_1 / a), fma(-0.5, Float64(Float64(t_3 / Float64(a * (b ^ 4.0))) + Float64(t_2 / Float64(a * Float64(b * b)))), Float64(Float64(0.5 * fma(Float64(t_1 * 0.5), t_3, Float64(0.25 * Float64(t_2 * t_2)))) / Float64(a * (b ^ 6.0)))))) / fma(b, b, fma(b, sqrt(t_0), t_0))) / -3.0)
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[(a * c), $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[(N[(b * N[(-0.5 * N[(t$95$1 / a), $MachinePrecision] + N[(-0.5 * N[(N[(t$95$3 / N[(a * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(N[(t$95$1 * 0.5), $MachinePrecision] * t$95$3 + N[(0.25 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]]]]]

Derivation

1. Initial program 56.7%

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

4. Applied egg-rr 56.7%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-inv N/A

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

   7. lift--.f64 N/A

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

   8. flip3-- N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in b around inf

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

9. Simplified 92.5%

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

10. Final simplification 92.5%

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

Alternative 4: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := \left(a \cdot a\right) \cdot 6.75\\ \frac{\frac{b \cdot \left({c}^{4} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(a \cdot 4.5, \mathsf{fma}\left(27, a \cdot \left(a \cdot a\right), \left(a \cdot -4.5\right) \cdot t\_1\right), 0.25 \cdot \left(t\_1 \cdot t\_1\right)\right)}{a \cdot {b}^{6}}, \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(27, \frac{a \cdot a}{{b}^{4}}, \frac{-4.5 \cdot t\_1}{{b}^{4}}\right), \frac{\mathsf{fma}\left(-0.5, 6.75 \cdot \frac{a}{b \cdot b}, \frac{4.5}{c}\right)}{-c}\right)}{-c}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{t\_0}, t\_0\right)\right)}}{-3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))) (t_1 (* (* a a) 6.75)))
   (/
    (/
     (*
      b
      (*
       (pow c 4.0)
       (fma
        0.5
        (/
         (fma
          (* a 4.5)
          (fma 27.0 (* a (* a a)) (* (* a -4.5) t_1))
          (* 0.25 (* t_1 t_1)))
         (* a (pow b 6.0)))
        (/
         (fma
          -0.5
          (fma 27.0 (/ (* a a) (pow b 4.0)) (/ (* -4.5 t_1) (pow b 4.0)))
          (/ (fma -0.5 (* 6.75 (/ a (* b b))) (/ 4.5 c)) (- c)))
         (- c)))))
     (fma b b (fma b (sqrt t_0) t_0)))
    -3.0)))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = (a * a) * 6.75;
	return ((b * (pow(c, 4.0) * fma(0.5, (fma((a * 4.5), fma(27.0, (a * (a * a)), ((a * -4.5) * t_1)), (0.25 * (t_1 * t_1))) / (a * pow(b, 6.0))), (fma(-0.5, fma(27.0, ((a * a) / pow(b, 4.0)), ((-4.5 * t_1) / pow(b, 4.0))), (fma(-0.5, (6.75 * (a / (b * b))), (4.5 / c)) / -c)) / -c)))) / fma(b, b, fma(b, sqrt(t_0), t_0))) / -3.0;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(Float64(a * a) * 6.75)
	return Float64(Float64(Float64(b * Float64((c ^ 4.0) * fma(0.5, Float64(fma(Float64(a * 4.5), fma(27.0, Float64(a * Float64(a * a)), Float64(Float64(a * -4.5) * t_1)), Float64(0.25 * Float64(t_1 * t_1))) / Float64(a * (b ^ 6.0))), Float64(fma(-0.5, fma(27.0, Float64(Float64(a * a) / (b ^ 4.0)), Float64(Float64(-4.5 * t_1) / (b ^ 4.0))), Float64(fma(-0.5, Float64(6.75 * Float64(a / Float64(b * b))), Float64(4.5 / c)) / Float64(-c))) / Float64(-c))))) / fma(b, b, fma(b, sqrt(t_0), t_0))) / -3.0)
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[(a * a), $MachinePrecision] * 6.75), $MachinePrecision]}, N[(N[(N[(b * N[(N[Power[c, 4.0], $MachinePrecision] * N[(0.5 * N[(N[(N[(a * 4.5), $MachinePrecision] * N[(27.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -4.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(27.0 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-4.5 * t$95$1), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(6.75 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.5 / c), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]]]

