Cubic critical, narrow range

Percentage Accurate: 55.3% → 91.5%

Time: 15.7s

Alternatives: 19

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{t\_0}\right)}^{-1} \cdot {\left(\frac{3 \cdot a}{b \cdot b - t\_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, -0.5625 \cdot \left({c}^{3} \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
     (*
      (pow (- (- b) (sqrt t_0)) -1.0)
      (pow (/ (* 3.0 a) (- (* b b) t_0)) -1.0))
     (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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = pow((-b - sqrt(t_0)), -1.0) * pow(((3.0 * a) / ((b * b) - t_0)), -1.0);
	} 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64((Float64(Float64(-b) - sqrt(t_0)) ^ -1.0) * (Float64(Float64(3.0 * a) / Float64(Float64(b * b) - t_0)) ^ -1.0));
	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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], N[(N[Power[N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(3.0 * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $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)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      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 83.2%

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

   6. Applied rewrites 84.5%

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

   if -0.044999999999999998 < (/.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 49.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}^{-1} \cdot {\left(\frac{3 \cdot a}{b \cdot b - \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, -0.5625 \cdot \left({c}^{3} \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b \cdot b} \cdot a\\ t_1 := -9 \cdot \left(c \cdot a\right)\\ t_2 := t\_1 \cdot 0.5\\ t_3 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_4 := \mathsf{fma}\left(t\_3, 27, {t\_1}^{2} \cdot -0.25\right)\\ t_5 := \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(t\_4 \cdot t\_1\right) \cdot -0.5\right)\\ \frac{\frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(t\_2, t\_5, {t\_4}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{t\_4}{b \cdot b} + \frac{t\_5}{{b}^{4}}, t\_2\right)\right) \cdot b}{\left(\mathsf{fma}\left(-3, t\_0, \mathsf{fma}\left(-1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{6}}, \mathsf{fma}\left(-1.125, \frac{t\_3}{{b}^{4}}, -1.5 \cdot t\_0\right)\right)\right) + 3\right) \cdot \left(b \cdot b\right)}}{a}}{3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c (* b b)) a))
        (t_1 (* -9.0 (* c a)))
        (t_2 (* t_1 0.5))
        (t_3 (* (* c c) (* a a)))
        (t_4 (fma t_3 27.0 (* (pow t_1 2.0) -0.25)))
        (t_5 (fma (* (pow a 3.0) -27.0) (pow c 3.0) (* (* t_4 t_1) -0.5))))
   (/
    (/
     (/
      (*
       (fma
        -0.5
        (/ (fma t_2 t_5 (* (pow t_4 2.0) 0.25)) (pow b 6.0))
        (fma 0.5 (+ (/ t_4 (* b b)) (/ t_5 (pow b 4.0))) t_2))
       b)
      (*
       (+
        (fma
         -3.0
         t_0
         (fma
          -1.6875
          (/ (* (pow c 3.0) (pow a 3.0)) (pow b 6.0))
          (fma -1.125 (/ t_3 (pow b 4.0)) (* -1.5 t_0))))
        3.0)
       (* b b)))
     a)
    3.0)))

C

double code(double a, double b, double c) {
	double t_0 = (c / (b * b)) * a;
	double t_1 = -9.0 * (c * a);
	double t_2 = t_1 * 0.5;
	double t_3 = (c * c) * (a * a);
	double t_4 = fma(t_3, 27.0, (pow(t_1, 2.0) * -0.25));
	double t_5 = fma((pow(a, 3.0) * -27.0), pow(c, 3.0), ((t_4 * t_1) * -0.5));
	return (((fma(-0.5, (fma(t_2, t_5, (pow(t_4, 2.0) * 0.25)) / pow(b, 6.0)), fma(0.5, ((t_4 / (b * b)) + (t_5 / pow(b, 4.0))), t_2)) * b) / ((fma(-3.0, t_0, fma(-1.6875, ((pow(c, 3.0) * pow(a, 3.0)) / pow(b, 6.0)), fma(-1.125, (t_3 / pow(b, 4.0)), (-1.5 * t_0)))) + 3.0) * (b * b))) / a) / 3.0;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / Float64(b * b)) * a)
	t_1 = Float64(-9.0 * Float64(c * a))
	t_2 = Float64(t_1 * 0.5)
	t_3 = Float64(Float64(c * c) * Float64(a * a))
	t_4 = fma(t_3, 27.0, Float64((t_1 ^ 2.0) * -0.25))
	t_5 = fma(Float64((a ^ 3.0) * -27.0), (c ^ 3.0), Float64(Float64(t_4 * t_1) * -0.5))
	return Float64(Float64(Float64(Float64(fma(-0.5, Float64(fma(t_2, t_5, Float64((t_4 ^ 2.0) * 0.25)) / (b ^ 6.0)), fma(0.5, Float64(Float64(t_4 / Float64(b * b)) + Float64(t_5 / (b ^ 4.0))), t_2)) * b) / Float64(Float64(fma(-3.0, t_0, fma(-1.6875, Float64(Float64((c ^ 3.0) * (a ^ 3.0)) / (b ^ 6.0)), fma(-1.125, Float64(t_3 / (b ^ 4.0)), Float64(-1.5 * t_0)))) + 3.0) * Float64(b * b))) / a) / 3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(-9.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * 27.0 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Power[a, 3.0], $MachinePrecision] * -27.0), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$4 * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(-0.5 * N[(N[(t$95$2 * t$95$5 + N[(N[Power[t$95$4, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$4 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(-3.0 * t$95$0 + N[(-1.6875 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Initial program 56.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
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 56.7%

