Cubic critical, narrow range

Percentage Accurate: 55.4% → 92.1%

Time: 11.1s

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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := 6.75 \cdot \left(c \cdot c\right)\\ t_3 := \left(t\_2 \cdot c\right) \cdot 4.5\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(t\_1 + b\right)}^{-1} \cdot {\left(\frac{a}{t\_0 - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4.5, c, \mathsf{fma}\left(0.5, 6.75 \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-4.5, \mathsf{fma}\left(-27, {c}^{3}, t\_3\right) \cdot c, {t\_2}^{2} \cdot 0.25\right) \cdot a}{{b}^{6}}, \mathsf{fma}\left(-27, \frac{{c}^{3}}{{b}^{4}}, \frac{t\_3}{{b}^{4}}\right) \cdot 0.5\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot b}{\mathsf{fma}\left(b, b, t\_1 \cdot b + t\_0\right) \cdot \left(a \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (* 6.75 (* c c)))
        (t_3 (* (* t_2 c) 4.5)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (/ (* (pow (+ t_1 b) -1.0) (pow (/ a (- t_0 (* b b))) -1.0)) 3.0)
     (/
      (*
       (*
        (fma
         -4.5
         c
         (*
          (fma
           0.5
           (* 6.75 (/ (* c c) (* b b)))
           (*
            (fma
             -0.5
             (/
              (*
               (fma
                -4.5
                (* (fma -27.0 (pow c 3.0) t_3) c)
                (* (pow t_2 2.0) 0.25))
               a)
              (pow b 6.0))
             (*
              (fma -27.0 (/ (pow c 3.0) (pow b 4.0)) (/ t_3 (pow b 4.0)))
              0.5))
            a))
          a))
        a)
       b)
      (* (fma b b (+ (* t_1 b) t_0)) (* a 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = 6.75 * (c * c);
	double t_3 = (t_2 * c) * 4.5;
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = (pow((t_1 + b), -1.0) * pow((a / (t_0 - (b * b))), -1.0)) / 3.0;
	} else {
		tmp = ((fma(-4.5, c, (fma(0.5, (6.75 * ((c * c) / (b * b))), (fma(-0.5, ((fma(-4.5, (fma(-27.0, pow(c, 3.0), t_3) * c), (pow(t_2, 2.0) * 0.25)) * a) / pow(b, 6.0)), (fma(-27.0, (pow(c, 3.0) / pow(b, 4.0)), (t_3 / pow(b, 4.0))) * 0.5)) * a)) * a)) * a) * b) / (fma(b, b, ((t_1 * b) + t_0)) * (a * 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(6.75 * Float64(c * c))
	t_3 = Float64(Float64(t_2 * c) * 4.5)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64((Float64(t_1 + b) ^ -1.0) * (Float64(a / Float64(t_0 - Float64(b * b))) ^ -1.0)) / 3.0);
	else
		tmp = Float64(Float64(Float64(fma(-4.5, c, Float64(fma(0.5, Float64(6.75 * Float64(Float64(c * c) / Float64(b * b))), Float64(fma(-0.5, Float64(Float64(fma(-4.5, Float64(fma(-27.0, (c ^ 3.0), t_3) * c), Float64((t_2 ^ 2.0) * 0.25)) * a) / (b ^ 6.0)), Float64(fma(-27.0, Float64((c ^ 3.0) / (b ^ 4.0)), Float64(t_3 / (b ^ 4.0))) * 0.5)) * a)) * a)) * a) * b) / Float64(fma(b, b, Float64(Float64(t_1 * b) + t_0)) * Float64(a * 3.0)));
	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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(6.75 * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * c), $MachinePrecision] * 4.5), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[Power[N[(t$95$1 + b), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(-4.5 * c + N[(N[(0.5 * N[(6.75 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(N[(N[(-4.5 * N[(N[(-27.0 * N[Power[c, 3.0], $MachinePrecision] + t$95$3), $MachinePrecision] * c), $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-27.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * b + N[(N[(t$95$1 * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $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)) < -4.5999999999999996

   1. Initial program 91.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. unpow-1 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