Derivation

1. Initial program 56.7%

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

4. Applied egg-rr 56.7%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-inv N/A

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

   7. lift--.f64 N/A

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

   8. flip3-- N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in b around inf

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

9. Simplified 92.5%

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

10. Taylor expanded in c around -inf

    \[\leadsto \frac{\frac{b \cdot \color{blue}{\left({c}^{4} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\frac{-81}{4} \cdot \frac{a}{{b}^{2}} + 27 \cdot \frac{a}{{b}^{2}}\right) + \frac{9}{2} \cdot \frac{1}{c}}{c} + \frac{-1}{2} \cdot \left(\frac{-9}{2} \cdot \frac{\frac{-81}{4} \cdot {a}^{2} + 27 \cdot {a}^{2}}{{b}^{4}} + 27 \cdot \frac{{a}^{2}}{{b}^{4}}\right)}{c} + \frac{1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\frac{-81}{4} \cdot {a}^{2} + 27 \cdot {a}^{2}\right)}^{2} + \frac{9}{2} \cdot \left(a \cdot \left(\frac{-9}{2} \cdot \left(a \cdot \left(\frac{-81}{4} \cdot {a}^{2} + 27 \cdot {a}^{2}\right)\right) + 27 \cdot {a}^{3}\right)\right)}{a \cdot {b}^{6}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}}{-3} \]

12. Simplified 92.4%

    \[\leadsto \frac{\frac{b \cdot \color{blue}{\left({c}^{4} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(4.5 \cdot a, \mathsf{fma}\left(27, a \cdot \left(a \cdot a\right), \left(-4.5 \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right), 0.25 \cdot \left(\left(\left(a \cdot a\right) \cdot 6.75\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)\right)}{a \cdot {b}^{6}}, -\frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(27, \frac{a \cdot a}{{b}^{4}}, \frac{-4.5 \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)}{{b}^{4}}\right), -\frac{\mathsf{fma}\left(-0.5, \frac{a}{b \cdot b} \cdot 6.75, \frac{4.5}{c}\right)}{c}\right)}{c}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}}{-3} \]

13. Final simplification 92.4%

    \[\leadsto \frac{\frac{b \cdot \left({c}^{4} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(a \cdot 4.5, \mathsf{fma}\left(27, a \cdot \left(a \cdot a\right), \left(a \cdot -4.5\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right), 0.25 \cdot \left(\left(\left(a \cdot a\right) \cdot 6.75\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)\right)}{a \cdot {b}^{6}}, \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(27, \frac{a \cdot a}{{b}^{4}}, \frac{-4.5 \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)}{{b}^{4}}\right), \frac{\mathsf{fma}\left(-0.5, 6.75 \cdot \frac{a}{b \cdot b}, \frac{4.5}{c}\right)}{-c}\right)}{-c}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}}{-3} \]
14. Add Preprocessing

