Cubic critical, narrow range

Percentage Accurate: 55.3% → 91.9%

Time: 21.1s

Alternatives: 13

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)\\ t_1 := {\left(a \cdot c\right)}^{2}\\ t_2 := b - \sqrt{a \cdot \left(3 \cdot c\right)}\\ t_3 := t\_0 \cdot t\_2\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\frac{{t\_3}^{1.5} - {b}^{3}}{{b}^{2} + \mathsf{fma}\left(t\_0, t\_2, b \cdot \sqrt{t\_3}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{t\_1}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left(t\_1 \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (sqrt (* 3.0 a)) (sqrt c) b))
        (t_1 (pow (* a c) 2.0))
        (t_2 (- b (sqrt (* a (* 3.0 c)))))
        (t_3 (* t_0 t_2)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -3.5)
     (/
      (/
       (- (pow t_3 1.5) (pow b 3.0))
       (+ (pow b 2.0) (fma t_0 t_2 (* b (sqrt t_3)))))
      (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (*
       -0.16666666666666666
       (+
        (* (/ (pow a 3.0) (pow b 7.0)) (* 6.328125 (pow c 4.0)))
        (+
         (* (/ t_1 a) (/ 2.25 (pow b 3.0)))
         (/ (* (* t_1 2.25) (* a (* c 1.5))) (* a (pow b 5.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(sqrt((3.0 * a)), sqrt(c), b);
	double t_1 = pow((a * c), 2.0);
	double t_2 = b - sqrt((a * (3.0 * c)));
	double t_3 = t_0 * t_2;
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5) {
		tmp = ((pow(t_3, 1.5) - pow(b, 3.0)) / (pow(b, 2.0) + fma(t_0, t_2, (b * sqrt(t_3))))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.16666666666666666 * (((pow(a, 3.0) / pow(b, 7.0)) * (6.328125 * pow(c, 4.0))) + (((t_1 / a) * (2.25 / pow(b, 3.0))) + (((t_1 * 2.25) * (a * (c * 1.5))) / (a * pow(b, 5.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(sqrt(Float64(3.0 * a)), sqrt(c), b)
	t_1 = Float64(a * c) ^ 2.0
	t_2 = Float64(b - sqrt(Float64(a * Float64(3.0 * c))))
	t_3 = Float64(t_0 * t_2)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -3.5)
		tmp = Float64(Float64(Float64((t_3 ^ 1.5) - (b ^ 3.0)) / Float64((b ^ 2.0) + fma(t_0, t_2, Float64(b * sqrt(t_3))))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * Float64(6.328125 * (c ^ 4.0))) + Float64(Float64(Float64(t_1 / a) * Float64(2.25 / (b ^ 3.0))) + Float64(Float64(Float64(t_1 * 2.25) * Float64(a * Float64(c * 1.5))) / Float64(a * (b ^ 5.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(b - N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $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], -3.5], N[(N[(N[(N[Power[t$95$3, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$2 + N[(b * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 / a), $MachinePrecision] * N[(2.25 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 * 2.25), $MachinePrecision] * N[(a * N[(c * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 3 a) c)))) (*.f64 3 a)) < -3.5

   1. Initial program 87.0%

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

      1. add-sqr-sqrt 87.0%

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

      2. difference-of-squares 86.7%

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

      3. associate-*l* 86.7%

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

      4. associate-*l* 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. associate-*r* 86.6%

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

      2. *-commutative 86.6%

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

      3. associate-*r* 86.7%

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

      4. *-commutative 86.7%

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

   6. Simplified 86.7%

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

      1. flip3-+ 86.4%

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

   8. Applied egg-rr 87.4%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(\sqrt{a \cdot 3}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\mathsf{fma}\left(\sqrt{a \cdot 3}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(\sqrt{a \cdot 3}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}\right)}}}{3 \cdot a} \]
   9. Step-by-step derivation