   4. Applied rewrites 52.6%

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

   6. Applied rewrites 53.6%

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

   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 93.3%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 93.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}^{-1} \cdot {\left(\frac{a}{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4.5, c, \mathsf{fma}\left(0.5, 6.75 \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-4.5, \mathsf{fma}\left(-27, {c}^{3}, \left(\left(6.75 \cdot \left(c \cdot c\right)\right) \cdot c\right) \cdot 4.5\right) \cdot c, {\left(6.75 \cdot \left(c \cdot c\right)\right)}^{2} \cdot 0.25\right) \cdot a}{{b}^{6}}, \mathsf{fma}\left(-27, \frac{{c}^{3}}{{b}^{4}}, \frac{\left(\left(6.75 \cdot \left(c \cdot c\right)\right) \cdot c\right) \cdot 4.5}{{b}^{4}}\right) \cdot 0.5\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot b}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} \cdot b + \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot \left(a \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{t\_0} + b\right)}^{-1} \cdot {\left(\frac{a}{t\_0 - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5625, \frac{c \cdot a}{{b}^{5}}, \frac{0.375}{{b}^{3}}\right) \cdot \left(a \cdot a\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (/ (* (pow (+ (sqrt t_0) b) -1.0) (pow (/ a (- t_0 (* b b))) -1.0)) 3.0)
     (*
      (pow
       (/
        (fma
         (fma
          (*
           (fma 0.5625 (/ (* c a) (pow b 5.0)) (/ 0.375 (pow b 3.0)))
           (* a a))
          c
          (* (/ a b) 0.5))
         c
         (* -0.6666666666666666 b))
        c)
       -1.0)
      0.3333333333333333))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = (pow((sqrt(t_0) + b), -1.0) * pow((a / (t_0 - (b * b))), -1.0)) / 3.0;
	} else {
		tmp = pow((fma(fma((fma(0.5625, ((c * a) / pow(b, 5.0)), (0.375 / pow(b, 3.0))) * (a * a)), c, ((a / b) * 0.5)), c, (-0.6666666666666666 * b)) / c), -1.0) * 0.3333333333333333;
	}
	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(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64((Float64(sqrt(t_0) + b) ^ -1.0) * (Float64(a / Float64(t_0 - Float64(b * b))) ^ -1.0)) / 3.0);
	else
		tmp = Float64((Float64(fma(fma(Float64(fma(0.5625, Float64(Float64(c * a) / (b ^ 5.0)), Float64(0.375 / (b ^ 3.0))) * Float64(a * a)), c, Float64(Float64(a / b) * 0.5)), c, Float64(-0.6666666666666666 * b)) / c) ^ -1.0) * 0.3333333333333333);
	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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[Power[N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[Power[N[(N[(N[(N[(N[(0.5625 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * c + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $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)) < -4.5999999999999996

   1. Initial program 91.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. unpow-1 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

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

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.3%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{a \cdot a}{{b}^{3}} \cdot -0.375\right) \cdot a}{b}, \mathsf{fma}\left(b \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a}, -0.2222222222222222, \frac{0.5625 \cdot {a}^{3}}{{b}^{5}}\right)\right), \frac{0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}\right)}}^{-1} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.3%

       \[\leadsto 0.3333333333333333 \cdot {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5625, \frac{a \cdot c}{{b}^{5}}, \frac{0.375}{{b}^{3}}\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}\right)}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}^{-1} \cdot {\left(\frac{a}{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5625, \frac{c \cdot a}{{b}^{5}}, \frac{0.375}{{b}^{3}}\right) \cdot \left(a \cdot a\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{t\_0} + b\right)}^{-1} \cdot {\left(\frac{a}{t\_0 - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, -0.5625 \cdot \left({c}^{3} \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (/ (* (pow (+ (sqrt t_0) b) -1.0) (pow (/ a (- t_0 (* b b))) -1.0)) 3.0)
     (fma
      (/
       (fma
        (* -1.0546875 (* a a))
        (pow c 4.0)
        (*
         (fma (* -0.375 (* b b)) (* c c) (* -0.5625 (* (pow c 3.0) a)))
         (* b b)))
       (pow b 7.0))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = (pow((sqrt(t_0) + b), -1.0) * pow((a / (t_0 - (b * b))), -1.0)) / 3.0;
	} else {
		tmp = fma((fma((-1.0546875 * (a * a)), pow(c, 4.0), (fma((-0.375 * (b * b)), (c * c), (-0.5625 * (pow(c, 3.0) * a))) * (b * b))) / pow(b, 7.0)), a, ((c / b) * -0.5));
	}
	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(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64((Float64(sqrt(t_0) + b) ^ -1.0) * (Float64(a / Float64(t_0 - Float64(b * b))) ^ -1.0)) / 3.0);
	else
		tmp = fma(Float64(fma(Float64(-1.0546875 * Float64(a * a)), (c ^ 4.0), Float64(fma(Float64(-0.375 * Float64(b * b)), Float64(c * c), Float64(-0.5625 * Float64((c ^ 3.0) * a))) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(c / b) * -0.5));
	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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[Power[N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(-1.0546875 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(-0.375 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. 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)) < -4.5999999999999996

