Cubic critical, narrow range

Percentage Accurate: 55.1% → 91.3%

Time: 13.5s

Alternatives: 20

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: 55.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 54.0%

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

6. Applied rewrites 55.2%

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

7. Taylor expanded in b around inf

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

9. Applied rewrites 93.7%

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

10. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

12. Applied rewrites 93.8%

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

Alternative 2: 91.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 54.0%

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

6. Applied rewrites 55.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 93.7%

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

Alternative 3: 91.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 54.0%

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

6. Applied rewrites 55.2%

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

7. Taylor expanded in b around inf

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

9. Applied rewrites 93.7%

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

    1. lift-+.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. associate-+l+ N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. *-commutative N/A

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

    7. distribute-rgt-in N/A

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

    8. +-commutative N/A

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

    9. lift-+.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. associate-*l* N/A

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

    13. lift-*.f64 N/A

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

    14. lower-fma.f64 N/A

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

11. Applied rewrites 93.7%

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

Alternative 4: 91.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 54.0%

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

6. Applied rewrites 55.2%

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

7. Taylor expanded in b around inf

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

9. Applied rewrites 93.7%

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

10. Taylor expanded in a around 0

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

12. Applied rewrites 93.7%

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

Alternative 5: 89.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{{a}^{-1} \cdot \frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
     (/ (* (pow a -1.0) (/ (- t_0 (* b b)) (+ (sqrt t_0) b))) 3.0)
     (/
      (fma
       (/ (* -0.375 a) b)
       (/ (* c c) b)
       (fma (/ (* (* (pow c 3.0) a) a) (pow b 4.0)) -0.5625 (* -0.5 c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (pow(a, -1.0) * ((t_0 - (b * b)) / (sqrt(t_0) + b))) / 3.0;
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), fma((((pow(c, 3.0) * a) * a) / pow(b, 4.0)), -0.5625, (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64((a ^ -1.0) * Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b))) / 3.0);
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), fma(Float64(Float64(Float64((c ^ 3.0) * a) * a) / (b ^ 4.0)), -0.5625, Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -0.1], N[(N[(N[Power[a, -1.0], $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 80.8%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 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. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

   6. Applied rewrites 82.9%

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

   if -0.10000000000000001 < (/.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 48.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{{a}^{-1} \cdot \frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5625 \cdot a, \frac{c}{{b}^{5}}, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
     (/ (* (pow a -1.0) (/ (- t_0 (* b b)) (+ (sqrt t_0) b))) 3.0)
     (fma
      (* (fma (* -0.5625 a) (/ c (pow b 5.0)) (/ -0.375 (pow b 3.0))) (* c c))
      a
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (pow(a, -1.0) * ((t_0 - (b * b)) / (sqrt(t_0) + b))) / 3.0;
	} else {
		tmp = fma((fma((-0.5625 * a), (c / pow(b, 5.0)), (-0.375 / pow(b, 3.0))) * (c * c)), a, (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64((a ^ -1.0) * Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b))) / 3.0);
	else
		tmp = fma(Float64(fma(Float64(-0.5625 * a), Float64(c / (b ^ 5.0)), Float64(-0.375 / (b ^ 3.0))) * Float64(c * c)), a, Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -0.1], N[(N[(N[Power[a, -1.0], $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(-0.5625 * a), $MachinePrecision] * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 80.8%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 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. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

   6. Applied rewrites 82.9%

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

   if -0.10000000000000001 < (/.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 48.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 96.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 94.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5625 \cdot a, \frac{c}{{b}^{5}}, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(-1.0546875 * (c ^ 4.0)), Float64(a * a), Float64(Float64(Float64(c * c) * fma(-0.375, Float64(b * b), Float64(-0.5625 * Float64(a * c)))) * Float64(b * b))) / (b ^ 7.0)), a, Float64(-0.5 * Float64(c / b)))
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[(c * c), $MachinePrecision] * N[(-0.375 * N[(b * b), $MachinePrecision] + N[(-0.5625 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in a around 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 93.5%

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

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 93.5%

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

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 93.5%

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

Alternative 8: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{{a}^{-1} \cdot \frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{{b}^{4}}, 0.5625, \mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
     (/ (* (pow a -1.0) (/ (- t_0 (* b b)) (+ (sqrt t_0) b))) 3.0)
     (*
      (/
       (fma
        (/ (* (* (* a a) c) c) (pow b 4.0))
        0.5625
        (fma (* 0.375 a) (/ c (* b b)) 0.5))
       (- b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (pow(a, -1.0) * ((t_0 - (b * b)) / (sqrt(t_0) + b))) / 3.0;
	} else {
		tmp = (fma(((((a * a) * c) * c) / pow(b, 4.0)), 0.5625, fma((0.375 * a), (c / (b * b)), 0.5)) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64((a ^ -1.0) * Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b))) / 3.0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(a * a) * c) * c) / (b ^ 4.0)), 0.5625, fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5)) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -0.1], N[(N[(N[Power[a, -1.0], $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 0.5625 + N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision] * c), $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)) < -0.10000000000000001

   1. Initial program 80.8%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 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. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