      1. +-commutative 87.4%

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

      2. cube-neg 87.4%

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

      3. unsub-neg 87.4%

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

      4. *-commutative 87.4%

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

   10. Simplified 87.4%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.6%

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

      1. flip3-- 53.3%

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

      2. pow2 53.3%

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

      3. pow-pow 53.2%

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

      4. metadata-eval 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. associate-*l* 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      6. unpow-prod-down 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      7. metadata-eval 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      8. pow2 53.2%

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

      9. pow2 53.2%

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

      10. pow-prod-up 53.5%

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

      11. metadata-eval 53.5%

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

      12. distribute-rgt-out 53.5%

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

      13. associate-*l* 53.5%

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

      14. +-commutative 53.5%

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

      15. fma-define 53.5%

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

      16. associate-*l* 53.5%

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in b around inf 92.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   7. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, a \cdot \left(c \cdot \mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)\right), 0\right)}{a \cdot {b}^{7}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)} \]

   8. Taylor expanded in a around 0 92.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}\right)}{{b}^{7}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/l* 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      2. distribute-rgt-out 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(1.265625 + 5.0625\right)}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      3. metadata-eval 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   10. Simplified 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   11. Step-by-step derivation

       1. distribute-lft-in 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)}\right) \]

       2. associate-/r/ 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right)} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

       3. +-rgt-identity 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right) + -0.16666666666666666 \cdot \left(\frac{\color{blue}{{\left(a \cdot c\right)}^{2} \cdot 2.25}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

   12. Applied egg-rr 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right) + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)}\right) \]
   13. Step-by-step derivation

       1. distribute-lft-out 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right) + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)}\right) \]

       2. *-commutative 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \color{blue}{\left(6.328125 \cdot {c}^{4}\right)} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

       3. times-frac 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\color{blue}{\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

       4. +-lft-identity 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

       5. *-commutative 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(\left(a \cdot 1.5\right) \cdot c\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

       6. associate-*l* 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \color{blue}{\left(a \cdot \left(1.5 \cdot c\right)\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

   14. Simplified 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(1.5 \cdot c\right)\right)}{a \cdot {b}^{5}}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)\right)}^{1.5} - {b}^{3}}{{b}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right), b - \sqrt{a \cdot \left(3 \cdot c\right)}, b \cdot \sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{2}\\ t_1 := \mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)\\ t_2 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\frac{{b}^{2} + t\_1 \cdot \left(t\_2 - b\right)}{\left(-b\right) - \sqrt{t\_1 \cdot \left(b - t\_2\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{t\_0}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left(t\_0 \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 2.0))
        (t_1 (fma (sqrt (* 3.0 a)) (sqrt c) b))
        (t_2 (sqrt (* a (* 3.0 c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -3.5)
     (/
      (/ (+ (pow b 2.0) (* t_1 (- t_2 b))) (- (- b) (sqrt (* t_1 (- b t_2)))))
      (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (*
       -0.16666666666666666
       (+
        (* (/ (pow a 3.0) (pow b 7.0)) (* 6.328125 (pow c 4.0)))
        (+
         (* (/ t_0 a) (/ 2.25 (pow b 3.0)))
         (/ (* (* t_0 2.25) (* a (* c 1.5))) (* a (pow b 5.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 2.0);
	double t_1 = fma(sqrt((3.0 * a)), sqrt(c), b);
	double t_2 = sqrt((a * (3.0 * c)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5) {
		tmp = ((pow(b, 2.0) + (t_1 * (t_2 - b))) / (-b - sqrt((t_1 * (b - t_2))))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.16666666666666666 * (((pow(a, 3.0) / pow(b, 7.0)) * (6.328125 * pow(c, 4.0))) + (((t_0 / a) * (2.25 / pow(b, 3.0))) + (((t_0 * 2.25) * (a * (c * 1.5))) / (a * pow(b, 5.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 2.0
	t_1 = fma(sqrt(Float64(3.0 * a)), sqrt(c), b)
	t_2 = sqrt(Float64(a * Float64(3.0 * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -3.5)
		tmp = Float64(Float64(Float64((b ^ 2.0) + Float64(t_1 * Float64(t_2 - b))) / Float64(Float64(-b) - sqrt(Float64(t_1 * Float64(b - t_2))))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * Float64(6.328125 * (c ^ 4.0))) + Float64(Float64(Float64(t_0 / a) * Float64(2.25 / (b ^ 3.0))) + Float64(Float64(Float64(t_0 * 2.25) * Float64(a * Float64(c * 1.5))) / Float64(a * (b ^ 5.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $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], -3.5], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$2 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(t$95$1 * N[(b - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 / a), $MachinePrecision] * N[(2.25 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * 2.25), $MachinePrecision] * N[(a * N[(c * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 3 a) c)))) (*.f64 3 a)) < -3.5