   1. Initial program 91.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. unpow-1 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

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

   3. Taylor expanded in a around 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 93.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 93.2%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}^{-1} \cdot {\left(\frac{a}{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, -0.5625 \cdot \left({c}^{3} \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{t\_0} + b\right)}^{-1} \cdot {\left(\frac{a}{t\_0 - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right) \cdot c\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (/ (* (pow (+ (sqrt t_0) b) -1.0) (pow (/ a (- t_0 (* b b))) -1.0)) 3.0)
     (/
      1.0
      (/
       (fma
        -2.0
        b
        (*
         (fma (* -3.0 c) (* (/ (* a a) (pow b 3.0)) -0.375) (* (/ a b) 1.5))
         c))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = (pow((sqrt(t_0) + b), -1.0) * pow((a / (t_0 - (b * b))), -1.0)) / 3.0;
	} else {
		tmp = 1.0 / (fma(-2.0, b, (fma((-3.0 * c), (((a * a) / pow(b, 3.0)) * -0.375), ((a / b) * 1.5)) * c)) / 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(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64((Float64(sqrt(t_0) + b) ^ -1.0) * (Float64(a / Float64(t_0 - Float64(b * b))) ^ -1.0)) / 3.0);
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(fma(Float64(-3.0 * c), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375), Float64(Float64(a / b) * 1.5)) * c)) / 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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[Power[N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(-3.0 * c), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -4.5999999999999996

   1. Initial program 91.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. unpow-1 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 52.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.6

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 91.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}^{-1} \cdot {\left(\frac{a}{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}\right)}^{-1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right) \cdot c\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right) \cdot c\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (* (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 0.3333333333333333)
     (/
      1.0
      (/
       (fma
        -2.0
        b
        (*
         (fma (* -3.0 c) (* (/ (* a a) (pow b 3.0)) -0.375) (* (/ a b) 1.5))
         c))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (fma(-2.0, b, (fma((-3.0 * c), (((a * a) / pow(b, 3.0)) * -0.375), ((a / b) * 1.5)) * c)) / 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(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(fma(Float64(-3.0 * c), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375), Float64(Float64(a / b) * 1.5)) * c)) / 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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(-3.0 * c), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -4.5999999999999996

   1. Initial program 91.3%

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

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 52.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.6

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 91.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right) \cdot c\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(-3, \left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, \frac{1.5}{b}\right) \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (* (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 0.3333333333333333)
     (/
      1.0
      (fma
       -2.0
       (/ b c)
       (* (fma -3.0 (* (* (/ c (pow b 3.0)) -0.375) a) (/ 1.5 b)) a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) * 0.3333333333333333;
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), (fma(-3.0, (((c / pow(b, 3.0)) * -0.375) * a), (1.5 / b)) * a));
	}
	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(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(fma(-3.0, Float64(Float64(Float64(c / (b ^ 3.0)) * -0.375) * a), Float64(1.5 / b)) * a)));
	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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(N[(-3.0 * N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * a), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] * a), $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)) < -4.5999999999999996

   1. Initial program 91.3%

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

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 52.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.6

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 91.1

          \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(-3, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.375\right), \color{blue}{\frac{1.5}{b}}\right)\right)} \]

   7. Applied rewrites 91.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(-3, \left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, \frac{1.5}{b}\right) \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.6)
     (* (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 0.3333333333333333)
     (*
      (fma
       (/ (fma (* a (* b 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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.6) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) * 0.3333333333333333;
	} else {
		tmp = fma((fma((a * (b * 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(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.6)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(Float64(a * Float64(b * 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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.6], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(b * b), $MachinePrecision]), $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)) < -4.5999999999999996

   1. Initial program 91.3%

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

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.5999999999999996 < (/.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 52.7%

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

   3. Taylor expanded in 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 90.7%

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

   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 90.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, \left(-b\right) \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (/
    1.0
    (/
     (* a 3.0)
     (/
      (fma b b (fma (* -3.0 a) c (* (- b) b)))
      (+ (sqrt (fma (* -3.0 c) a (* b b))) b))))
   (/ 1.0 (/ (fma -2.0 b (* (/ (* c a) b) 1.5)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = 1.0 / ((a * 3.0) / (fma(b, b, fma((-3.0 * a), c, (-b * b))) / (sqrt(fma((-3.0 * c), a, (b * b))) + b)));
	} else {
		tmp = 1.0 / (fma(-2.0, b, (((c * a) / b) * 1.5)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(1.0 / Float64(Float64(a * 3.0) / Float64(fma(b, b, fma(Float64(-3.0 * a), c, Float64(Float64(-b) * b))) / Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b))));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c * a) / b) * 1.5)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], N[(1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 80.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.4

