Cubic critical, narrow range

Percentage Accurate: 55.6% → 92.0%

Time: 15.0s

Alternatives: 14

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Initial Program: 55.6% 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: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\\ t_1 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - t\_1}{b + \sqrt{t\_1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.75, c \cdot \frac{t\_0}{b \cdot b}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)}{c \cdot c}, \frac{\left(c \cdot c\right) \cdot 0.5625}{{b}^{5}}\right)\right), t\_0\right), \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c (* b (* b b))) -0.375)) (t_1 (fma -3.0 (* a c) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.25)
     (/ -1.0 (/ a (* 0.3333333333333333 (/ (- (* b b) t_1) (+ b (sqrt t_1))))))
     (/
      1.0
      (fma
       a
       (fma
        a
        (*
         -3.0
         (fma
          a
          (fma
           -0.75
           (* c (/ t_0 (* b b)))
           (fma
            -0.2222222222222222
            (/ (* b (* (/ (pow c 4.0) (pow b 6.0)) 6.328125)) (* c c))
            (/ (* (* c c) 0.5625) (pow b 5.0))))
          t_0))
        (/ 1.5 b))
       (/ (* b -2.0) c))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / (b * (b * b))) * -0.375;
	double t_1 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.25) {
		tmp = -1.0 / (a / (0.3333333333333333 * (((b * b) - t_1) / (b + sqrt(t_1)))));
	} else {
		tmp = 1.0 / fma(a, fma(a, (-3.0 * fma(a, fma(-0.75, (c * (t_0 / (b * b))), fma(-0.2222222222222222, ((b * ((pow(c, 4.0) / pow(b, 6.0)) * 6.328125)) / (c * c)), (((c * c) * 0.5625) / pow(b, 5.0)))), t_0)), (1.5 / b)), ((b * -2.0) / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / Float64(b * Float64(b * b))) * -0.375)
	t_1 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.25)
		tmp = Float64(-1.0 / Float64(a / Float64(0.3333333333333333 * Float64(Float64(Float64(b * b) - t_1) / Float64(b + sqrt(t_1))))));
	else
		tmp = Float64(1.0 / fma(a, fma(a, Float64(-3.0 * fma(a, fma(-0.75, Float64(c * Float64(t_0 / Float64(b * b))), fma(-0.2222222222222222, Float64(Float64(b * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125)) / Float64(c * c)), Float64(Float64(Float64(c * c) * 0.5625) / (b ^ 5.0)))), t_0)), Float64(1.5 / b)), Float64(Float64(b * -2.0) / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]}, Block[{t$95$1 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.25], N[(-1.0 / N[(a / N[(0.3333333333333333 * N[(N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(a * N[(-3.0 * N[(a * N[(-0.75 * N[(c * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.2222222222222222 * N[(N[(b * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * 0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * -2.0), $MachinePrecision] / c), $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)) < -0.25

   1. Initial program 86.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 86.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 87.5%

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

   if -0.25 < (/.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 46.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 46.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.75, c \cdot \frac{\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375}{b \cdot b}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)}{c \cdot c}, \frac{\left(c \cdot c\right) \cdot 0.5625}{{b}^{5}}\right)\right), \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right), \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - t\_0}{b + \sqrt{t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(t\_1 \cdot t\_1\right)}, -0.16666666666666666, \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_1}\right), a, \frac{c \cdot \left(c \cdot -0.375\right)}{t\_1}\right), -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))) (t_1 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.25)
     (/ -1.0 (/ a (* 0.3333333333333333 (/ (- (* b b) t_0) (+ b (sqrt t_0))))))
     (fma
      a
      (fma
       (fma
        (/ (* (* c (* c (* c c))) (* a 6.328125)) (* b (* t_1 t_1)))
        -0.16666666666666666
        (/ (* c (* (* c c) -0.5625)) (* (* b b) t_1)))
       a
       (/ (* c (* c -0.375)) t_1))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double t_1 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.25) {
		tmp = -1.0 / (a / (0.3333333333333333 * (((b * b) - t_0) / (b + sqrt(t_0)))));
	} else {
		tmp = fma(a, fma(fma((((c * (c * (c * c))) * (a * 6.328125)) / (b * (t_1 * t_1))), -0.16666666666666666, ((c * ((c * c) * -0.5625)) / ((b * b) * t_1))), a, ((c * (c * -0.375)) / t_1)), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.25)
		tmp = Float64(-1.0 / Float64(a / Float64(0.3333333333333333 * Float64(Float64(Float64(b * b) - t_0) / Float64(b + sqrt(t_0))))));
	else
		tmp = fma(a, fma(fma(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 6.328125)) / Float64(b * Float64(t_1 * t_1))), -0.16666666666666666, Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_1))), a, Float64(Float64(c * Float64(c * -0.375)) / t_1)), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.25], N[(-1.0 / N[(a / N[(0.3333333333333333 * N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / t$95$1), $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)) < -0.25