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

6. Applied rewrites 58.1%

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

7. Taylor expanded in b around inf

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

9. Applied rewrites 90.1%

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

10. Taylor expanded in b around inf

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 N/A

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

12. Applied rewrites 90.3%

    \[\leadsto \frac{\frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-3, a \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, -1.5 \cdot \left(a \cdot \frac{c}{b \cdot b}\right)\right)\right)\right) + 3\right)}}}{a}}{3} \]

13. Final simplification 90.3%

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\left(-9 \cdot \left(c \cdot a\right)\right) \cdot 0.5, \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(c \cdot a\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(c \cdot a\right)\right)\right) \cdot -0.5\right), {\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(c \cdot a\right)\right)}^{2} \cdot -0.25\right)\right)}^{2} \cdot 0.25\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(c \cdot a\right)\right)}^{2} \cdot -0.25\right)}{b \cdot b} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(c \cdot a\right)\right)}^{2} \cdot -0.25\right) \cdot \left(-9 \cdot \left(c \cdot a\right)\right)\right) \cdot -0.5\right)}{{b}^{4}}, \left(-9 \cdot \left(c \cdot a\right)\right) \cdot 0.5\right)\right) \cdot b}{\left(\mathsf{fma}\left(-3, \frac{c}{b \cdot b} \cdot a, \mathsf{fma}\left(-1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{6}}, \mathsf{fma}\left(-1.125, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{4}}, -1.5 \cdot \left(\frac{c}{b \cdot b} \cdot a\right)\right)\right)\right) + 3\right) \cdot \left(b \cdot b\right)}}{a}}{3} \]
14. Add Preprocessing

Alternative 3: 91.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
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 56.7%

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

6. Applied rewrites 58.1%

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

7. Taylor expanded in b around inf

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

9. Applied rewrites 90.1%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

12. Applied rewrites 90.2%

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

13. Final simplification 90.2%

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

Alternative 4: 91.1% accurate, 0.1× speedup

Math

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

FPCore

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

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	t_1 = Float64(6.75 * Float64(c * c))
	t_2 = Float64(t_1 * c)
	t_3 = Float64((c ^ 3.0) * -27.0)
	return Float64(Float64(Float64(Float64(Float64(fma(a, fma(0.5, Float64(6.75 * Float64(Float64(c * c) / Float64(b * b))), Float64(fma(0.5, fma(4.5, Float64(t_2 / (b ^ 4.0)), Float64(t_3 / (b ^ 4.0))), Float64(Float64(Float64(fma(0.25, (t_1 ^ 2.0), Float64(Float64(fma(4.5, t_2, t_3) * c) * -4.5)) * a) * -0.5) / (b ^ 6.0))) * a)), Float64(-4.5 * c)) * a) * b) / fma(b, b, Float64(Float64(sqrt(t_0) * b) + t_0))) / a) / 3.0)
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[(6.75 * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * c), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[c, 3.0], $MachinePrecision] * -27.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(a * N[(0.5 * N[(6.75 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(4.5 * N[(t$95$2 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[(N[(4.5 * t$95$2 + t$95$3), $MachinePrecision] * c), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.5), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(-4.5 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision]]]]]

Derivation

1. Initial program 56.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
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 56.7%

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

6. Applied rewrites 58.1%

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

7. Taylor expanded in b around inf

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

9. Applied rewrites 90.1%

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

10. Taylor expanded in a around 0

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

12. Applied rewrites 90.2%

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

13. Final simplification 90.2%

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

Alternative 5: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-1.6875 \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot 1.125\right), c, \frac{1.5}{b}\right), c, -2 \cdot \frac{b}{a}\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (pow a -1.0)
  (/
   (fma
    (fma
     (fma
      (- c)
      (/ (* -1.6875 (* a a)) (pow b 5.0))
      (* (/ a (pow b 3.0)) 1.125))
     c
     (/ 1.5 b))
    c
    (* -2.0 (/ b a)))
   c)))