   1. Initial program 87.0%

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

      1. add-sqr-sqrt 87.0%

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

      2. difference-of-squares 86.7%

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

      3. associate-*l* 86.7%

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

      4. associate-*l* 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. associate-*r* 86.6%

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

      2. *-commutative 86.6%

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

      3. associate-*r* 86.7%

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

      4. *-commutative 86.7%

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

   6. Simplified 86.7%

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

      1. flip-+ 86.3%

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

      2. pow2 86.3%

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

      3. add-sqr-sqrt 87.4%

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

      4. +-commutative 87.4%

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

      5. sqrt-prod 87.4%

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

      6. fma-define 87.5%

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

      7. associate-*l* 87.4%

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

   8. Applied egg-rr 87.4%

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

      1. unpow2 87.4%

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

      2. sqr-neg 87.4%

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

      3. unpow2 87.4%

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

      4. *-commutative 87.4%

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

      5. *-commutative 87.4%

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

   10. Simplified 87.4%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.6%

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

      1. flip3-- 53.3%

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

      2. pow2 53.3%

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

      3. pow-pow 53.2%

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

      4. metadata-eval 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. associate-*l* 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      6. unpow-prod-down 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      7. metadata-eval 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      8. pow2 53.2%

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

      9. pow2 53.2%

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

      10. pow-prod-up 53.5%

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

      11. metadata-eval 53.5%

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

      12. distribute-rgt-out 53.5%

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

      13. associate-*l* 53.5%

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

      14. +-commutative 53.5%

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

      15. fma-define 53.5%

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

      16. associate-*l* 53.5%

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in b around inf 92.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   7. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, a \cdot \left(c \cdot \mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)\right), 0\right)}{a \cdot {b}^{7}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)} \]

   8. Taylor expanded in a around 0 92.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}\right)}{{b}^{7}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/l* 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      2. distribute-rgt-out 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(1.265625 + 5.0625\right)}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      3. metadata-eval 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   10. Simplified 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   11. Step-by-step derivation

       1. distribute-lft-in 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)}\right) \]

       2. associate-/r/ 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right)} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

       3. +-rgt-identity 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right) + -0.16666666666666666 \cdot \left(\frac{\color{blue}{{\left(a \cdot c\right)}^{2} \cdot 2.25}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

   12. Applied egg-rr 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right) + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)}\right) \]
   13. Step-by-step derivation

       1. distribute-lft-out 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right) + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)}\right) \]

       2. *-commutative 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \color{blue}{\left(6.328125 \cdot {c}^{4}\right)} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

       3. times-frac 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\color{blue}{\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

       4. +-lft-identity 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

       5. *-commutative 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(\left(a \cdot 1.5\right) \cdot c\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

       6. associate-*l* 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \color{blue}{\left(a \cdot \left(1.5 \cdot c\right)\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

