Cubic critical, narrow range

Percentage Accurate: 55.2% → 91.3%

Time: 14.0s

Alternatives: 19

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 91.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	t_0 = Float64(-9.0 * Float64(a * c))
	t_1 = fma(Float64(Float64(c * c) * Float64(a * a)), 27.0, Float64((t_0 ^ 2.0) * -0.25))
	t_2 = fma(Float64((a ^ 3.0) * -27.0), (c ^ 3.0), Float64(Float64(t_1 * t_0) * -0.5))
	t_3 = Float64(t_0 * 0.5)
	return Float64(Float64(Float64(fma(-0.5, Float64(fma(t_3, t_2, Float64((t_1 ^ 2.0) * 0.25)) / (b ^ 6.0)), fma(0.5, Float64(Float64(t_1 / Float64(b * b)) + Float64(t_2 / (b ^ 4.0))), t_3)) * b) / Float64(fma(2.0, Float64(b * b), fma(b, b, Float64(fma(a, -4.5, Float64(fma(Float64(-1.125 / b), Float64(Float64(a * a) / b), Float64(Float64(Float64((a ^ 3.0) * c) * -1.6875) / (b ^ 4.0))) * c)) * c))) * a)) / 3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-9.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[a, 3.0], $MachinePrecision] * -27.0), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * 0.5), $MachinePrecision]}, N[(N[(N[(N[(-0.5 * N[(N[(t$95$3 * t$95$2 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(2.0 * N[(b * b), $MachinePrecision] + N[(b * b + N[(N[(a * -4.5 + N[(N[(N[(-1.125 / b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] * -1.6875), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]

Derivation

1. Initial program 55.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

6. Applied rewrites 55.8%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}}{3} \]

7. Taylor expanded in b around inf

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

9. Applied rewrites 91.2%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

10. Taylor expanded in c around 0

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

    1. lower-fma.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 N/A

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

    4. +-commutative N/A

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

    5. unpow2 N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. associate-+r+ N/A

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

12. Applied rewrites 91.3%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(2, b \cdot b, \mathsf{fma}\left(b, b, c \cdot \mathsf{fma}\left(a, -4.5, c \cdot \mathsf{fma}\left(\frac{-1.125}{b}, \frac{a \cdot a}{b}, \frac{-1.6875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{4}}\right)\right)\right)\right)} \cdot a}}{3} \]

13. Final simplification 91.3%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5, \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right), {\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)\right)}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)}{b \cdot b} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{4}}, \left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5\right)\right) \cdot b}{\mathsf{fma}\left(2, b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, -4.5, \mathsf{fma}\left(\frac{-1.125}{b}, \frac{a \cdot a}{b}, \frac{\left({a}^{3} \cdot c\right) \cdot -1.6875}{{b}^{4}}\right) \cdot c\right) \cdot c\right)\right) \cdot a}}{3} \]
14. Add Preprocessing