   1. Initial program 86.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 86.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 87.5%

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

   if -0.25 < (/.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 46.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. Applied rewrites 96.2%

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

   7. Applied rewrites 96.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -0.16666666666666666, \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), a, \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - t\_0}{b + \sqrt{t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{b \cdot b}\right) + \mathsf{fma}\left(-0.5625, \frac{c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{t\_1}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot t\_1\right)}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))) (t_1 (* b (* b (* b b)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.25)
     (/ -1.0 (/ a (* 0.3333333333333333 (/ (- (* b b) t_0) (+ b (sqrt t_0))))))
     (/
      (+
       (fma c -0.5 (/ (* -0.375 (* c (* a c))) (* b b)))
       (fma
        -0.5625
        (/ (* c (* c (* c (* a a)))) t_1)
        (/
         (* (* c (* c (* c c))) (* -1.0546875 (* a (* a a))))
         (* b (* b t_1)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double t_1 = b * (b * (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.25) {
		tmp = -1.0 / (a / (0.3333333333333333 * (((b * b) - t_0) / (b + sqrt(t_0)))));
	} else {
		tmp = (fma(c, -0.5, ((-0.375 * (c * (a * c))) / (b * b))) + fma(-0.5625, ((c * (c * (c * (a * a)))) / t_1), (((c * (c * (c * c))) * (-1.0546875 * (a * (a * a)))) / (b * (b * t_1))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	t_1 = Float64(b * Float64(b * Float64(b * b)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.25)
		tmp = Float64(-1.0 / Float64(a / Float64(0.3333333333333333 * Float64(Float64(Float64(b * b) - t_0) / Float64(b + sqrt(t_0))))));
	else
		tmp = Float64(Float64(fma(c, -0.5, Float64(Float64(-0.375 * Float64(c * Float64(a * c))) / Float64(b * b))) + fma(-0.5625, Float64(Float64(c * Float64(c * Float64(c * Float64(a * a)))) / t_1), Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(-1.0546875 * Float64(a * Float64(a * a)))) / Float64(b * Float64(b * t_1))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.25], N[(-1.0 / N[(a / N[(0.3333333333333333 * N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5 + N[(N[(-0.375 * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(c * N[(c * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.25

   1. Initial program 86.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 86.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 87.5%

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

   if -0.25 < (/.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 46.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. Applied rewrites 96.2%

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.2%

       \[\leadsto \frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)\right)}{\color{blue}{b}} \]