Alternative 5: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(a \cdot c\right) \cdot 4.5\\ t_2 := a \cdot \left(c \cdot c\right)\\ t_3 := \mathsf{fma}\left(a \cdot t\_2, -20.25, a \cdot \left(27 \cdot t\_2\right)\right)\\ t_4 := \mathsf{fma}\left(t\_3, t\_1, \left(c \cdot c\right) \cdot \left(c \cdot \left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)\\ \frac{b \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(t\_3, 0.25 \cdot t\_3, t\_4 \cdot \left(\left(a \cdot c\right) \cdot -4.5\right)\right)}{t\_0 \cdot \left(a \cdot t\_0\right)}, \mathsf{fma}\left(-0.5, \frac{t\_4}{a \cdot \left(b \cdot t\_0\right)} + \frac{t\_3}{a \cdot \left(b \cdot b\right)}, \frac{t\_1}{a}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, c \cdot \left(a \cdot -3\right)\right)\right)} \cdot -0.3333333333333333 \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* (* a c) 4.5))
        (t_2 (* a (* c c)))
        (t_3 (fma (* a t_2) -20.25 (* a (* 27.0 t_2))))
        (t_4 (fma t_3 t_1 (* (* c c) (* c (* -27.0 (* a (* a a))))))))
   (*
    (/
     (*
      b
      (fma
       0.5
       (/ (fma t_3 (* 0.25 t_3) (* t_4 (* (* a c) -4.5))) (* t_0 (* a t_0)))
       (fma -0.5 (+ (/ t_4 (* a (* b t_0))) (/ t_3 (* a (* b b)))) (/ t_1 a))))
     (fma
      b
      b
      (fma b (+ b (sqrt (fma c (* a -3.0) (* b b)))) (* c (* a -3.0)))))
    -0.3333333333333333)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (a * c) * 4.5;
	double t_2 = a * (c * c);
	double t_3 = fma((a * t_2), -20.25, (a * (27.0 * t_2)));
	double t_4 = fma(t_3, t_1, ((c * c) * (c * (-27.0 * (a * (a * a))))));
	return ((b * fma(0.5, (fma(t_3, (0.25 * t_3), (t_4 * ((a * c) * -4.5))) / (t_0 * (a * t_0))), fma(-0.5, ((t_4 / (a * (b * t_0))) + (t_3 / (a * (b * b)))), (t_1 / a)))) / fma(b, b, fma(b, (b + sqrt(fma(c, (a * -3.0), (b * b)))), (c * (a * -3.0))))) * -0.3333333333333333;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(a * c) * 4.5)
	t_2 = Float64(a * Float64(c * c))
	t_3 = fma(Float64(a * t_2), -20.25, Float64(a * Float64(27.0 * t_2)))
	t_4 = fma(t_3, t_1, Float64(Float64(c * c) * Float64(c * Float64(-27.0 * Float64(a * Float64(a * a))))))
	return Float64(Float64(Float64(b * fma(0.5, Float64(fma(t_3, Float64(0.25 * t_3), Float64(t_4 * Float64(Float64(a * c) * -4.5))) / Float64(t_0 * Float64(a * t_0))), fma(-0.5, Float64(Float64(t_4 / Float64(a * Float64(b * t_0))) + Float64(t_3 / Float64(a * Float64(b * b)))), Float64(t_1 / a)))) / fma(b, b, fma(b, Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))), Float64(c * Float64(a * -3.0))))) * -0.3333333333333333)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a * t$95$2), $MachinePrecision] * -20.25 + N[(a * N[(27.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$1 + N[(N[(c * c), $MachinePrecision] * N[(c * N[(-27.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(0.5 * N[(N[(t$95$3 * N[(0.25 * t$95$3), $MachinePrecision] + N[(t$95$4 * N[(N[(a * c), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(t$95$4 / N[(a * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(b * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Initial program 56.7%

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

4. Applied egg-rr 56.7%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-inv N/A

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

   7. lift--.f64 N/A

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

   8. flip3-- N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in b around inf

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

9. Simplified 92.5%

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

11. Applied egg-rr 92.3%

    \[\leadsto \color{blue}{\frac{b \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot 27\right)\right), \mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot 27\right)\right) \cdot 0.25, \mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot 27\right)\right), 4.5 \cdot \left(a \cdot c\right), \left(c \cdot c\right) \cdot \left(c \cdot \left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right) \cdot \left(\left(a \cdot c\right) \cdot -4.5\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot 27\right)\right), 4.5 \cdot \left(a \cdot c\right), \left(c \cdot c\right) \cdot \left(c \cdot \left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)}{a \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot 27\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{4.5 \cdot \left(a \cdot c\right)}{a}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, c \cdot \left(a \cdot -3\right)\right)\right)} \cdot -0.3333333333333333} \]