Alternative 2: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ t_1 := -9 \cdot \left(a \cdot c\right)\\ t_2 := \mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {t\_1}^{2} \cdot -0.25\right)\\ t_3 := \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(t\_2 \cdot t\_1\right) \cdot -0.5\right)\\ t_4 := t\_1 \cdot 0.5\\ \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(t\_4, t\_3, {t\_2}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{t\_2}{b \cdot b} + \frac{t\_3}{{b}^{4}}, t\_4\right)\right) \cdot b}{\left(\mathsf{fma}\left(b, b, t\_0\right) + \sqrt{t\_0} \cdot b\right) \cdot a}}{3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b)))
        (t_1 (* -9.0 (* a c)))
        (t_2 (fma (* (* c c) (* a a)) 27.0 (* (pow t_1 2.0) -0.25)))
        (t_3 (fma (* (pow a 3.0) -27.0) (pow c 3.0) (* (* t_2 t_1) -0.5)))
        (t_4 (* t_1 0.5)))
   (/
    (/
     (*
      (fma
       -0.5
       (/ (fma t_4 t_3 (* (pow t_2 2.0) 0.25)) (pow b 6.0))
       (fma 0.5 (+ (/ t_2 (* b b)) (/ t_3 (pow b 4.0))) t_4))
      b)
     (* (+ (fma b b t_0) (* (sqrt t_0) b)) a))
    3.0)))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double t_1 = -9.0 * (a * c);
	double t_2 = fma(((c * c) * (a * a)), 27.0, (pow(t_1, 2.0) * -0.25));
	double t_3 = fma((pow(a, 3.0) * -27.0), pow(c, 3.0), ((t_2 * t_1) * -0.5));
	double t_4 = t_1 * 0.5;
	return ((fma(-0.5, (fma(t_4, t_3, (pow(t_2, 2.0) * 0.25)) / pow(b, 6.0)), fma(0.5, ((t_2 / (b * b)) + (t_3 / pow(b, 4.0))), t_4)) * b) / ((fma(b, b, t_0) + (sqrt(t_0) * b)) * a)) / 3.0;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	t_1 = Float64(-9.0 * Float64(a * c))
	t_2 = fma(Float64(Float64(c * c) * Float64(a * a)), 27.0, Float64((t_1 ^ 2.0) * -0.25))
	t_3 = fma(Float64((a ^ 3.0) * -27.0), (c ^ 3.0), Float64(Float64(t_2 * t_1) * -0.5))
	t_4 = Float64(t_1 * 0.5)
	return Float64(Float64(Float64(fma(-0.5, Float64(fma(t_4, t_3, Float64((t_2 ^ 2.0) * 0.25)) / (b ^ 6.0)), fma(0.5, Float64(Float64(t_2 / Float64(b * b)) + Float64(t_3 / (b ^ 4.0))), t_4)) * b) / Float64(Float64(fma(b, b, t_0) + Float64(sqrt(t_0) * b)) * a)) / 3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-9.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[a, 3.0], $MachinePrecision] * -27.0), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$2 * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * 0.5), $MachinePrecision]}, N[(N[(N[(N[(-0.5 * N[(N[(t$95$4 * t$95$3 + N[(N[Power[t$95$2, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(b * b + t$95$0), $MachinePrecision] + N[(N[Sqrt[t$95$0], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Initial program 55.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

6. Applied rewrites 55.8%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}}{3} \]

7. Taylor expanded in b around inf

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

9. Applied rewrites 91.2%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. distribute-rgt-in N/A

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

    4. lift-*.f64 N/A

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

    5. associate-+l+ N/A

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

    6. lower-+.f64 N/A

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

11. Applied rewrites 91.2%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} \cdot b + \mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \cdot a}}{3} \]

12. Final simplification 91.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5, \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right), {\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)\right)}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)}{b \cdot b} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{4}}, \left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5\right)\right) \cdot b}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right) + \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} \cdot b\right) \cdot a}}{3} \]
13. Add Preprocessing