C

double code(double a, double b, double c) {
	return pow(a, -1.0) / (fma(fma(fma(-c, ((-1.6875 * (a * a)) / pow(b, 5.0)), ((a / pow(b, 3.0)) * 1.125)), c, (1.5 / b)), c, (-2.0 * (b / a))) / c);
}

Julia

function code(a, b, c)
	return Float64((a ^ -1.0) / Float64(fma(fma(fma(Float64(-c), Float64(Float64(-1.6875 * Float64(a * a)) / (b ^ 5.0)), Float64(Float64(a / (b ^ 3.0)) * 1.125)), c, Float64(1.5 / b)), c, Float64(-2.0 * Float64(b / a))) / c))
end

Wolfram

code[a_, b_, c_] := N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[((-c) * N[(N[(-1.6875 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 1.125), $MachinePrecision]), $MachinePrecision] * c + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] * c + N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $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. div-inv N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l/ N/A

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

   8. frac-times N/A

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

4. Applied rewrites 56.7%

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

5. Taylor expanded in c around 0

   \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{-2 \cdot \frac{b}{a} + c \cdot \left(c \cdot \left(-1 \cdot \left(c \cdot \left(\frac{-3}{4} \cdot \frac{a \cdot \left(\frac{-9}{4} \cdot \frac{a}{{b}^{3}} + \frac{9}{8} \cdot \frac{a}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{a}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{a}^{2}} + \frac{27}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{a}{{b}^{3}} + \frac{9}{8} \cdot \frac{a}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{c}}} \]

7. Applied rewrites 89.9%

   \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(-1.125 \cdot \frac{a}{{b}^{3}}\right) \cdot a}{b}, \mathsf{fma}\left(\frac{-0.6666666666666666}{a}, \frac{\left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{a}, \frac{1.6875 \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right), 1.125 \cdot \frac{a}{{b}^{3}}\right), c, \frac{1.5}{b}\right), c, \frac{b}{a} \cdot -2\right)}{c}}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-27}{16} \cdot \frac{{a}^{2}}{{b}^{5}}, \frac{9}{8} \cdot \frac{a}{{b}^{3}}\right), c, \frac{\frac{3}{2}}{b}\right), c, \frac{b}{a} \cdot -2\right)}{c}} \]
9. Step-by-step derivation

10. Applied rewrites 89.9%

    \[\leadsto \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-1.6875 \cdot \left(a \cdot a\right)}{{b}^{5}}, 1.125 \cdot \frac{a}{{b}^{3}}\right), c, \frac{1.5}{b}\right), c, \frac{b}{a} \cdot -2\right)}{c}} \]

11. Final simplification 89.9%

    \[\leadsto \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-1.6875 \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot 1.125\right), c, \frac{1.5}{b}\right), c, -2 \cdot \frac{b}{a}\right)}{c}} \]
12. Add Preprocessing

Alternative 6: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{t\_0}\right)}^{-1} \cdot {\left(\frac{3 \cdot a}{b \cdot b - t\_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right), -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
     (*
      (pow (- (- b) (sqrt t_0)) -1.0)
      (pow (/ (* 3.0 a) (- (* b b) t_0)) -1.0))
     (/
      1.0
      (/
       (fma
        c
        (fma (* -3.0 c) (* (/ (* a a) (pow b 3.0)) -0.375) (* (/ a b) 1.5))
        (* -2.0 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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = pow((-b - sqrt(t_0)), -1.0) * pow(((3.0 * a) / ((b * b) - t_0)), -1.0);
	} else {
		tmp = 1.0 / (fma(c, fma((-3.0 * c), (((a * a) / pow(b, 3.0)) * -0.375), ((a / b) * 1.5)), (-2.0 * 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64((Float64(Float64(-b) - sqrt(t_0)) ^ -1.0) * (Float64(Float64(3.0 * a) / Float64(Float64(b * b) - t_0)) ^ -1.0));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(Float64(-3.0 * c), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375), Float64(Float64(a / b) * 1.5)), Float64(-2.0 * 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], N[(N[Power[N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(3.0 * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * 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] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      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 83.2%

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

   6. Applied rewrites 84.5%

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

   if -0.044999999999999998 < (/.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 49.7%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.7