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 80.4

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

   4. Applied rewrites 80.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 81.7%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, \left(-b\right) \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 90.0)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (/ 1.0 (/ (fma -2.0 b (* (/ (* c a) b) 1.5)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 90.0) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (((c * a) / b) * 1.5)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c * a) / b) * 1.5)) / 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[b, 90.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

   4. Applied rewrites 80.4%

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

   6. Applied rewrites 81.7%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, \left(-b\right) \cdot b\right)\right)}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (/
    (fma b b (fma (* -3.0 a) c (* (- b) b)))
    (* (+ (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0)))
   (/ 1.0 (/ (fma -2.0 b (* (/ (* c a) b) 1.5)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = fma(b, b, fma((-3.0 * a), c, (-b * b))) / ((sqrt(fma((-3.0 * c), a, (b * b))) + b) * (a * 3.0));
	} else {
		tmp = 1.0 / (fma(-2.0, b, (((c * a) / b) * 1.5)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(fma(b, b, fma(Float64(-3.0 * a), c, Float64(Float64(-b) * b))) / Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b) * Float64(a * 3.0)));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c * a) / b) * 1.5)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], N[(N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 80.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.4

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 80.4

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

   4. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 81.7%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, \left(-b\right) \cdot b\right)\right)}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 90.0)
     (* (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 0.3333333333333333)
     (/ 1.0 (/ (fma -2.0 b (* (/ (* c a) b) 1.5)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 90.0) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (fma(-2.0, b, (((c * a) / b) * 1.5)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c * a) / b) * 1.5)) / 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[b, 90.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

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

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

   6. Applied rewrites 81.6%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (/ 0.3333333333333333 (/ a (- (sqrt (fma (* -3.0 c) a (* b b))) b)))
   (/ 1.0 (/ (fma -2.0 b (* (/ (* c a) b) 1.5)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = 0.3333333333333333 / (a / (sqrt(fma((-3.0 * c), a, (b * b))) - b));
	} else {
		tmp = 1.0 / (fma(-2.0, b, (((c * a) / b) * 1.5)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(0.3333333333333333 / Float64(a / Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b)));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c * a) / b) * 1.5)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], N[(0.3333333333333333 / N[(a / N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.5

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

   4. Applied rewrites 80.5%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (/ (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) 0.3333333333333333) a)
   (/ 1.0 (/ (fma -2.0 b (* (/ (* c a) b) 1.5)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = 1.0 / (fma(-2.0, b, (((c * a) / b) * 1.5)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c * a) / b) * 1.5)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 80.5%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, \frac{c \cdot a}{b} \cdot 1.5\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a}{b} \cdot 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (/ (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) 0.3333333333333333) a)
   (/ 1.0 (fma -2.0 (/ b c) (* (/ a b) 1.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), ((a / b) * 1.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(Float64(a / b) * 1.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 80.5%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a}{b} \cdot 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a}{b} \cdot 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (/ 1.0 (fma -2.0 (/ b c) (* (/ a b) 1.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), ((a / b) * 1.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(Float64(a / b) * 1.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], 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[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 80.5%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a}{b} \cdot 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a}{b} \cdot 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 90.0)
   (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma -2.0 (/ b c) (* (/ a b) 1.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 90.0) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), ((a / b) * 1.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 90.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(Float64(a / b) * 1.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 90.0], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 90

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 80.5

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

      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 80.5

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

   4. Applied rewrites 80.5%

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

   if 90 < b

   1. Initial program 45.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 45.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 90:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a}{b} \cdot 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 82.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. 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. lower-/.f64 N/A

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

   4. lower-/.f64 56.7

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.7

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

4. Applied rewrites 56.7%

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

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 81.4

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

7. Applied rewrites 81.4%

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

8. Final simplification 81.4%

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

Alternative 18: 81.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in 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 86.6%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 63.4%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 80.7%

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

12. Final simplification 80.7%

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

Alternative 19: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 63.5

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

5. Applied rewrites 63.5%

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

Alternative 20: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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) / b) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in 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 86.6%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2024276 
(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)))