Alternative 3: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ t_1 := -9 \cdot \left(a \cdot c\right)\\ t_2 := \mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {t\_1}^{2} \cdot -0.25\right)\\ t_3 := \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(t\_2 \cdot t\_1\right) \cdot -0.5\right)\\ t_4 := t\_1 \cdot 0.5\\ \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(t\_4, t\_3, {t\_2}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{t\_2}{b \cdot b} + \frac{t\_3}{{b}^{4}}, t\_4\right)\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{t\_0} + b, t\_0\right) \cdot a}}{3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b)))
        (t_1 (* -9.0 (* a c)))
        (t_2 (fma (* (* c c) (* a a)) 27.0 (* (pow t_1 2.0) -0.25)))
        (t_3 (fma (* (pow a 3.0) -27.0) (pow c 3.0) (* (* t_2 t_1) -0.5)))
        (t_4 (* t_1 0.5)))
   (/
    (/
     (*
      (fma
       -0.5
       (/ (fma t_4 t_3 (* (pow t_2 2.0) 0.25)) (pow b 6.0))
       (fma 0.5 (+ (/ t_2 (* b b)) (/ t_3 (pow b 4.0))) t_4))
      b)
     (* (fma b (+ (sqrt t_0) b) t_0) a))
    3.0)))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double t_1 = -9.0 * (a * c);
	double t_2 = fma(((c * c) * (a * a)), 27.0, (pow(t_1, 2.0) * -0.25));
	double t_3 = fma((pow(a, 3.0) * -27.0), pow(c, 3.0), ((t_2 * t_1) * -0.5));
	double t_4 = t_1 * 0.5;
	return ((fma(-0.5, (fma(t_4, t_3, (pow(t_2, 2.0) * 0.25)) / pow(b, 6.0)), fma(0.5, ((t_2 / (b * b)) + (t_3 / pow(b, 4.0))), t_4)) * b) / (fma(b, (sqrt(t_0) + b), t_0) * a)) / 3.0;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	t_1 = Float64(-9.0 * Float64(a * c))
	t_2 = fma(Float64(Float64(c * c) * Float64(a * a)), 27.0, Float64((t_1 ^ 2.0) * -0.25))
	t_3 = fma(Float64((a ^ 3.0) * -27.0), (c ^ 3.0), Float64(Float64(t_2 * t_1) * -0.5))
	t_4 = Float64(t_1 * 0.5)
	return Float64(Float64(Float64(fma(-0.5, Float64(fma(t_4, t_3, Float64((t_2 ^ 2.0) * 0.25)) / (b ^ 6.0)), fma(0.5, Float64(Float64(t_2 / Float64(b * b)) + Float64(t_3 / (b ^ 4.0))), t_4)) * b) / Float64(fma(b, Float64(sqrt(t_0) + b), t_0) * a)) / 3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-9.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[a, 3.0], $MachinePrecision] * -27.0), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$2 * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * 0.5), $MachinePrecision]}, N[(N[(N[(N[(-0.5 * N[(N[(t$95$4 * t$95$3 + N[(N[Power[t$95$2, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] + t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Initial program 55.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

6. Applied rewrites 55.8%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}}{3} \]

7. Taylor expanded in b around inf

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

9. Applied rewrites 91.2%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

10. Final simplification 91.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5, \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right), {\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)\right)}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)}{b \cdot b} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{4}}, \left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5\right)\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right) \cdot a}}{3} \]
11. Add Preprocessing

Alternative 4: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9 \cdot \left(a \cdot c\right)\\ t_1 := \mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {t\_0}^{2} \cdot -0.25\right)\\ t_2 := 6.75 \cdot \left(a \cdot a\right)\\ t_3 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {t\_2}^{2}, \left(\mathsf{fma}\left(-4.5, t\_2 \cdot a, 27 \cdot {a}^{3}\right) \cdot a\right) \cdot 4.5\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{t\_1}{b \cdot b} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(t\_1 \cdot t\_0\right) \cdot -0.5\right)}{{b}^{4}}, t\_0 \cdot 0.5\right)\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{t\_3} + b, t\_3\right) \cdot a}}{3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -9.0 (* a c)))
        (t_1 (fma (* (* c c) (* a a)) 27.0 (* (pow t_0 2.0) -0.25)))
        (t_2 (* 6.75 (* a a)))
        (t_3 (fma (* c -3.0) a (* b b))))
   (/
    (/
     (*
      (fma
       -0.5
       (/
        (*
         (fma
          0.25
          (pow t_2 2.0)
          (* (* (fma -4.5 (* t_2 a) (* 27.0 (pow a 3.0))) a) 4.5))
         (pow c 4.0))
        (pow b 6.0))
       (fma
        0.5
        (+
         (/ t_1 (* b b))
         (/
          (fma (* (pow a 3.0) -27.0) (pow c 3.0) (* (* t_1 t_0) -0.5))
          (pow b 4.0)))
        (* t_0 0.5)))
      b)
     (* (fma b (+ (sqrt t_3) b) t_3) a))
    3.0)))

C

double code(double a, double b, double c) {
	double t_0 = -9.0 * (a * c);
	double t_1 = fma(((c * c) * (a * a)), 27.0, (pow(t_0, 2.0) * -0.25));
	double t_2 = 6.75 * (a * a);
	double t_3 = fma((c * -3.0), a, (b * b));
	return ((fma(-0.5, ((fma(0.25, pow(t_2, 2.0), ((fma(-4.5, (t_2 * a), (27.0 * pow(a, 3.0))) * a) * 4.5)) * pow(c, 4.0)) / pow(b, 6.0)), fma(0.5, ((t_1 / (b * b)) + (fma((pow(a, 3.0) * -27.0), pow(c, 3.0), ((t_1 * t_0) * -0.5)) / pow(b, 4.0))), (t_0 * 0.5))) * b) / (fma(b, (sqrt(t_3) + b), t_3) * a)) / 3.0;
}