   14. Simplified 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(1.5 \cdot c\right)\right)}{a \cdot {b}^{5}}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\frac{{b}^{2} + \mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(\sqrt{a \cdot \left(3 \cdot c\right)} - b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{t\_0}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left(t\_0 \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 2.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -3.5)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (*
       -0.16666666666666666
       (+
        (* (/ (pow a 3.0) (pow b 7.0)) (* 6.328125 (pow c 4.0)))
        (+
         (* (/ t_0 a) (/ 2.25 (pow b 3.0)))
         (/ (* (* t_0 2.25) (* a (* c 1.5))) (* a (pow b 5.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 2.0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.16666666666666666 * (((pow(a, 3.0) / pow(b, 7.0)) * (6.328125 * pow(c, 4.0))) + (((t_0 / a) * (2.25 / pow(b, 3.0))) + (((t_0 * 2.25) * (a * (c * 1.5))) / (a * pow(b, 5.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -3.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * Float64(6.328125 * (c ^ 4.0))) + Float64(Float64(Float64(t_0 / a) * Float64(2.25 / (b ^ 3.0))) + Float64(Float64(Float64(t_0 * 2.25) * Float64(a * Float64(c * 1.5))) / Float64(a * (b ^ 5.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 / a), $MachinePrecision] * N[(2.25 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * 2.25), $MachinePrecision] * N[(a * N[(c * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 3 a) c)))) (*.f64 3 a)) < -3.5

   1. Initial program 87.0%

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

   3. Taylor expanded in a around 0 87.1%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.6%

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

      1. flip3-- 53.3%

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

      2. pow2 53.3%

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

      3. pow-pow 53.2%

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

      4. metadata-eval 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. associate-*l* 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      6. unpow-prod-down 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      7. metadata-eval 53.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      8. pow2 53.2%

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

      9. pow2 53.2%

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

      10. pow-prod-up 53.5%

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

      11. metadata-eval 53.5%

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

      12. distribute-rgt-out 53.5%

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

      13. associate-*l* 53.5%

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

      14. +-commutative 53.5%

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

      15. fma-define 53.5%

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

      16. associate-*l* 53.5%

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in b around inf 92.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   7. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, a \cdot \left(c \cdot \mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)\right), 0\right)}{a \cdot {b}^{7}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)} \]

   8. Taylor expanded in a around 0 92.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}\right)}{{b}^{7}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/l* 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      2. distribute-rgt-out 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(1.265625 + 5.0625\right)}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      3. metadata-eval 92.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   10. Simplified 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   11. Step-by-step derivation

       1. distribute-lft-in 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)}\right) \]

       2. associate-/r/ 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right)} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

       3. +-rgt-identity 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right) + -0.16666666666666666 \cdot \left(\frac{\color{blue}{{\left(a \cdot c\right)}^{2} \cdot 2.25}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

   12. Applied egg-rr 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right)\right) + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)}\right) \]
   13. Step-by-step derivation

       1. distribute-lft-out 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left({c}^{4} \cdot 6.328125\right) + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)}\right) \]

       2. *-commutative 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \color{blue}{\left(6.328125 \cdot {c}^{4}\right)} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

       3. times-frac 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\color{blue}{\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

       4. +-lft-identity 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

       5. *-commutative 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(\left(a \cdot 1.5\right) \cdot c\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

       6. associate-*l* 92.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \color{blue}{\left(a \cdot \left(1.5 \cdot c\right)\right)}}{a \cdot {b}^{5}}\right)\right)\right) \]

   14. Simplified 92.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(1.5 \cdot c\right)\right)}{a \cdot {b}^{5}}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot \left(6.328125 \cdot {c}^{4}\right) + \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -3.5)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+
     (* -0.5 (/ c b))
     (+
      (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
      (*
       -0.16666666666666666
       (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 6.328125 a))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / pow(b, 7.0)) * (6.328125 / a)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-3.5d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((a * c) ** 4.0d0) / (b ** 7.0d0)) * (6.328125d0 / a)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((a * c), 4.0) / Math.pow(b, 7.0)) * (6.328125 / a)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((a * c), 4.0) / math.pow(b, 7.0)) * (6.328125 / a)))))
	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)) <= -3.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.5)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((a * c) ^ 4.0) / (b ^ 7.0)) * (6.328125 / a)))));
	end
	tmp_2 = 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], -3.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $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 3 a) c)))) (*.f64 3 a)) < -3.5

