Cubic critical, narrow range

Percentage Accurate: 55.5% → 91.9%

Time: 13.8s

Alternatives: 13

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: 55.5% 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: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := b + \sqrt{t\_0}\\ t_2 := t\_1 \cdot t\_1\\ t_3 := b \cdot \left(b \cdot b\right)\\ t_4 := b \cdot t\_3\\ t_5 := b \cdot t\_4\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\frac{\frac{t\_0 \cdot t\_0}{t\_2} - \frac{t\_4}{t\_2}}{\frac{t\_0}{t\_1} + \frac{b \cdot b}{t\_1}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot t\_5\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{t\_5}\right), \frac{c \cdot c}{t\_3 \cdot -2.6666666666666665}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b)))
        (t_1 (+ b (sqrt t_0)))
        (t_2 (* t_1 t_1))
        (t_3 (* b (* b b)))
        (t_4 (* b t_3))
        (t_5 (* b t_4)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7.0)
     (/
      (/ (- (/ (* t_0 t_0) t_2) (/ t_4 t_2)) (+ (/ t_0 t_1) (/ (* b b) t_1)))
      (* 3.0 a))
     (fma
      (/ c b)
      -0.5
      (*
       a
       (fma
        a
        (*
         c
         (fma
          (* c (* c (* a c)))
          (/ -1.0546875 (* b (* b t_5)))
          (/ (* (* c c) -0.5625) t_5)))
        (/ (* c c) (* t_3 -2.6666666666666665))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = b + sqrt(t_0);
	double t_2 = t_1 * t_1;
	double t_3 = b * (b * b);
	double t_4 = b * t_3;
	double t_5 = b * t_4;
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7.0) {
		tmp = ((((t_0 * t_0) / t_2) - (t_4 / t_2)) / ((t_0 / t_1) + ((b * b) / t_1))) / (3.0 * a);
	} else {
		tmp = fma((c / b), -0.5, (a * fma(a, (c * fma((c * (c * (a * c))), (-1.0546875 / (b * (b * t_5))), (((c * c) * -0.5625) / t_5))), ((c * c) / (t_3 * -2.6666666666666665)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(b + sqrt(t_0))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(b * Float64(b * b))
	t_4 = Float64(b * t_3)
	t_5 = Float64(b * t_4)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7.0)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * t_0) / t_2) - Float64(t_4 / t_2)) / Float64(Float64(t_0 / t_1) + Float64(Float64(b * b) / t_1))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(c / b), -0.5, Float64(a * fma(a, Float64(c * fma(Float64(c * Float64(c * Float64(a * c))), Float64(-1.0546875 / Float64(b * Float64(b * t_5))), Float64(Float64(Float64(c * c) * -0.5625) / t_5))), Float64(Float64(c * c) / Float64(t_3 * -2.6666666666666665)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(b * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(b * t$95$4), $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], -7.0], N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(t$95$4 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 / t$95$1), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5 + N[(a * N[(a * N[(c * N[(N[(c * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / N[(b * N[(b * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(t$95$3 * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -7