Julia

function code(a, b, c)
	t_0 = Float64(-9.0 * Float64(a * c))
	t_1 = fma(Float64(Float64(c * c) * Float64(a * a)), 27.0, Float64((t_0 ^ 2.0) * -0.25))
	t_2 = Float64(6.75 * Float64(a * a))
	t_3 = fma(Float64(c * -3.0), a, Float64(b * b))
	return Float64(Float64(Float64(fma(-0.5, Float64(Float64(fma(0.25, (t_2 ^ 2.0), Float64(Float64(fma(-4.5, Float64(t_2 * a), Float64(27.0 * (a ^ 3.0))) * a) * 4.5)) * (c ^ 4.0)) / (b ^ 6.0)), fma(0.5, Float64(Float64(t_1 / Float64(b * b)) + Float64(fma(Float64((a ^ 3.0) * -27.0), (c ^ 3.0), Float64(Float64(t_1 * t_0) * -0.5)) / (b ^ 4.0))), Float64(t_0 * 0.5))) * b) / Float64(fma(b, Float64(sqrt(t_3) + b), t_3) * a)) / 3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-9.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(6.75 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(-0.5 * N[(N[(N[(0.25 * N[Power[t$95$2, 2.0], $MachinePrecision] + N[(N[(N[(-4.5 * N[(t$95$2 * a), $MachinePrecision] + N[(27.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * -27.0), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * N[(N[Sqrt[t$95$3], $MachinePrecision] + b), $MachinePrecision] + t$95$3), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]

Derivation

1. Initial program 55.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

6. Applied rewrites 55.8%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}}{3} \]

7. Taylor expanded in b around inf

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

9. Applied rewrites 91.2%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

10. Taylor expanded in c around -inf

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

12. Applied rewrites 91.2%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{{c}^{4} \cdot \mathsf{fma}\left(0.25, {\left(\left(a \cdot a\right) \cdot 6.75\right)}^{2}, 4.5 \cdot \left(a \cdot \mathsf{fma}\left(-4.5, a \cdot \left(\left(a \cdot a\right) \cdot 6.75\right), 27 \cdot {a}^{3}\right)\right)\right)}{{\color{blue}{b}}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

13. Final simplification 91.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(6.75 \cdot \left(a \cdot a\right)\right)}^{2}, \left(\mathsf{fma}\left(-4.5, \left(6.75 \cdot \left(a \cdot a\right)\right) \cdot a, 27 \cdot {a}^{3}\right) \cdot a\right) \cdot 4.5\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right)}{b \cdot b} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(a \cdot c\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(a \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{4}}, \left(-9 \cdot \left(a \cdot c\right)\right) \cdot 0.5\right)\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right) \cdot a}}{3} \]
14. Add Preprocessing

Alternative 5: 90.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375, {\left(b \cdot c\right)}^{2}, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right) \cdot {b}^{-7}\right) \cdot a\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ -0.5 b)
  c
  (*
   (*
    (fma
     (* -1.0546875 (* a a))
     (pow c 4.0)
     (* (fma -0.375 (pow (* b c) 2.0) (* (* (pow c 3.0) a) -0.5625)) (* b b)))
    (pow b -7.0))
   a)))

C

double code(double a, double b, double c) {
	return fma((-0.5 / b), c, ((fma((-1.0546875 * (a * a)), pow(c, 4.0), (fma(-0.375, pow((b * c), 2.0), ((pow(c, 3.0) * a) * -0.5625)) * (b * b))) * pow(b, -7.0)) * a));
}