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

   4. Applied rewrites 49.6%

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

   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 91.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}^{-1} \cdot {\left(\frac{3 \cdot a}{b \cdot b - \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right), -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right), -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
     (/ 1.0 (/ 3.0 (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) a)))
     (/
      1.0
      (/
       (fma
        c
        (fma (* -3.0 c) (* (/ (* a a) (pow b 3.0)) -0.375) (* (/ a b) 1.5))
        (* -2.0 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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = 1.0 / (3.0 / (((t_0 - (b * b)) / (sqrt(t_0) + b)) / a));
	} else {
		tmp = 1.0 / (fma(c, fma((-3.0 * c), (((a * a) / pow(b, 3.0)) * -0.375), ((a / b) * 1.5)), (-2.0 * 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64(1.0 / Float64(3.0 / Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / a)));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(Float64(-3.0 * c), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375), Float64(Float64(a / b) * 1.5)), Float64(-2.0 * 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], N[(1.0 / N[(3.0 / N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * 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] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-+.f64 84.5

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

   6. Applied rewrites 84.5%

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

   if -0.044999999999999998 < (/.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 49.7%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.7

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

   4. Applied rewrites 49.6%

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

   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 91.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot 1.5\right), -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(-3, \left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, \frac{1.5}{b}\right) \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
     (/ 1.0 (/ 3.0 (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) a)))
     (/
      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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = 1.0 / (3.0 / (((t_0 - (b * b)) / (sqrt(t_0) + b)) / a));
	} 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64(1.0 / Float64(3.0 / Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / a)));
	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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], N[(1.0 / N[(3.0 / N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $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)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-+.f64 84.5

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

   6. Applied rewrites 84.5%

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

   if -0.044999999999999998 < (/.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 49.7%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.7

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

   4. Applied rewrites 49.6%

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

   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 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(-3, \left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, \frac{1.5}{b}\right) \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(t\_0 - b \cdot b\right)}{t\_1 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\mathsf{fma}\left(c \cdot 0.5, \frac{a \cdot a}{b \cdot b} \cdot 6.75, -4.5 \cdot a\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, t\_1 \cdot b + t\_0\right)}}{a}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0028)
     (/ (* (/ 0.3333333333333333 a) (- t_0 (* b b))) (+ t_1 b))
     (/
      (/
       (/
        (* (* (fma (* c 0.5) (* (/ (* a a) (* b b)) 6.75) (* -4.5 a)) c) 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 tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0028) {
		tmp = ((0.3333333333333333 / a) * (t_0 - (b * b))) / (t_1 + b);
	} else {
		tmp = ((((fma((c * 0.5), (((a * a) / (b * b)) * 6.75), (-4.5 * a)) * c) * 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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0028)
		tmp = Float64(Float64(Float64(0.3333333333333333 / a) * Float64(t_0 - Float64(b * b))) / Float64(t_1 + b));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(c * 0.5), Float64(Float64(Float64(a * a) / Float64(b * b)) * 6.75), Float64(-4.5 * a)) * c) * b) / fma(b, b, Float64(Float64(t_1 * b) + t_0))) / 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]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0028], N[(N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(c * 0.5), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * 6.75), $MachinePrecision] + N[(-4.5 * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00279999999999999997

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.7%

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

   6. Applied rewrites 81.3%

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

   if -0.00279999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.8%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in b around inf

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

   9. Applied rewrites 94.3%

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

   10. Taylor expanded in c around 0

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

   12. Applied rewrites 88.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\mathsf{fma}\left(c \cdot 0.5, \frac{a \cdot a}{b \cdot b} \cdot 6.75, -4.5 \cdot a\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} \cdot b + \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}}{a}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(t\_0 - b \cdot b\right)}{t\_1 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\mathsf{fma}\left(a \cdot 0.5, 6.75 \cdot \frac{c \cdot c}{b \cdot b}, -4.5 \cdot c\right) \cdot a\right) \cdot b}{\mathsf{fma}\left(b, b, t\_1 \cdot b + t\_0\right)}}{a}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0028)
     (/ (* (/ 0.3333333333333333 a) (- t_0 (* b b))) (+ t_1 b))
     (/
      (/
       (/
        (* (* (fma (* a 0.5) (* 6.75 (/ (* c c) (* b b))) (* -4.5 c)) 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 tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0028) {
		tmp = ((0.3333333333333333 / a) * (t_0 - (b * b))) / (t_1 + b);
	} else {
		tmp = ((((fma((a * 0.5), (6.75 * ((c * c) / (b * b))), (-4.5 * c)) * 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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0028)
		tmp = Float64(Float64(Float64(0.3333333333333333 / a) * Float64(t_0 - Float64(b * b))) / Float64(t_1 + b));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(a * 0.5), Float64(6.75 * Float64(Float64(c * c) / Float64(b * b))), Float64(-4.5 * c)) * a) * b) / fma(b, b, Float64(Float64(t_1 * b) + t_0))) / 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]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0028], N[(N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * 0.5), $MachinePrecision] * N[(6.75 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.5 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00279999999999999997