   12. Applied rewrites 96.2%

       \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(c \cdot a\right)\right)}{b \cdot b}\right) + \mathsf{fma}\left(-0.5625, \frac{c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{b \cdot b}\right) + \mathsf{fma}\left(-0.5625, \frac{c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - t\_0}{b + \sqrt{t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{b \cdot \left(b \cdot t\_1\right)}, -1.0546875, \mathsf{fma}\left(-0.5625, \frac{c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{t\_1}, \mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))) (t_1 (* b (* b (* b b)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.25)
     (/ -1.0 (/ a (* 0.3333333333333333 (/ (- (* b b) t_0) (+ b (sqrt t_0))))))
     (/
      (fma
       (/ (* (* a a) (* a (* c (* c (* c c))))) (* b (* b t_1)))
       -1.0546875
       (fma
        -0.5625
        (/ (* c (* c (* c (* a a)))) t_1)
        (fma c -0.5 (/ (* -0.375 (* c (* a c))) (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double t_1 = b * (b * (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.25) {
		tmp = -1.0 / (a / (0.3333333333333333 * (((b * b) - t_0) / (b + sqrt(t_0)))));
	} else {
		tmp = fma((((a * a) * (a * (c * (c * (c * c))))) / (b * (b * t_1))), -1.0546875, fma(-0.5625, ((c * (c * (c * (a * a)))) / t_1), fma(c, -0.5, ((-0.375 * (c * (a * c))) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	t_1 = Float64(b * Float64(b * Float64(b * b)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.25)
		tmp = Float64(-1.0 / Float64(a / Float64(0.3333333333333333 * Float64(Float64(Float64(b * b) - t_0) / Float64(b + sqrt(t_0))))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * Float64(a * Float64(c * Float64(c * Float64(c * c))))) / Float64(b * Float64(b * t_1))), -1.0546875, fma(-0.5625, Float64(Float64(c * Float64(c * Float64(c * Float64(a * a)))) / t_1), fma(c, -0.5, Float64(Float64(-0.375 * Float64(c * Float64(a * c))) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.25], N[(-1.0 / N[(a / N[(0.3333333333333333 * N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0546875 + N[(-0.5625 * N[(N[(c * N[(c * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(c * -0.5 + N[(N[(-0.375 * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.25

   1. Initial program 86.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 86.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 87.5%

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

   if -0.25 < (/.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 46.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. Applied rewrites 96.2%

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.2%

       \[\leadsto \frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)\right)}{\color{blue}{b}} \]

   12. Applied rewrites 96.2%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -1.0546875, \mathsf{fma}\left(-0.5625, \frac{c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(c \cdot a\right)\right)}{b \cdot b}\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.25:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -1.0546875, \mathsf{fma}\left(-0.5625, \frac{c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - t\_0}{b + \sqrt{t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -3, -0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, 1.5 \cdot \frac{a}{b}\right), b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.05)
     (/ -1.0 (/ a (* 0.3333333333333333 (/ (- (* b b) t_0) (+ b (sqrt t_0))))))
     (/
      1.0
      (/
       (fma
        c
        (fma (* c -3.0) (* -0.375 (/ (* a a) (* b (* b b)))) (* 1.5 (/ a b)))
        (* b -2.0))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.05) {
		tmp = -1.0 / (a / (0.3333333333333333 * (((b * b) - t_0) / (b + sqrt(t_0)))));
	} else {
		tmp = 1.0 / (fma(c, fma((c * -3.0), (-0.375 * ((a * a) / (b * (b * b)))), (1.5 * (a / b))), (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.05)
		tmp = Float64(-1.0 / Float64(a / Float64(0.3333333333333333 * Float64(Float64(Float64(b * b) - t_0) / Float64(b + sqrt(t_0))))));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(Float64(c * -3.0), Float64(-0.375 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))), Float64(1.5 * Float64(a / b))), Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.05], N[(-1.0 / N[(a / N[(0.3333333333333333 * N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(N[(c * -3.0), $MachinePrecision] * N[(-0.375 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.050000000000000003

   1. Initial program 85.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 85.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 87.1%

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

   if -0.050000000000000003 < (/.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 45.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 94.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{-1}{\frac{a}{0.3333333333333333 \cdot \frac{b \cdot b - \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -3, -0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, 1.5 \cdot \frac{a}{b}\right), b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -3, -0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, 1.5 \cdot \frac{a}{b}\right), b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.05)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ b (sqrt t_0))))
     (/
      1.0
      (/
       (fma
        c
        (fma (* c -3.0) (* -0.375 (/ (* a a) (* b (* b b)))) (* 1.5 (/ a b)))
        (* b -2.0))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.05) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (b + sqrt(t_0)));
	} else {
		tmp = 1.0 / (fma(c, fma((c * -3.0), (-0.375 * ((a * a) / (b * (b * b)))), (1.5 * (a / b))), (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.05)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(Float64(c * -3.0), Float64(-0.375 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))), Float64(1.5 * Float64(a / b))), Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(N[(c * -3.0), $MachinePrecision] * N[(-0.375 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.050000000000000003