Julia

function code(a, b, c)
	return fma(Float64(-0.5 / b), c, Float64(Float64(fma(Float64(-1.0546875 * Float64(a * a)), (c ^ 4.0), Float64(fma(-0.375, (Float64(b * c) ^ 2.0), Float64(Float64((c ^ 3.0) * a) * -0.5625)) * Float64(b * b))) * (b ^ -7.0)) * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c + N[(N[(N[(N[(-1.0546875 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(-0.375 * N[Power[N[(b * c), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 90.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{-0.5625 \cdot {c}^{3}}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, -0.5 \cdot \frac{c}{b}\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 90.8%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, -0.5 \cdot \frac{c}{b}\right) \]

10. Applied rewrites 90.8%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, \color{blue}{c}, \left({b}^{-7} \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot -1.0546875, {c}^{4}, \mathsf{fma}\left(-0.375, {\left(c \cdot b\right)}^{2}, -0.5625 \cdot \left(a \cdot {c}^{3}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \cdot a\right) \]

11. Final simplification 90.8%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375, {\left(b \cdot c\right)}^{2}, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right) \cdot {b}^{-7}\right) \cdot a\right) \]
12. Add Preprocessing

Alternative 6: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(-0.375, b \cdot b, -0.5625 \cdot \left(a \cdot c\right)\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma
    (* -1.0546875 (pow c 4.0))
    (* a a)
    (* (* (fma -0.375 (* b b) (* -0.5625 (* a c))) (* c c)) (* b b)))
   (pow b 7.0))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma((fma((-1.0546875 * pow(c, 4.0)), (a * a), ((fma(-0.375, (b * b), (-0.5625 * (a * c))) * (c * c)) * (b * b))) / pow(b, 7.0)), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(-1.0546875 * (c ^ 4.0)), Float64(a * a), Float64(Float64(fma(-0.375, Float64(b * b), Float64(-0.5625 * Float64(a * c))) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-1.0546875 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(-0.375 * N[(b * b), $MachinePrecision] + N[(-0.5625 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 90.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{-0.5625 \cdot {c}^{3}}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, -0.5 \cdot \frac{c}{b}\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 90.8%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, -0.5 \cdot \frac{c}{b}\right) \]

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot {c}^{4}, a \cdot a, \left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
10. Step-by-step derivation

11. Applied rewrites 90.8%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.375, b \cdot b, -0.5625 \cdot \left(a \cdot c\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, -0.5 \cdot \frac{c}{b}\right) \]

12. Final simplification 90.8%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(-0.375, b \cdot b, -0.5625 \cdot \left(a \cdot c\right)\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]
13. Add Preprocessing

Alternative 7: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {t\_0}^{1.5}\right)}{\left(\left(c \cdot -3\right) \cdot a + \mathsf{fma}\left(\sqrt{t\_0} + b, b, b \cdot b\right)\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.3)
     (/
      (/
       (fma (* (- b) b) b (pow t_0 1.5))
       (* (+ (* (* c -3.0) a) (fma (+ (sqrt t_0) b) b (* b b))) a))
      3.0)
     (/
      (fma
       (* (/ (fma (/ (* -0.5625 (* a a)) b) (/ c b) (* -0.375 a)) (* b b)) c)
       c
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.3) {
		tmp = (fma((-b * b), b, pow(t_0, 1.5)) / ((((c * -3.0) * a) + fma((sqrt(t_0) + b), b, (b * b))) * a)) / 3.0;
	} else {
		tmp = fma(((fma(((-0.5625 * (a * a)) / b), (c / b), (-0.375 * a)) / (b * b)) * c), c, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, 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)) <= -0.3)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, (t_0 ^ 1.5)) / Float64(Float64(Float64(Float64(c * -3.0) * a) + fma(Float64(sqrt(t_0) + b), b, Float64(b * b))) * a)) / 3.0);
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-0.5625 * Float64(a * a)) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)) * c), c, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.3], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * b + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.299999999999999989

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

   6. Applied rewrites 83.3%

      \[\leadsto \frac{\color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}}{3} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.4

         \[\leadsto \frac{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\left(\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, b, b \cdot b\right) + \color{blue}{\left(-3 \cdot c\right) \cdot a}\right) \cdot a}}{3} \]

   8. Applied rewrites 83.4%

      \[\leadsto \frac{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, b, b \cdot b\right) + \left(-3 \cdot c\right) \cdot a\right)} \cdot a}}{3} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 83.4