   1. Initial program 86.3%

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

   4. Applied rewrites 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

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

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 87.6%

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

   8. Applied rewrites 88.6%

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

   if -7 < (/.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 49.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   11. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(\frac{c}{b}, \color{blue}{-0.5}, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(c \cdot a\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) \cdot \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} - \frac{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} + \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ t_2 := b \cdot \left(b \cdot t\_1\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{-1}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot t\_2\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{t\_2}\right), \frac{c \cdot c}{t\_1 \cdot -2.6666666666666665}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b)))
        (t_1 (* b (* b b)))
        (t_2 (* b (* b t_1))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7.0)
     (/ (* (- (* b b) t_0) (/ -1.0 (+ b (sqrt t_0)))) (* 3.0 a))
     (fma
      (/ c b)
      -0.5
      (*
       a
       (fma
        a
        (*
         c
         (fma
          (* c (* c (* a c)))
          (/ -1.0546875 (* b (* b t_2)))
          (/ (* (* c c) -0.5625) t_2)))
        (/ (* c c) (* t_1 -2.6666666666666665))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = b * (b * b);
	double t_2 = b * (b * t_1);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7.0) {
		tmp = (((b * b) - t_0) * (-1.0 / (b + sqrt(t_0)))) / (3.0 * a);
	} else {
		tmp = fma((c / b), -0.5, (a * fma(a, (c * fma((c * (c * (a * c))), (-1.0546875 / (b * (b * t_2))), (((c * c) * -0.5625) / t_2))), ((c * c) / (t_1 * -2.6666666666666665)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(b * Float64(b * b))
	t_2 = Float64(b * Float64(b * t_1))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(-1.0 / Float64(b + sqrt(t_0)))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(c / b), -0.5, Float64(a * fma(a, Float64(c * fma(Float64(c * Float64(c * Float64(a * c))), Float64(-1.0546875 / Float64(b * Float64(b * t_2))), Float64(Float64(Float64(c * c) * -0.5625) / t_2))), Float64(Float64(c * c) / Float64(t_1 * -2.6666666666666665)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * t$95$1), $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], -7.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(-1.0 / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5 + N[(a * N[(a * N[(c * N[(N[(c * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / N[(b * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(t$95$1 * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -7

   1. Initial program 86.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 88.6%

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

   if -7 < (/.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 49.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   11. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(\frac{c}{b}, \color{blue}{-0.5}, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(c \cdot a\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot \frac{-1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ t_2 := b \cdot \left(b \cdot t\_1\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{-1}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot t\_2\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{t\_2}\right), \frac{c \cdot c}{t\_1 \cdot -2.6666666666666665}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b)))
        (t_1 (* b (* b b)))
        (t_2 (* b (* b t_1))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7.0)
     (/ (* (- (* b b) t_0) (/ -1.0 (+ b (sqrt t_0)))) (* 3.0 a))
     (fma
      (/ -0.5 b)
      c
      (*
       a
       (fma
        a
        (*
         c
         (fma
          (* c (* c (* a c)))
          (/ -1.0546875 (* b (* b t_2)))
          (/ (* (* c c) -0.5625) t_2)))
        (/ (* c c) (* t_1 -2.6666666666666665))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = b * (b * b);
	double t_2 = b * (b * t_1);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7.0) {
		tmp = (((b * b) - t_0) * (-1.0 / (b + sqrt(t_0)))) / (3.0 * a);
	} else {
		tmp = fma((-0.5 / b), c, (a * fma(a, (c * fma((c * (c * (a * c))), (-1.0546875 / (b * (b * t_2))), (((c * c) * -0.5625) / t_2))), ((c * c) / (t_1 * -2.6666666666666665)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(b * Float64(b * b))
	t_2 = Float64(b * Float64(b * t_1))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(-1.0 / Float64(b + sqrt(t_0)))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(-0.5 / b), c, Float64(a * fma(a, Float64(c * fma(Float64(c * Float64(c * Float64(a * c))), Float64(-1.0546875 / Float64(b * Float64(b * t_2))), Float64(Float64(Float64(c * c) * -0.5625) / t_2))), Float64(Float64(c * c) / Float64(t_1 * -2.6666666666666665)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * t$95$1), $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], -7.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(-1.0 / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c + N[(a * N[(a * N[(c * N[(N[(c * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / N[(b * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(t$95$1 * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -7

   1. Initial program 86.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 88.6%

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

   if -7 < (/.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 49.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   11. Applied rewrites 93.9%