   1. Initial program 85.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 87.0%

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

   if -0.050000000000000003 < (/.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 45.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 94.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -3, -0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, 1.5 \cdot \frac{a}{b}\right), b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -3, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.05)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ b (sqrt t_0))))
     (/
      1.0
      (fma
       a
       (fma (* a -3.0) (* (/ c (* b (* b b))) -0.375) (/ 1.5 b))
       (/ (* b -2.0) c))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.05) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (b + sqrt(t_0)));
	} else {
		tmp = 1.0 / fma(a, fma((a * -3.0), ((c / (b * (b * b))) * -0.375), (1.5 / b)), ((b * -2.0) / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.05)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(1.0 / fma(a, fma(Float64(a * -3.0), Float64(Float64(c / Float64(b * Float64(b * b))) * -0.375), Float64(1.5 / b)), Float64(Float64(b * -2.0) / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(N[(a * -3.0), $MachinePrecision] * N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * -2.0), $MachinePrecision] / c), $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)) < -0.050000000000000003

   1. Initial program 85.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 87.0%

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

   if -0.050000000000000003 < (/.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 45.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 94.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -3, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{a \cdot c}{b}, b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.038)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ b (sqrt t_0))))
     (/ 1.0 (/ (fma 1.5 (/ (* a c) b) (* b -2.0)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.038) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (b + sqrt(t_0)));
	} else {
		tmp = 1.0 / (fma(1.5, ((a * c) / b), (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.038)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(a * c) / b), Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.038], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.0379999999999999991

   1. Initial program 85.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 86.9%

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

   if -0.0379999999999999991 < (/.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 44.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 91.2

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{a \cdot c}{b}, \color{blue}{-2 \cdot b}\right)}{c}} \]

   10. Applied rewrites 91.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{a \cdot c}{b}, b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{a \cdot c}{b}, b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.038)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ 1.0 (/ (fma 1.5 (/ (* a c) b) (* b -2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.038) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(1.5, ((a * c) / b), (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.038)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(a * c) / b), Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.038], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.0379999999999999991

   1. Initial program 85.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 85.8

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

   4. Applied rewrites 85.8%

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

   if -0.0379999999999999991 < (/.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 44.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 91.2

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{a \cdot c}{b}, \color{blue}{-2 \cdot b}\right)}{c}} \]

   10. Applied rewrites 91.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{a \cdot c}{b}, b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.038)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ 1.0 (fma -2.0 (/ b c) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.038) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.038)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.038], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0379999999999999991

   1. Initial program 85.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 85.8

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

   4. Applied rewrites 85.8%

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

   if -0.0379999999999999991 < (/.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 44.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 91.1

         \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

   10. Applied rewrites 91.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.038)
   (* 0.3333333333333333 (/ (- (sqrt (fma -3.0 (* a c) (* b b))) b) a))
   (/ 1.0 (fma -2.0 (/ b c) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.038) {
		tmp = 0.3333333333333333 * ((sqrt(fma(-3.0, (a * c), (b * b))) - b) / a);
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.038)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(-3.0, Float64(a * c), Float64(b * b))) - b) / a));
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.038], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0379999999999999991

   1. Initial program 85.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if -0.0379999999999999991 < (/.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 44.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 91.1

         \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

   10. Applied rewrites 91.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.038:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma -2.0 (/ b c) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-2.0, (b / c), (1.5 * (a / b)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(1.5 * Float64(a / b))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 51.7

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

5. Applied rewrites 51.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 51.9%

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

8. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 84.9

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

10. Applied rewrites 84.9%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}} \]
11. Add Preprocessing

Alternative 13: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma -0.375 (/ (* a c) (* b b)) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma(-0.375, ((a * c) / (b * b)), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 51.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Applied rewrites 92.3%

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

7. Applied rewrites 92.3%

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

8. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{\color{blue}{b}} \]
9. Step-by-step derivation

10. Applied rewrites 92.3%

    \[\leadsto \frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)\right)}{\color{blue}{b}} \]

11. Taylor expanded in c around 0

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

13. Applied rewrites 84.5%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \]
14. Add Preprocessing

Alternative 14: 64.3% 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 51.8%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 67.7

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

5. Applied rewrites 67.7%

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

Reproduce

herbie shell --seed 2024233 
(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)))