Cubic critical, narrow range

Percentage Accurate: 55.7% → 91.4%

Time: 16.6s

Alternatives: 12

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: 55.7% 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: 91.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ t_1 := \frac{{c}^{4}}{{b}^{8}}\\ t_2 := \frac{{c}^{3}}{{b}^{6}} \cdot -1.4765625\\ t_3 := \frac{{c}^{2}}{{b}^{4}} \cdot -0.84375\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, 0.3333333333333333 \cdot \left(a \cdot \mathsf{fma}\left(0.5625, \frac{{c}^{2}}{{b}^{3}}, 2 \cdot \left(b \cdot t_3\right)\right) + \left({a}^{2} \cdot \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{t_3}}, 2 \cdot \left(b \cdot t_2\right)\right) + {a}^{3} \cdot \mathsf{fma}\left(-1.5, \frac{c \cdot t_2}{b}, \mathsf{fma}\left(2, b \cdot \left(t_1 \cdot -5.37890625 + t_1 \cdot 2.33349609375\right), b \cdot {t_3}^{2}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0)))
        (t_1 (/ (pow c 4.0) (pow b 8.0)))
        (t_2 (* (/ (pow c 3.0) (pow b 6.0)) -1.4765625))
        (t_3 (* (/ (pow c 2.0) (pow b 4.0)) -0.84375)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (*
       0.3333333333333333
       (+
        (* a (fma 0.5625 (/ (pow c 2.0) (pow b 3.0)) (* 2.0 (* b t_3))))
        (+
         (* (pow a 2.0) (fma -1.5 (/ c (/ b t_3)) (* 2.0 (* b t_2))))
         (*
          (pow a 3.0)
          (fma
           -1.5
           (/ (* c t_2) b)
           (fma
            2.0
            (* b (+ (* t_1 -5.37890625) (* t_1 2.33349609375)))
            (* b (pow t_3 2.0))))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double t_1 = pow(c, 4.0) / pow(b, 8.0);
	double t_2 = (pow(c, 3.0) / pow(b, 6.0)) * -1.4765625;
	double t_3 = (pow(c, 2.0) / pow(b, 4.0)) * -0.84375;
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.03) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (0.3333333333333333 * ((a * fma(0.5625, (pow(c, 2.0) / pow(b, 3.0)), (2.0 * (b * t_3)))) + ((pow(a, 2.0) * fma(-1.5, (c / (b / t_3)), (2.0 * (b * t_2)))) + (pow(a, 3.0) * fma(-1.5, ((c * t_2) / b), fma(2.0, (b * ((t_1 * -5.37890625) + (t_1 * 2.33349609375))), (b * pow(t_3, 2.0)))))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	t_1 = Float64((c ^ 4.0) / (b ^ 8.0))
	t_2 = Float64(Float64((c ^ 3.0) / (b ^ 6.0)) * -1.4765625)
	t_3 = Float64(Float64((c ^ 2.0) / (b ^ 4.0)) * -0.84375)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.03)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(0.3333333333333333 * Float64(Float64(a * fma(0.5625, Float64((c ^ 2.0) / (b ^ 3.0)), Float64(2.0 * Float64(b * t_3)))) + Float64(Float64((a ^ 2.0) * fma(-1.5, Float64(c / Float64(b / t_3)), Float64(2.0 * Float64(b * t_2)))) + Float64((a ^ 3.0) * fma(-1.5, Float64(Float64(c * t_2) / b), fma(2.0, Float64(b * Float64(Float64(t_1 * -5.37890625) + Float64(t_1 * 2.33349609375))), Float64(b * (t_3 ^ 2.0)))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -1.4765625), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.84375), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.03], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(a * N[(0.5625 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(-1.5 * N[(c / N[(b / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(-1.5 * N[(N[(c * t$95$2), $MachinePrecision] / b), $MachinePrecision] + N[(2.0 * N[(b * N[(N[(t$95$1 * -5.37890625), $MachinePrecision] + N[(t$95$1 * 2.33349609375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. flip-- 83.0%