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}}}{\left(\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, b, b \cdot b\right) + \left(-3 \cdot c\right) \cdot a\right) \cdot a}}{3} \]

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-pow.f64 N/A

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

      6. cube-neg N/A

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

      7. +-commutative N/A

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

      8. cube-neg N/A

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

      9. unpow3 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 84.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(-b\right)} \cdot b, b, {\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5}\right)}{\left(\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, b, b \cdot b\right) + \left(-3 \cdot c\right) \cdot a\right) \cdot a}}{3} \]

   10. Applied rewrites 84.5%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5}\right)}}{\left(\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, b, b \cdot b\right) + \left(-3 \cdot c\right) \cdot a\right) \cdot a}}{3} \]

   if -0.299999999999999989 < (/.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 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \left(\left(-0.5625 \cdot a\right) \cdot a\right) \cdot \frac{c}{{b}^{4}}\right), c, -0.5\right) \cdot c}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 91.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right)}^{1.5}\right)}{\left(\left(c \cdot -3\right) \cdot a + \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, b, b \cdot b\right)\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, \sqrt{t\_0} + b, t\_0\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.3)
     (/
      (/ (fma (* (- b) b) b (pow t_0 1.5)) (* (fma b (+ (sqrt t_0) b) t_0) a))
      3.0)
     (/
      (fma
       (* (/ (fma (/ (* -0.5625 (* a a)) b) (/ c b) (* -0.375 a)) (* b b)) c)
       c
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.3) {
		tmp = (fma((-b * b), b, pow(t_0, 1.5)) / (fma(b, (sqrt(t_0) + b), t_0) * a)) / 3.0;
	} else {
		tmp = fma(((fma(((-0.5625 * (a * a)) / b), (c / b), (-0.375 * a)) / (b * b)) * c), c, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, 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)) <= -0.3)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, (t_0 ^ 1.5)) / Float64(fma(b, Float64(sqrt(t_0) + b), t_0) * a)) / 3.0);
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-0.5625 * Float64(a * a)) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)) * c), c, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.3], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(b * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] + t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.299999999999999989

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

   6. Applied rewrites 83.3%

      \[\leadsto \frac{\color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot 1}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}}{3} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 83.3

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 84.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(-b\right)} \cdot b, b, {\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]

   if -0.299999999999999989 < (/.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 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \left(\left(-0.5625 \cdot a\right) \cdot a\right) \cdot \frac{c}{{b}^{4}}\right), c, -0.5\right) \cdot c}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 91.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, {\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{\left(t\_0 - b \cdot b\right) \cdot {a}^{-1}}{\sqrt{t\_0} + b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.3)
     (/ (/ (* (- t_0 (* b b)) (pow a -1.0)) (+ (sqrt t_0) b)) 3.0)
     (/
      (fma
       (* (/ (fma (/ (* -0.5625 (* a a)) b) (/ c b) (* -0.375 a)) (* b b)) c)
       c
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.3) {
		tmp = (((t_0 - (b * b)) * pow(a, -1.0)) / (sqrt(t_0) + b)) / 3.0;
	} else {
		tmp = fma(((fma(((-0.5625 * (a * a)) / b), (c / b), (-0.375 * a)) / (b * b)) * c), c, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, 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)) <= -0.3)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) * (a ^ -1.0)) / Float64(sqrt(t_0) + b)) / 3.0);
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-0.5625 * Float64(a * a)) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)) * c), c, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.3], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.299999999999999989

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 84.2%

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

   if -0.299999999999999989 < (/.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 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \left(\left(-0.5625 \cdot a\right) \cdot a\right) \cdot \frac{c}{{b}^{4}}\right), c, -0.5\right) \cdot c}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 91.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot {a}^{-1}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.3)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (/
      (fma
       (* (/ (fma (/ (* -0.5625 (* a a)) b) (/ c b) (* -0.375 a)) (* b b)) c)
       c
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.3) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma(((fma(((-0.5625 * (a * a)) / b), (c / b), (-0.375 * a)) / (b * b)) * c), c, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, 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)) <= -0.3)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-0.5625 * Float64(a * a)) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)) * c), c, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.3], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.299999999999999989