       \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, \color{blue}{c}, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(c \cdot a\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot \frac{-1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(a, c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ t_2 := b \cdot \left(b \cdot t\_1\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{-1}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot \mathsf{fma}\left(c \cdot \left(c \cdot \left(a \cdot c\right)\right), \frac{-1.0546875}{b \cdot \left(b \cdot t\_2\right)}, \frac{\left(c \cdot c\right) \cdot -0.5625}{t\_2}\right), a \cdot a, c \cdot \mathsf{fma}\left(c \cdot \frac{-0.375}{t\_1}, a, \frac{-0.5}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b)))
        (t_1 (* b (* b b)))
        (t_2 (* b (* b t_1))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7.0)
     (/ (* (- (* b b) t_0) (/ -1.0 (+ b (sqrt t_0)))) (* 3.0 a))
     (fma
      (*
       c
       (fma
        (* c (* c (* a c)))
        (/ -1.0546875 (* b (* b t_2)))
        (/ (* (* c c) -0.5625) t_2)))
      (* a a)
      (* c (fma (* c (/ -0.375 t_1)) a (/ -0.5 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double t_1 = b * (b * b);
	double t_2 = b * (b * t_1);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7.0) {
		tmp = (((b * b) - t_0) * (-1.0 / (b + sqrt(t_0)))) / (3.0 * a);
	} else {
		tmp = fma((c * fma((c * (c * (a * c))), (-1.0546875 / (b * (b * t_2))), (((c * c) * -0.5625) / t_2))), (a * a), (c * fma((c * (-0.375 / t_1)), a, (-0.5 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_1 = Float64(b * Float64(b * b))
	t_2 = Float64(b * Float64(b * t_1))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(-1.0 / Float64(b + sqrt(t_0)))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(c * fma(Float64(c * Float64(c * Float64(a * c))), Float64(-1.0546875 / Float64(b * Float64(b * t_2))), Float64(Float64(Float64(c * c) * -0.5625) / t_2))), Float64(a * a), Float64(c * fma(Float64(c * Float64(-0.375 / t_1)), a, Float64(-0.5 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * t$95$1), $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], -7.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(-1.0 / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / N[(b * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(c * N[(N[(c * N[(-0.375 / t$95$1), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -7

   1. Initial program 86.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 88.6%

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

   if -7 < (/.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 49.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   11. Applied rewrites 93.8%

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

4. Final simplification 93.3%

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

Alternative 5: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.004)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ b (sqrt t_0))))
     (fma
      a
      (/
       (fma -0.5625 (/ (* a (* c (* c c))) (* b b)) (* (* c c) -0.375))
       (* b (* b b)))
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, (fma(-0.5625, ((a * (c * (c * c))) / (b * b)), ((c * c) * -0.375)) / (b * (b * b))), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), 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)) <= -0.004)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(a, Float64(fma(-0.5625, Float64(Float64(a * Float64(c * Float64(c * c))) / Float64(b * b)), Float64(Float64(c * c) * -0.375)) / Float64(b * Float64(b * b))), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-0.5625 * N[(N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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)) < -0.0040000000000000001

   1. Initial program 79.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 94.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.004)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ b (sqrt t_0))))
     (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), 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)) <= -0.004)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 79.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.004)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ b (sqrt t_0))))
     (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, 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], -0.004], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 79.5%

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sub-div N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 81.3%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.004:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.004)
     t_0
     (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.004) {
		tmp = t_0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.004)
		tmp = t_0;
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -0.004], t$95$0, N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 79.5%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.004)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* 3.0 a))
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (3.0 * a);
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 79.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 79.5

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 79.5

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

   4. Applied rewrites 79.5%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.004)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* 3.0 a))
   (/ (* c (fma -0.375 (/ (* a c) (* b b)) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c * fma(-0.375, ((a * c) / (b * b)), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 79.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 79.5

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 79.5

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

   4. Applied rewrites 79.5%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 90.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.004)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (/ (* c (fma -0.375 (/ (* a c) (* b b)) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = (c * fma(-0.375, ((a * c) / (b * b)), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 79.5%

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

   4. Applied rewrites 79.5%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 90.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 82.7%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 82.6%

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

Alternative 13: 64.2% 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 53.8%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 66.2

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

5. Applied rewrites 66.2%

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

6. Final simplification 66.2%

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

Reproduce

herbie shell --seed 2024226 
(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)))