12. Final simplification 92.3%

    \[\leadsto \frac{b \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(27 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right), 0.25 \cdot \mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(27 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(27 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right), \left(a \cdot c\right) \cdot 4.5, \left(c \cdot c\right) \cdot \left(c \cdot \left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right) \cdot \left(\left(a \cdot c\right) \cdot -4.5\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(27 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right), \left(a \cdot c\right) \cdot 4.5, \left(c \cdot c\right) \cdot \left(c \cdot \left(-27 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)}{a \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right), -20.25, a \cdot \left(27 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{a \cdot \left(b \cdot b\right)}, \frac{\left(a \cdot c\right) \cdot 4.5}{a}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, c \cdot \left(a \cdot -3\right)\right)\right)} \cdot -0.3333333333333333 \]
13. Add Preprocessing

Alternative 6: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := \left(c \cdot c\right) \cdot 6.75\\ t_2 := \mathsf{fma}\left(c \cdot -4.5, t\_1, \left(c \cdot \left(c \cdot c\right)\right) \cdot 27\right)\\ \frac{\frac{b \cdot \left(\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, t\_1 \cdot t\_1, \left(c \cdot 4.5\right) \cdot t\_2\right)}{{b}^{6}}, \frac{\mathsf{fma}\left(0.5, \frac{t\_2}{{b}^{4}}, \frac{\mathsf{fma}\left(-4.5, \frac{c}{a}, \frac{0.5 \cdot t\_1}{b \cdot b}\right)}{-a}\right)}{-a}\right) \cdot \left(a \cdot \left(a \cdot \left(-a\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{t\_0}, t\_0\right)\right)}}{-3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b)))
        (t_1 (* (* c c) 6.75))
        (t_2 (fma (* c -4.5) t_1 (* (* c (* c c)) 27.0))))
   (/
    (/
     (*
      b
      (*
       (fma
        -0.5
        (/ (fma 0.25 (* t_1 t_1) (* (* c 4.5) t_2)) (pow b 6.0))
        (/
         (fma
          0.5
          (/ t_2 (pow b 4.0))
          (/ (fma -4.5 (/ c a) (/ (* 0.5 t_1) (* b b))) (- a)))
         (- a)))
       (* a (* a (- a)))))
     (fma b b (fma b (sqrt t_0) t_0)))
    -3.0)))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = (c * c) * 6.75;
	double t_2 = fma((c * -4.5), t_1, ((c * (c * c)) * 27.0));
	return ((b * (fma(-0.5, (fma(0.25, (t_1 * t_1), ((c * 4.5) * t_2)) / pow(b, 6.0)), (fma(0.5, (t_2 / pow(b, 4.0)), (fma(-4.5, (c / a), ((0.5 * t_1) / (b * b))) / -a)) / -a)) * (a * (a * -a)))) / fma(b, b, fma(b, sqrt(t_0), t_0))) / -3.0;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(Float64(c * c) * 6.75)
	t_2 = fma(Float64(c * -4.5), t_1, Float64(Float64(c * Float64(c * c)) * 27.0))
	return Float64(Float64(Float64(b * Float64(fma(-0.5, Float64(fma(0.25, Float64(t_1 * t_1), Float64(Float64(c * 4.5) * t_2)) / (b ^ 6.0)), Float64(fma(0.5, Float64(t_2 / (b ^ 4.0)), Float64(fma(-4.5, Float64(c / a), Float64(Float64(0.5 * t_1) / Float64(b * b))) / Float64(-a))) / Float64(-a))) * Float64(a * Float64(a * Float64(-a))))) / fma(b, b, fma(b, sqrt(t_0), t_0))) / -3.0)
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 * c), $MachinePrecision] * 6.75), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * -4.5), $MachinePrecision] * t$95$1 + N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(N[(-0.5 * N[(N[(0.25 * N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(c * 4.5), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(t$95$2 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-4.5 * N[(c / a), $MachinePrecision] + N[(N[(0.5 * t$95$1), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]]]]