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

   6. Applied rewrites 84.2%

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

   if -0.299999999999999989 < (/.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 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \left(\left(-0.5625 \cdot a\right) \cdot a\right) \cdot \frac{c}{{b}^{4}}\right), c, -0.5\right) \cdot c}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 91.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b} \cdot c, c, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b}, c, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.3)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (/
      (*
       (fma
        (/ (fma (/ (* -0.5625 (* a a)) b) (/ c b) (* -0.375 a)) (* b b))
        c
        -0.5)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.3) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = (fma((fma(((-0.5625 * (a * a)) / b), (c / b), (-0.375 * a)) / (b * b)), c, -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, 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)) <= -0.3)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(Float64(fma(Float64(fma(Float64(Float64(-0.5625 * Float64(a * a)) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)), c, -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.3], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $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)) < -0.299999999999999989

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

   6. Applied rewrites 84.2%

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

   if -0.299999999999999989 < (/.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 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 91.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b}, c, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b}, c, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.3)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ (sqrt t_0) b)))
     (/
      (*
       (fma
        (/ (fma (/ (* -0.5625 (* a a)) b) (/ c b) (* -0.375 a)) (* b b))
        c
        -0.5)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.3) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (sqrt(t_0) + b));
	} else {
		tmp = (fma((fma(((-0.5625 * (a * a)) / b), (c / b), (-0.375 * a)) / (b * b)), c, -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, 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)) <= -0.3)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(Float64(fma(Float64(fma(Float64(Float64(-0.5625 * Float64(a * a)) / b), Float64(c / b), Float64(-0.375 * a)) / Float64(b * b)), c, -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.3], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $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)) < -0.299999999999999989

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

   6. Applied rewrites 84.2%

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

   if -0.299999999999999989 < (/.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 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 91.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -0.375 \cdot a\right)}{b \cdot b}, c, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 220:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 220.0)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ (sqrt t_0) b)))
     (/ (fma (* -0.375 a) (/ (* c c) (* b b)) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 220.0) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (sqrt(t_0) + b));
	} else {
		tmp = fma((-0.375 * a), ((c * c) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 220.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(fma(Float64(-0.375 * a), Float64(Float64(c * c) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 220.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 220

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 77.8%

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

   6. Applied rewrites 78.8%

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

   if 220 < b

   1. Initial program 45.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 220:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 219.0)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ (fma (* -0.375 a) (/ (* c c) (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 219.0) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = fma((-0.375 * a), ((c * c) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 219.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(-0.375 * a), Float64(Float64(c * c) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 219.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 219

   1. Initial program 77.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 77.9

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

   4. Applied rewrites 77.9%

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

   if 219 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 219.0)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ (* (fma (* (/ a (* b b)) -0.375) c -0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 219.0) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = (fma(((a / (b * b)) * -0.375), c, -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 219.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(a / Float64(b * b)) * -0.375), c, -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 219.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 219

   1. Initial program 77.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 77.9

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

   4. Applied rewrites 77.9%

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

   if 219 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 93.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 89.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 219.0)
   (* (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a) 0.3333333333333333)
   (/ (* (fma (* (/ a (* b b)) -0.375) c -0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 219.0) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (fma(((a / (b * b)) * -0.375), c, -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 219.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(Float64(a / Float64(b * b)) * -0.375), c, -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 219.0], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 219

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   if 219 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 93.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 89.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 219.0)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ (* (fma (* (/ a (* b b)) -0.375) c -0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 219.0) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (fma(((a / (b * b)) * -0.375), c, -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 219.0)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(a / Float64(b * b)) * -0.375), c, -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 219.0], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * c + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 219

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 77.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 77.8

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

   4. Applied rewrites 77.9%

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

   if 219 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 93.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 89.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 219:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b \cdot b} \cdot -0.375, c, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 88.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 88.1%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 82.2%

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

12. Final simplification 82.2%

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

Alternative 19: 64.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 65.2

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

5. Applied rewrites 65.2%

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

6. Final simplification 65.2%

   \[\leadsto \frac{c}{b} \cdot -0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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)))