   1. Initial program 87.0%

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

   3. Taylor expanded in a around 0 87.1%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.6%

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

   3. Taylor expanded in b around inf 92.7%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   4. Taylor expanded in c around 0 92.7%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   5. Step-by-step derivation

      1. distribute-rgt-in 92.7%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left(1.265625 \cdot {a}^{4}\right) \cdot {c}^{4} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 92.7%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}}{a \cdot {b}^{7}}\right)\right) \]

      3. associate-*r* 92.7%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}}{a \cdot {b}^{7}}\right)\right) \]

      4. distribute-rgt-out 92.7%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      5. *-commutative 92.7%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}{\color{blue}{{b}^{7} \cdot a}}\right)\right) \]

      6. times-frac 92.7%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot \frac{1.265625 + 5.0625}{a}\right)}\right)\right) \]

   6. Simplified 92.7%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(a \cdot c\right) \cdot -1.5\right)}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{t\_0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot t\_0, 0\right)}{a \cdot {b}^{5}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* (* a c) -1.5) 2.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.8)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (fma
      -0.16666666666666666
      (+
       (/ t_0 (* a (pow b 3.0)))
       (/ (fma 1.5 (* (* a c) t_0) 0.0) (* a (pow b 5.0))))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(((a * c) * -1.5), 2.0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.8) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.16666666666666666, ((t_0 / (a * pow(b, 3.0))) + (fma(1.5, ((a * c) * t_0), 0.0) / (a * pow(b, 5.0)))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -1.5) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.16666666666666666, Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(fma(1.5, Float64(Float64(a * c) * t_0), 0.0) / Float64(a * (b ^ 5.0)))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(N[(a * c), $MachinePrecision] * -1.5), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.7999999999999998

   1. Initial program 85.6%

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

   3. Taylor expanded in a around 0 85.7%

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

   if -2.7999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.3%

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

      1. flip3-- 53.0%

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

      2. pow2 53.0%

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

      3. pow-pow 53.0%

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

      4. metadata-eval 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. associate-*l* 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      6. unpow-prod-down 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      7. metadata-eval 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      8. pow2 53.0%

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

      9. pow2 53.0%

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

      10. pow-prod-up 53.2%

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

      11. metadata-eval 53.2%

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

      12. distribute-rgt-out 53.2%

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

      13. associate-*l* 53.2%

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

      14. +-commutative 53.2%

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

      15. fma-define 53.2%

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

      16. associate-*l* 53.2%

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

   4. Applied egg-rr 53.2%

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

      1. expm1-log1p-u 51.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a}\right)\right)} \]

      2. expm1-undefine 50.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a}\right)} - 1} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right), {b}^{4}\right)}}\right)}{a \cdot 3}\right)} - 1} \]

   7. Taylor expanded in b around inf 89.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]

   9. Simplified 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{0 + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, 0 + \left(a \cdot c\right) \cdot {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}, 0\right)}{a \cdot {b}^{5}}, -0.5 \cdot \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(a \cdot c\right) \cdot -1.5\right)}^{2}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot {\left(\left(a \cdot c\right) \cdot -1.5\right)}^{2}, 0\right)}{a \cdot {b}^{5}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.8)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (fma
    -0.5
    (/ c b)
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
     (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.8) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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 3 a) c)))) (*.f64 3 a)) < -2.7999999999999998

   1. Initial program 85.6%

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

   3. Taylor expanded in a around 0 85.7%

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

   if -2.7999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.3%

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

      1. flip3-- 53.0%

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

      2. pow2 53.0%

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

      3. pow-pow 53.0%

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

      4. metadata-eval 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. associate-*l* 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      6. unpow-prod-down 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      7. metadata-eval 53.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      8. pow2 53.0%