Derivation

1. Initial program 56.7%

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

4. Applied egg-rr 56.7%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-inv N/A

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

   7. lift--.f64 N/A

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

   8. flip3-- N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in b around inf

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

9. Simplified 92.5%

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

10. Taylor expanded in a around -inf

    \[\leadsto \frac{\frac{b \cdot \color{blue}{\left(-1 \cdot \left({a}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-9}{2} \cdot \frac{c}{a} + \frac{1}{2} \cdot \frac{\frac{-81}{4} \cdot {c}^{2} + 27 \cdot {c}^{2}}{{b}^{2}}}{a} + \frac{1}{2} \cdot \frac{\frac{-9}{2} \cdot \left(c \cdot \left(\frac{-81}{4} \cdot {c}^{2} + 27 \cdot {c}^{2}\right)\right) + 27 \cdot {c}^{3}}{{b}^{4}}}{a} + \frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\frac{-81}{4} \cdot {c}^{2} + 27 \cdot {c}^{2}\right)}^{2} + \frac{9}{2} \cdot \left(c \cdot \left(\frac{-9}{2} \cdot \left(c \cdot \left(\frac{-81}{4} \cdot {c}^{2} + 27 \cdot {c}^{2}\right)\right) + 27 \cdot {c}^{3}\right)\right)}{{b}^{6}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}}{-3} \]

12. Simplified 92.3%

    \[\leadsto \frac{\frac{b \cdot \color{blue}{\left(-\left(a \cdot \left(a \cdot a\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, \left(\left(c \cdot c\right) \cdot 6.75\right) \cdot \left(\left(c \cdot c\right) \cdot 6.75\right), \left(4.5 \cdot c\right) \cdot \mathsf{fma}\left(-4.5 \cdot c, \left(c \cdot c\right) \cdot 6.75, 27 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{{b}^{6}}, -\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-4.5 \cdot c, \left(c \cdot c\right) \cdot 6.75, 27 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{{b}^{4}}, -\frac{\mathsf{fma}\left(-4.5, \frac{c}{a}, \frac{0.5 \cdot \left(\left(c \cdot c\right) \cdot 6.75\right)}{b \cdot b}\right)}{a}\right)}{a}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}}{-3} \]

13. Final simplification 92.3%

    \[\leadsto \frac{\frac{b \cdot \left(\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, \left(\left(c \cdot c\right) \cdot 6.75\right) \cdot \left(\left(c \cdot c\right) \cdot 6.75\right), \left(c \cdot 4.5\right) \cdot \mathsf{fma}\left(c \cdot -4.5, \left(c \cdot c\right) \cdot 6.75, \left(c \cdot \left(c \cdot c\right)\right) \cdot 27\right)\right)}{{b}^{6}}, \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(c \cdot -4.5, \left(c \cdot c\right) \cdot 6.75, \left(c \cdot \left(c \cdot c\right)\right) \cdot 27\right)}{{b}^{4}}, \frac{\mathsf{fma}\left(-4.5, \frac{c}{a}, \frac{0.5 \cdot \left(\left(c \cdot c\right) \cdot 6.75\right)}{b \cdot b}\right)}{-a}\right)}{-a}\right) \cdot \left(a \cdot \left(a \cdot \left(-a\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)\right)}}{-3} \]
14. Add Preprocessing