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

      2. cbrt-unprod 84.2%

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

      3. pow-prod-up 84.6%

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

      4. metadata-eval 84.6%

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

      5. pow3 84.7%

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

      6. add-cbrt-cube 86.0%

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

      7. unpow2 86.0%

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

      8. pow1/3 85.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} + b}}{3 \cdot a} \]

      9. pow-pow 86.0%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      10. metadata-eval 86.0%

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

      11. pow1/2 86.0%

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

   9. Applied egg-rr 86.0%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   3. Simplified 44.9%

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

      1. add-sqr-sqrt 44.4%

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

      2. pow2 44.4%

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

      3. pow1/2 44.4%

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

      4. sqrt-pow1 44.8%

         \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b}{3 \cdot a} \]

      5. pow2 44.8%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b}{3 \cdot a} \]

      6. metadata-eval 44.8%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b}{3 \cdot a} \]

   5. Applied egg-rr 44.8%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}} - b}{3 \cdot a} \]

   6. Taylor expanded in a around 0 96.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(0.3333333333333333 \cdot \left(a \cdot \left(0.5625 \cdot \frac{{c}^{2}}{{b}^{3}} + 2 \cdot \left(b \cdot \left(-1.125 \cdot \frac{{c}^{2}}{{b}^{4}} + 0.28125 \cdot \frac{{c}^{2}}{{b}^{4}}\right)\right)\right)\right) + \left(0.3333333333333333 \cdot \left({a}^{2} \cdot \left(-1.5 \cdot \frac{c \cdot \left(-1.125 \cdot \frac{{c}^{2}}{{b}^{4}} + 0.28125 \cdot \frac{{c}^{2}}{{b}^{4}}\right)}{b} + 2 \cdot \left(b \cdot \left(-2.25 \cdot \frac{{c}^{3}}{{b}^{6}} + \left(-0.0703125 \cdot \frac{{c}^{3}}{{b}^{6}} + 0.84375 \cdot \frac{{c}^{3}}{{b}^{6}}\right)\right)\right)\right)\right) + 0.3333333333333333 \cdot \left({a}^{3} \cdot \left(-1.5 \cdot \frac{c \cdot \left(-2.25 \cdot \frac{{c}^{3}}{{b}^{6}} + \left(-0.0703125 \cdot \frac{{c}^{3}}{{b}^{6}} + 0.84375 \cdot \frac{{c}^{3}}{{b}^{6}}\right)\right)}{b} + \left(2 \cdot \left(b \cdot \left(-5.0625 \cdot \frac{{c}^{4}}{{b}^{8}} + \left(-0.31640625 \cdot \frac{{c}^{4}}{{b}^{8}} + \left(0.01318359375 \cdot \frac{{c}^{4}}{{b}^{8}} + \left(0.6328125 \cdot \frac{{c}^{4}}{{b}^{8}} + 1.6875 \cdot \frac{{c}^{4}}{{b}^{8}}\right)\right)\right)\right)\right) + b \cdot {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{4}} + 0.28125 \cdot \frac{{c}^{2}}{{b}^{4}}\right)}^{2}\right)\right)\right)\right)\right)} \]