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

      9. pow2 53.0%

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

      10. pow-prod-up 53.2%

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

      11. metadata-eval 53.2%

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

      12. distribute-rgt-out 53.2%

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

      13. associate-*l* 53.2%

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

      14. +-commutative 53.2%

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

      15. fma-define 53.2%

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

      16. associate-*l* 53.2%

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

   4. Applied egg-rr 53.2%

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

   5. Taylor expanded in b around inf 92.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   7. Simplified 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, a \cdot \left(c \cdot \mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)\right), 0\right)}{a \cdot {b}^{7}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)} \]

   8. Taylor expanded in a around 0 92.9%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}\right)}{{b}^{7}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/l* 92.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      2. distribute-rgt-out 92.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(1.265625 + 5.0625\right)}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

      3. metadata-eval 92.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   10. Simplified 92.9%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}} + \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot 0, \left(1.5 \cdot a\right) \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   11. Taylor expanded in a around 0 89.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.8)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.8) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-2.8d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.8) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.8:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.8)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	tmp_2 = 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], -2.8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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 3 a) c)))) (*.f64 3 a)) < -2.7999999999999998

   1. Initial program 85.6%

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

   3. Taylor expanded in a around 0 85.7%

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

   if -2.7999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.3%

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

   3. Taylor expanded in b around inf 89.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.001:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.001)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.001) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.001)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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 3 a) c)))) (*.f64 3 a)) < -1e-3

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

      2. sqr-neg 77.0%

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

      3. unsub-neg 77.0%

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

      4. div-sub 75.8%

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

      5. --rgt-identity 75.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 77.0%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 77.1%

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

   if -1e-3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 43.1%

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

   3. Taylor expanded in b around inf 91.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.001:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ (* 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)) <= -1.5e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (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)) <= -1.5e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(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], -1.5e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $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 3 a) c)))) (*.f64 3 a)) < -1.50000000000000004e-5

   1. Initial program 74.7%

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

      1. +-commutative 74.7%

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

      2. sqr-neg 74.7%

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

      3. unsub-neg 74.7%

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

      4. div-sub 73.6%

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

      5. --rgt-identity 73.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 74.7%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 74.8%

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

   if -1.50000000000000004e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 37.8%

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

   3. Taylor expanded in b around inf 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -1.5e-5) t_0 (/ (* 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 <= -1.5e-5) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-1.5d-5)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -1.5e-5) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -1.5e-5:
		tmp = t_0
	else:
		tmp = (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 <= -1.5e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -1.5e-5)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = 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, -1.5e-5], t$95$0, N[(N[(c * -0.5), $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 3 a) c)))) (*.f64 3 a)) < -1.50000000000000004e-5

   1. Initial program 74.7%

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

   if -1.50000000000000004e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 37.8%

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

   3. Taylor expanded in b around inf 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2850:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2850.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2850.0) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2850.0d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2850.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2850.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2850.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2850.0)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2850.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2850

   1. Initial program 73.3%

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

   3. Taylor expanded in a around 0 73.3%

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

   if 2850 < b

   1. Initial program 42.2%

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

   3. Taylor expanded in b around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*l/ 75.4%

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

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2850:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in b around inf 63.1%

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

   1. *-commutative 63.1%

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

   2. associate-*l/ 63.1%

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

5. Simplified 63.1%

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

6. Final simplification 63.1%

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

Alternative 13: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. add-sqr-sqrt 57.0%

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

   2. difference-of-squares 56.9%

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

   3. associate-*l* 56.9%

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

   4. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. associate-*r* 56.9%

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

   2. *-commutative 56.9%

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

   3. associate-*r* 56.9%

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

   4. *-commutative 56.9%

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

6. Simplified 56.9%

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

7. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))