Alternative 7: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, b \cdot b, -t\_0 \cdot t\_1\right)}{a}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + t\_1, c \cdot \left(a \cdot -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
     (/
      (* -0.3333333333333333 (/ (fma b (* b b) (- (* t_0 t_1))) a))
      (fma b b (fma b (+ b t_1) (* c (* a -3.0)))))
     (fma
      a
      (/ (* (* c c) (fma -0.5625 (* a (/ c (* b b))) -0.375)) (* b (* b b)))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = (-0.3333333333333333 * (fma(b, (b * b), -(t_0 * t_1)) / a)) / fma(b, b, fma(b, (b + t_1), (c * (a * -3.0))));
	} else {
		tmp = fma(a, (((c * c) * fma(-0.5625, (a * (c / (b * b))), -0.375)) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(fma(b, Float64(b * b), Float64(-Float64(t_0 * t_1))) / a)) / fma(b, b, fma(b, Float64(b + t_1), Float64(c * Float64(a * -3.0)))));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
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[Sqrt[t$95$0], $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[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(-0.3333333333333333 * N[(N[(b * N[(b * b), $MachinePrecision] + (-N[(t$95$0 * t$95$1), $MachinePrecision])), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(b * N[(b + t$95$1), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-inv N/A

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

      7. lift--.f64 N/A

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

      8. flip3-- N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 83.1%

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

   8. Applied egg-rr 82.9%

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

   10. Applied egg-rr 83.4%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2} + \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      21. lower-*.f64 95.1

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

   8. Simplified 95.1%

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

   9. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)}}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-3}{8}}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       6. lower-fma.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{-9}{16}, a \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-3}{8}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       11. lower-*.f64 95.1

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

   11. Simplified 95.1%

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, b \cdot b, -\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}{a}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, c \cdot \left(a \cdot -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b \cdot b, -t\_0 \cdot t\_1\right)}{a \cdot \left(-3 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + t\_1, c \cdot \left(a \cdot -3\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
     (/
      (fma b (* b b) (- (* t_0 t_1)))
      (* a (* -3.0 (fma b b (fma b (+ b t_1) (* c (* a -3.0)))))))
     (fma
      a
      (/ (* (* c c) (fma -0.5625 (* a (/ c (* b b))) -0.375)) (* b (* b b)))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = fma(b, (b * b), -(t_0 * t_1)) / (a * (-3.0 * fma(b, b, fma(b, (b + t_1), (c * (a * -3.0))))));
	} else {
		tmp = fma(a, (((c * c) * fma(-0.5625, (a * (c / (b * b))), -0.375)) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(fma(b, Float64(b * b), Float64(-Float64(t_0 * t_1))) / Float64(a * Float64(-3.0 * fma(b, b, fma(b, Float64(b + t_1), Float64(c * Float64(a * -3.0)))))));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
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[Sqrt[t$95$0], $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[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(b * N[(b * b), $MachinePrecision] + (-N[(t$95$0 * t$95$1), $MachinePrecision])), $MachinePrecision] / N[(a * N[(-3.0 * N[(b * b + N[(b * N[(b + t$95$1), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-inv N/A

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

      7. lift--.f64 N/A

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

      8. flip3-- N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 83.1%

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

   8. Applied egg-rr 82.9%

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

   10. Applied egg-rr 83.4%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2} + \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      21. lower-*.f64 95.1

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

   8. Simplified 95.1%

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

   9. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)}}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-3}{8}}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       6. lower-fma.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{-9}{16}, a \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-3}{8}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       11. lower-*.f64 95.1

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

   11. Simplified 95.1%

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b \cdot b, -\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}{a \cdot \left(-3 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}, c \cdot \left(a \cdot -3\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - t\_0}{b + \sqrt{t\_0}}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
     (/ (/ (/ (- (* b b) t_0) (+ b (sqrt t_0))) a) -3.0)
     (fma
      a
      (/ (* (* c c) (fma -0.5625 (* a (/ c (* b b))) -0.375)) (* b (* b b)))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = ((((b * b) - t_0) / (b + sqrt(t_0))) / a) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * fma(-0.5625, (a * (c / (b * b))), -0.375)) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(b + sqrt(t_0))) / a) / -3.0);
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $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[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   6. Applied egg-rr 83.3%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2} + \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      21. lower-*.f64 95.1