   8. Simplified 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, 0.3333333333333333 \cdot \left(a \cdot \mathsf{fma}\left(0.5625, \frac{{c}^{2}}{{b}^{3}}, 2 \cdot \left(b \cdot \left(\frac{{c}^{2}}{{b}^{4}} \cdot -0.84375\right)\right)\right) + \left({a}^{2} \cdot \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{\frac{{c}^{2}}{{b}^{4}} \cdot -0.84375}}, 2 \cdot \left(b \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -1.4765625\right)\right)\right) + {a}^{3} \cdot \mathsf{fma}\left(-1.5, \frac{c \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -1.4765625\right)}{b}, \mathsf{fma}\left(2, b \cdot \left(\frac{{c}^{4}}{{b}^{8}} \cdot -5.37890625 + \frac{{c}^{4}}{{b}^{8}} \cdot 2.33349609375\right), b \cdot {\left(\frac{{c}^{2}}{{b}^{4}} \cdot -0.84375\right)}^{2}\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, 0.3333333333333333 \cdot \left(a \cdot \mathsf{fma}\left(0.5625, \frac{{c}^{2}}{{b}^{3}}, 2 \cdot \left(b \cdot \left(\frac{{c}^{2}}{{b}^{4}} \cdot -0.84375\right)\right)\right) + \left({a}^{2} \cdot \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{\frac{{c}^{2}}{{b}^{4}} \cdot -0.84375}}, 2 \cdot \left(b \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -1.4765625\right)\right)\right) + {a}^{3} \cdot \mathsf{fma}\left(-1.5, \frac{c \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -1.4765625\right)}{b}, \mathsf{fma}\left(2, b \cdot \left(\frac{{c}^{4}}{{b}^{8}} \cdot -5.37890625 + \frac{{c}^{4}}{{b}^{8}} \cdot 2.33349609375\right), b \cdot {\left(\frac{{c}^{2}}{{b}^{4}} \cdot -0.84375\right)}^{2}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 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) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.03) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (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) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.03)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / 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) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.03], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $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] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. flip-- 83.0%

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

      2. cbrt-unprod 84.2%

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

      3. pow-prod-up 84.6%

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

      4. metadata-eval 84.6%

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

      5. pow3 84.7%

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

      6. add-cbrt-cube 86.0%

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

      7. unpow2 86.0%

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

      8. pow1/3 85.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} + b}}{3 \cdot a} \]

      9. pow-pow 86.0%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      10. metadata-eval 86.0%

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

      11. pow1/2 86.0%

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

   9. Applied egg-rr 86.0%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 96.3%

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

   3. Taylor expanded in c around 0 96.3%

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

      1. distribute-rgt-out 96.3%

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

      2. associate-*r* 96.3%

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

      3. *-commutative 96.3%

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

      4. times-frac 96.3%

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

   5. Simplified 96.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 3: 89.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, {c}^{2} \cdot \frac{a}{{b}^{3}}, \frac{{c}^{3} \cdot \left({a}^{2} \cdot -0.5625\right)}{{b}^{5}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (fma
       -0.375
       (* (pow c 2.0) (/ a (pow b 3.0)))
       (/ (* (pow c 3.0) (* (pow a 2.0) -0.5625)) (pow b 5.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.03) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.375, (pow(c, 2.0) * (a / pow(b, 3.0))), ((pow(c, 3.0) * (pow(a, 2.0) * -0.5625)) / pow(b, 5.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.03)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.375, Float64((c ^ 2.0) * Float64(a / (b ^ 3.0))), Float64(Float64((c ^ 3.0) * Float64((a ^ 2.0) * -0.5625)) / (b ^ 5.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.03], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. flip-- 83.0%

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

      2. cbrt-unprod 84.2%

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

      3. pow-prod-up 84.6%

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

      4. metadata-eval 84.6%

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

      5. pow3 84.7%

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

      6. add-cbrt-cube 86.0%

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

      7. unpow2 86.0%

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

      8. pow1/3 85.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} + b}}{3 \cdot a} \]

      9. pow-pow 86.0%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      10. metadata-eval 86.0%

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

      11. pow1/2 86.0%

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

   9. Applied egg-rr 86.0%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 94.6%

      \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. fma-def 94.6%

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

      2. cube-prod 94.6%

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

      3. fma-def 94.6%

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

      4. associate-/l* 94.6%

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

      5. associate-/l* 94.6%

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

   4. Simplified 94.6%

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

   5. Taylor expanded in a around 0 94.9%

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

      1. +-commutative 94.9%

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

      2. associate-+l+ 94.9%

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

      3. +-commutative 94.9%

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

      4. fma-def 94.9%

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

      5. +-commutative 94.9%

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

      6. fma-def 94.9%

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

      7. associate-/l* 94.9%

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

      8. associate-/r/ 94.9%

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

      9. associate-*r/ 94.9%

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

      10. associate-*r* 94.9%

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

   7. Simplified 94.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, {c}^{2} \cdot \frac{a}{{b}^{3}}, \frac{{c}^{3} \cdot \left({a}^{2} \cdot -0.5625\right)}{{b}^{5}}\right)\right)\\ \end{array} \]