   6. Applied rewrites 82.9%

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

   if -0.10000000000000001 < (/.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 48.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 94.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{{b}^{4}}, 0.5625, \mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)\right)}{-b} \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{{b}^{4}}, 0.5625, \mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
     (/ (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 3.0)
     (*
      (/
       (fma
        (/ (* (* (* a a) c) c) (pow b 4.0))
        0.5625
        (fma (* 0.375 a) (/ c (* b b)) 0.5))
       (- b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) / 3.0;
	} else {
		tmp = (fma(((((a * a) * c) * c) / pow(b, 4.0)), 0.5625, fma((0.375 * a), (c / (b * b)), 0.5)) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) / 3.0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(a * a) * c) * c) / (b ^ 4.0)), 0.5625, fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5)) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -0.1], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 0.5625 + N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision] * c), $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)) < -0.10000000000000001

   1. Initial program 80.8%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 82.9%

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

   if -0.10000000000000001 < (/.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 48.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 94.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{{b}^{4}}, 0.5625, \mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)\right)}{-b} \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{{b}^{4}}, 0.5625, \mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)\right)}{-b} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(b \cdot a\right) \cdot b, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
     (/ (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 3.0)
     (*
      (fma
       (/ (fma (* (* b a) b) -0.375 (* (* (* a a) c) -0.5625)) (pow b 5.0))
       c
       (/ -0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) / 3.0;
	} else {
		tmp = fma((fma(((b * a) * b), -0.375, (((a * a) * c) * -0.5625)) / pow(b, 5.0)), c, (-0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) / 3.0);
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(b * a) * b), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625)) / (b ^ 5.0)), c, Float64(-0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -0.1], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $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)) < -0.10000000000000001

   1. Initial program 80.8%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 82.9%

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

   if -0.10000000000000001 < (/.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 48.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 94.2%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(b \cdot a\right) \cdot b, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(b \cdot a\right) \cdot b, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.83)
     (/ (* (- (* b b) t_0) (/ 0.3333333333333333 a)) (- (- b) (sqrt t_0)))
     (/ (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((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.83) {
		tmp = (((b * b) - t_0) * (0.3333333333333333 / a)) / (-b - sqrt(t_0));
	} 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(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	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[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.83], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $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 < 0.82999999999999996

   1. Initial program 83.3%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 83.2%

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

   6. Applied rewrites 84.9%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

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

      5. unpow2 N/A

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

      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 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 88.6%

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

Alternative 12: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.83)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (/ (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((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.83) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} 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(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	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[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.83], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $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 < 0.82999999999999996

   1. Initial program 83.3%

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

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 83.2%

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

   6. Applied rewrites 84.8%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

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

      5. unpow2 N/A

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

      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 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 88.6%

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

Alternative 13: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c \cdot c}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.83)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 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 <= 0.83) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (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 <= 0.83)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / 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, 0.83], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < 0.82999999999999996

   1. Initial program 83.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 83.5

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

   4. Applied rewrites 83.5%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

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

      5. unpow2 N/A

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

      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 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 88.6%

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

Alternative 14: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.83)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* (/ (fma (* 0.375 a) (/ c (* b b)) 0.5) (- b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.83) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (fma((0.375 * a), (c / (b * b)), 0.5) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.83], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.82999999999999996

   1. Initial program 83.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 83.5

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

   4. Applied rewrites 83.5%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 88.5%

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

Alternative 15: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.83)
   (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
   (* (/ (fma (* 0.375 a) (/ c (* b b)) 0.5) (- b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.83) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (fma((0.375 * a), (c / (b * b)), 0.5) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.83], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.82999999999999996

   1. Initial program 83.3%

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

   4. Applied rewrites 83.2%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 88.5%

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

Alternative 16: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.83)
   (/ (* 0.3333333333333333 (- (sqrt (fma (* -3.0 c) a (* b b))) b)) a)
   (* (/ (fma (* 0.375 a) (/ c (* b b)) 0.5) (- b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.83) {
		tmp = (0.3333333333333333 * (sqrt(fma((-3.0 * c), a, (b * b))) - b)) / a;
	} else {
		tmp = (fma((0.375 * a), (c / (b * b)), 0.5) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b)) / a);
	else
		tmp = Float64(Float64(fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.83], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.82999999999999996

   1. Initial program 83.3%

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

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 83.2%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 88.5%

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

Alternative 17: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.83)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (* (/ (fma (* 0.375 a) (/ c (* b b)) 0.5) (- b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.83) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (fma((0.375 * a), (c / (b * b)), 0.5) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.83], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.82999999999999996

   1. Initial program 83.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 83.2%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 88.5%

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

Alternative 18: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.83:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.375 \cdot a, \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.83)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
   (* (/ (fma (* 0.375 a) (/ c (* b b)) 0.5) (- b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.83) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = (fma((0.375 * a), (c / (b * b)), 0.5) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.83)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(fma(Float64(0.375 * a), Float64(c / Float64(b * b)), 0.5) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.83], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.375 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.82999999999999996

   1. Initial program 83.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 83.3

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

      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 83.3

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

   4. Applied rewrites 83.2%

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

   if 0.82999999999999996 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 88.5%

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

Alternative 19: 81.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 90.4%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 83.9%

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

Alternative 20: 64.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in a around 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 66.4

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

5. Applied rewrites 66.4%

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

Reproduce

herbie shell --seed 2024309 
(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)))