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.7%

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

   6. Applied rewrites 81.3%

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

   if -0.00279999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.8%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in b around inf

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

   9. Applied rewrites 94.3%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\mathsf{fma}\left(a \cdot 0.5, 6.75 \cdot \frac{c \cdot c}{b \cdot b}, -4.5 \cdot c\right) \cdot a\right) \cdot b}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} \cdot b + \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}}{a}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(t\_0 - b \cdot b\right)}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0028)
     (/ (* (/ 0.3333333333333333 a) (- t_0 (* b b))) (+ (sqrt t_0) b))
     (/ 1.0 (/ (fma 1.5 (* (/ c b) a) (* -2.0 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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0028) {
		tmp = ((0.3333333333333333 / a) * (t_0 - (b * b))) / (sqrt(t_0) + b);
	} else {
		tmp = 1.0 / (fma(1.5, ((c / b) * a), (-2.0 * 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0028)
		tmp = Float64(Float64(Float64(0.3333333333333333 / a) * Float64(t_0 - Float64(b * b))) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(c / b) * a), Float64(-2.0 * 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0028], N[(N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00279999999999999997

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.7%

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

   6. Applied rewrites 81.3%

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

   if -0.00279999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.9%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.9

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

   4. Applied rewrites 46.8%

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

   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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0028)
     (/ (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 3.0)
     (/ 1.0 (/ (fma 1.5 (* (/ c b) a) (* -2.0 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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0028) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) / 3.0;
	} else {
		tmp = 1.0 / (fma(1.5, ((c / b) * a), (-2.0 * 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0028)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) / 3.0);
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(c / b) * a), Float64(-2.0 * 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0028], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00279999999999999997

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.7%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a}}{3}} \]
   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. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 81.3

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

   6. Applied rewrites 81.3%

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

   if -0.00279999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.9%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.9

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

   4. Applied rewrites 46.8%

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

   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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0028)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 3.0 a)))
     (/ 1.0 (/ (fma 1.5 (* (/ c b) a) (* -2.0 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) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0028) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (3.0 * a));
	} else {
		tmp = 1.0 / (fma(1.5, ((c / b) * a), (-2.0 * 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(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0028)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(3.0 * a)));
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(c / b) * a), Float64(-2.0 * 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0028], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00279999999999999997

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 81.2%

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

   if -0.00279999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.9%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.9

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

   4. Applied rewrites 46.8%

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

   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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
   (/ (- (sqrt (fma (* -3.0 a) c (* b b))) b) (* 3.0 a))
   (/ 1.0 (/ (fma 1.5 (* (/ c b) a) (* -2.0 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(1.5, ((c / b) * a), (-2.0 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(c / b) * a), Float64(-2.0 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 83.3

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

   4. Applied rewrites 83.3%

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

   if -0.044999999999999998 < (/.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 49.7%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.7

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

   4. Applied rewrites 49.6%

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

   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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
   (/ (- (sqrt (fma (* -3.0 a) c (* b b))) b) (* 3.0 a))
   (/ 1.0 (fma 1.5 (/ a b) (* (/ b c) -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma(1.5, (a / b), ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(1.5, Float64(a / b), Float64(Float64(b / c) * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -2.0), $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)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 83.3

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

   4. Applied rewrites 83.3%

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

   if -0.044999999999999998 < (/.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 49.7%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.7

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

   4. Applied rewrites 49.6%

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

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.045)
   (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma 1.5 (/ a b) (* (/ b c) -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.045) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma(1.5, (a / b), ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.045)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / fma(1.5, Float64(a / b), Float64(Float64(b / c) * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.045], 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[(1.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -2.0), $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)) < -0.044999999999999998

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 83.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -0.044999999999999998 < (/.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 49.7%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 49.7

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

   4. Applied rewrites 49.6%

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

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.045:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 82.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 56.8

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

4. Applied rewrites 56.7%

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

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. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 80.7

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

7. Applied rewrites 80.7%

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

8. Final simplification 80.7%

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

Alternative 18: 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 63.0

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

5. Applied rewrites 63.0%

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

Alternative 19: 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

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 63.0

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

5. Applied rewrites 63.0%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 62.9%

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

8. Final simplification 62.9%

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

Reproduce

herbie shell --seed 2024272 
(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)))