Alternative 4: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 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 t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.03) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (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;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.03)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / 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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.03], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. flip-- 83.0%

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

      2. cbrt-unprod 84.2%

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

      3. pow-prod-up 84.6%

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

      4. metadata-eval 84.6%

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

      5. pow3 84.7%

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

      6. add-cbrt-cube 86.0%

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

      7. unpow2 86.0%

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

      8. pow1/3 85.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} + b}}{3 \cdot a} \]

      9. pow-pow 86.0%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      10. metadata-eval 86.0%

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

      11. pow1/2 86.0%

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

   9. Applied egg-rr 86.0%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 94.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b}{3}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
   (/ 1.0 (/ a (/ (- (sqrt (fma a (* c -3.0) (pow b 2.0))) b) 3.0)))
   (+
    (* -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)) <= -0.03) {
		tmp = 1.0 / (a / ((sqrt(fma(a, (c * -3.0), pow(b, 2.0))) - b) / 3.0));
	} 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;
}

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)) <= -0.03)
		tmp = Float64(1.0 / Float64(a / Float64(Float64(sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0))) - b) / 3.0)));
	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

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], -0.03], N[(1.0 / N[(a / N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision]), $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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. *-commutative 83.1%

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

      4. pow1/3 80.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 84.8%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 84.8%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. unpow-1 84.8%

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

       2. associate-/l* 84.8%

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

   11. Simplified 84.8%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 94.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b}{3}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b}{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
   (/ 1.0 (/ a (/ (- (sqrt (fma a (* c -3.0) (pow b 2.0))) b) 3.0)))
   (/
    (fma
     -1.6875
     (/ (pow (* a c) 3.0) (pow b 5.0))
     (+ (* -1.5 (/ (* a c) b)) (* -1.125 (/ (* (* a c) (* a c)) (pow b 3.0)))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.03) {
		tmp = 1.0 / (a / ((sqrt(fma(a, (c * -3.0), pow(b, 2.0))) - b) / 3.0));
	} else {
		tmp = fma(-1.6875, (pow((a * c), 3.0) / pow(b, 5.0)), ((-1.5 * ((a * c) / b)) + (-1.125 * (((a * c) * (a * c)) / pow(b, 3.0))))) / (3.0 * 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)) <= -0.03)
		tmp = Float64(1.0 / Float64(a / Float64(Float64(sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0))) - b) / 3.0)));
	else
		tmp = Float64(fma(-1.6875, Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)), Float64(Float64(-1.5 * Float64(Float64(a * c) / b)) + Float64(-1.125 * Float64(Float64(Float64(a * c) * Float64(a * c)) / (b ^ 3.0))))) / Float64(3.0 * a));
	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], -0.03], N[(1.0 / N[(a / N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[(N[(a * c), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. *-commutative 83.1%

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

      4. pow1/3 80.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 84.8%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 84.8%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. unpow-1 84.8%

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

       2. associate-/l* 84.8%

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

   11. Simplified 84.8%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 94.6%

      \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. fma-def 94.6%

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

      2. cube-prod 94.6%

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

      3. fma-def 94.6%

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

      4. associate-/l* 94.6%

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

      5. associate-/l* 94.6%

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

   4. Simplified 94.6%

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

   5. Taylor expanded in a around 0 94.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}{3 \cdot a} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 94.6%