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

   8. Simplified 95.1%

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

   9. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)}}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-3}{8}}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       6. lower-fma.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{-9}{16}, a \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-3}{8}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       11. lower-*.f64 95.1

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

   11. Simplified 95.1%

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{a \cdot \left(b + \sqrt{t\_0}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
     (/ (/ (- (* b b) t_0) (* a (+ b (sqrt t_0)))) -3.0)
     (fma
      a
      (/ (* (* c c) (fma -0.5625 (* a (/ c (* b b))) -0.375)) (* b (* b b)))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = (((b * b) - t_0) / (a * (b + sqrt(t_0)))) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * fma(-0.5625, (a * (c / (b * b))), -0.375)) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(a * Float64(b + sqrt(t_0)))) / -3.0);
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $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[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied egg-rr 83.3%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2} + \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      21. lower-*.f64 95.1

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

   8. Simplified 95.1%

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

   9. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right)}}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-3}{8}}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       6. lower-fma.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{-9}{16}, a \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-3}{8}\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

       11. lower-*.f64 95.1

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

   11. Simplified 95.1%

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{a \cdot \left(b + \sqrt{t\_0}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
     (/ (/ (- (* b b) t_0) (* a (+ b (sqrt t_0)))) -3.0)
     (fma a (/ (* (* c c) -0.375) (* b (* b b))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = (((b * b) - t_0) / (a * (b + sqrt(t_0)))) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(a * Float64(b + sqrt(t_0)))) / -3.0);
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $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[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied egg-rr 83.3%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 90.7

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Simplified 90.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
     (/ (- (* b b) t_0) (* (* a -3.0) (+ b (sqrt t_0))))
     (fma a (/ (* (* c c) -0.375) (* b (* b b))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = ((b * b) - t_0) / ((a * -3.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(a * -3.0) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $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[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 83.3%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 90.7

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Simplified 90.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
   (/ (/ (- b (sqrt (fma b b (* c (* a -3.0))))) a) -3.0)
   (fma a (/ (* (* c c) -0.375) (* b (* b b))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = ((b - sqrt(fma(b, b, (c * (a * -3.0))))) / a) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / a) / -3.0);
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.2

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

   6. Applied egg-rr 82.2%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 96.8%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 90.7

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Simplified 90.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
   (/ (/ (- b (sqrt (fma b b (* c (* a -3.0))))) a) -3.0)
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = ((b - sqrt(fma(b, b, (c * (a * -3.0))))) / a) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / a) / -3.0);
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

   4. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.2

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

   6. Applied egg-rr 82.2%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 82.2

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

   4. Applied egg-rr 82.2%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.008)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (* c (fma a (* -0.375 (/ c (* b (* b b)))) (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.008) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * fma(a, (-0.375 * (c / (b * (b * b)))), (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.008)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * fma(a, Float64(-0.375 * Float64(c / Float64(b * Float64(b * b)))), Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.008], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * N[(-0.375 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0080000000000000002

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 82.2

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

   4. Applied egg-rr 82.2%

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

   if -0.0080000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{3}} \cdot \frac{-3}{8}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.008:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.5e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -1.5e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -1.5e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1.5e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.50000000000000004e-5

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 75.8

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

   4. Applied egg-rr 75.8%

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

   if -1.50000000000000004e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 39.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 78.9

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.5e-5)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -1.5e-5) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -1.5e-5)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1.5e-5], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.50000000000000004e-5

   1. Initial program 75.8%

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

   4. Applied egg-rr 75.8%

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

   if -1.50000000000000004e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 39.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 78.9

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

   2. lower-/.f64 63.9

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Simplified 63.9%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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)))