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

      2. pow2 94.6%

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

      3. cbrt-div 94.6%

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

      4. pow-prod-down 94.6%

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

      5. unpow3 94.6%

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

      6. add-cbrt-cube 94.6%

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

      7. cbrt-div 94.6%

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

      8. pow-prod-down 94.6%

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

      9. unpow3 94.6%

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

      10. add-cbrt-cube 94.6%

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

   7. Applied egg-rr 94.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{{\left(a \cdot c\right)}^{2}}}{b}\right)}^{2} \cdot \frac{\sqrt[3]{{\left(a \cdot c\right)}^{2}}}{b}\right)}\right)}{3 \cdot a} \]
   8. Step-by-step derivation

      1. pow-plus 94.6%

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

      2. metadata-eval 94.6%

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

      3. cube-div 94.6%

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

      4. rem-cube-cbrt 94.6%

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

   9. Simplified 94.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}}\right)}{3 \cdot a} \]
   10. Step-by-step derivation

       1. unpow2 94.6%

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

   11. Applied egg-rr 94.6%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}\right)}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b}{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 7: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b}{3}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
   (/ 1.0 (/ a (/ (- (sqrt (fma a (* c -3.0) (pow b 2.0))) b) 3.0)))
   (+ (* -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)) <= -0.03) {
		tmp = 1.0 / (a / ((sqrt(fma(a, (c * -3.0), pow(b, 2.0))) - b) / 3.0));
	} 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)) <= -0.03)
		tmp = Float64(1.0 / Float64(a / Float64(Float64(sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0))) - b) / 3.0)));
	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], -0.03], N[(1.0 / N[(a / N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $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]]

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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.8%

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

      7. add-cbrt-cube 83.0%

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

      8. pow1/3 80.3%

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

      9. pow3 80.3%

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

      10. sqrt-pow2 80.1%

          \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

      11. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      12. +-commutative 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      13. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{\left(a \cdot -3\right) \cdot c} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      14. associate-*r* 80.2%

          \[\leadsto \frac{{\left({\left(\color{blue}{a \cdot \left(-3 \cdot c\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      15. *-commutative 80.2%

          \[\leadsto \frac{{\left({\left(a \cdot \color{blue}{\left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      16. fma-udef 80.2%

          \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      17. pow2 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

      18. metadata-eval 80.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 80.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 83.1%

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

   7. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. *-commutative 83.1%

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

      4. pow1/3 80.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 84.8%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 84.8%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. unpow-1 84.8%

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

       2. associate-/l* 84.8%

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

   11. Simplified 84.8%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 90.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b}{3}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 8: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.03)
   (/ (- (sqrt (fma b b (* c (* a -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)) <= -0.03) {
		tmp = (sqrt(fma(b, b, (c * (a * -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)) <= -0.03)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -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], -0.03], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -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)) < -0.029999999999999999

   1. Initial program 84.7%

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

   3. Simplified 84.8%

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

   if -0.029999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 90.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.03:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 9: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7e-7)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7e-7) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / 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)) <= -7e-7)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / 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], -7e-7], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -6.99999999999999968e-7

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   if -6.99999999999999968e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.0%

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

   2. Taylor expanded in b around inf 85.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 76.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7e-7)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7e-7) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / 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)) <= -7e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / 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], -7e-7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -6.99999999999999968e-7

   1. Initial program 72.4%

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

   3. Simplified 72.5%

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

   if -6.99999999999999968e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.0%

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

   2. Taylor expanded in b around inf 85.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -7 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -7e-7) t_0 (* -0.5 (/ c 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 <= -7e-7) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / 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 <= (-7d-7)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / 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 <= -7e-7) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / 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 <= -7e-7:
		tmp = t_0
	else:
		tmp = -0.5 * (c / 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 <= -7e-7)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / 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 <= -7e-7)
		tmp = t_0;
	else
		tmp = -0.5 * (c / 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, -7e-7], t$95$0, N[(-0.5 * N[(c / b), $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)) < -6.99999999999999968e-7

   1. Initial program 72.4%

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

   if -6.99999999999999968e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.0%

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

   2. Taylor expanded in b around inf 85.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 64.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

2. Taylor expanded in b around inf 65.9%

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

3. Final simplification 65.9%

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

Reproduce

herbie shell --seed 2023307 
(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)))