Cubic critical, narrow range

Percentage Accurate: 55.2% → 91.7%

Time: 20.5s

Alternatives: 13

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ t_1 := {t_0}^{1.5}\\ t_2 := \left(t_0 + b \cdot \left(b + \sqrt[3]{t_1}\right)\right) \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{t_1}{t_2} - \frac{{b}^{3}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0)))
        (t_1 (pow t_0 1.5))
        (t_2 (* (+ t_0 (* b (+ b (cbrt t_1)))) (* a 3.0))))
   (if (<= b 0.009)
     (- (/ t_1 t_2) (/ (pow b 3.0) t_2))
     (+
      (* -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)))
        (/ (/ (* -1.0546875 (pow (* a c) 4.0)) (pow b 7.0)) a)))))))

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(t_0, 1.5);
	double t_2 = (t_0 + (b * (b + cbrt(t_1)))) * (a * 3.0);
	double tmp;
	if (b <= 0.009) {
		tmp = (t_1 / t_2) - (pow(b, 3.0) / t_2);
	} 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))) + (((-1.0546875 * pow((a * c), 4.0)) / pow(b, 7.0)) / a)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -3.0, (b ^ 2.0))
	t_1 = t_0 ^ 1.5
	t_2 = Float64(Float64(t_0 + Float64(b * Float64(b + cbrt(t_1)))) * Float64(a * 3.0))
	tmp = 0.0
	if (b <= 0.009)
		tmp = Float64(Float64(t_1 / t_2) - Float64((b ^ 3.0) / t_2));
	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(Float64(Float64(-1.0546875 * (Float64(a * c) ^ 4.0)) / (b ^ 7.0)) / a))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1.5], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + N[(b * N[(b + N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.009], N[(N[(t$95$1 / t$95$2), $MachinePrecision] - N[(N[Power[b, 3.0], $MachinePrecision] / t$95$2), $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[(N[(N[(-1.0546875 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00899999999999999932

   1. Initial program 91.7%

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

   3. Simplified 91.8%

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

      1. add-cbrt-cube 89.5%

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

      2. unpow3 89.5%

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

      3. fma-udef 89.3%

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

      4. +-commutative 89.3%

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

      5. fma-udef 89.2%

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

      6. sqrt-pow2 90.2%

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

      7. fma-udef 90.0%

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

      8. +-commutative 90.0%

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

      9. associate-*r* 90.0%

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

      10. fma-def 90.0%

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

      11. pow2 90.0%

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

      12. metadata-eval 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. flip3-- 89.9%

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

      2. div-inv 89.8%

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

      3. rem-cube-cbrt 92.0%

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

      4. cbrt-unprod 92.2%

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

      5. pow-sqr 92.2%

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

      6. metadata-eval 92.2%

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

   7. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{3}} + b \cdot \left(b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}\right)}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 92.2%

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

   9. Simplified 92.2%

      \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) + b \cdot \left(b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}\right)}}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. associate-/l* 92.3%

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

       2. div-sub 93.0%

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

   11. Applied egg-rr 93.0%

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

   if 0.00899999999999999932 < b

   1. Initial program 53.7%

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

   2. Taylor expanded in b around inf 93.9%

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

      1. div-inv 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\left(5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)}\right)\right) \]

      2. fma-def 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\color{blue}{\mathsf{fma}\left(5.0625, {a}^{4} \cdot {c}^{4}, {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right)} \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      3. pow-prod-down 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, \color{blue}{{\left(a \cdot c\right)}^{4}}, {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      4. unpow2 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      5. *-commutative 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -1.125\right)}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      6. *-commutative 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -1.125\right)} \cdot \left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      7. swap-sqr 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-1.125 \cdot -1.125\right)}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      8. pow-prod-down 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      9. pow-prod-down 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      10. pow-sqr 93.9%

          \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      11. metadata-eval 93.9%

          \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      12. metadata-eval 93.9%

          \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot \color{blue}{1.265625}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

   4. Applied egg-rr 93.9%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right) \cdot 1}{a \cdot {b}^{7}}}\right)\right) \]

      2. associate-*l/ 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}{a \cdot {b}^{7}} \cdot 1\right)}\right)\right) \]

      3. *-rgt-identity 93.9%

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

      4. fma-udef 93.9%

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

      5. *-commutative 93.9%

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

      6. distribute-rgt-out 93.9%

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

      7. metadata-eval 93.9%

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

   6. Simplified 93.9%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-/r* 93.9%

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

      4. *-commutative 93.9%

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

      5. associate-*r* 93.9%

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

      6. metadata-eval 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{\color{blue}{-1.0546875} \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}\right)\right) \]

   8. Applied egg-rr 93.9%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) + b \cdot \left(b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}\right)\right) \cdot \left(a \cdot 3\right)} - \frac{{b}^{3}}{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) + b \cdot \left(b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}\right)\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}\right)\right)\\ \end{array} \]

Alternative 2: 89.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(\left(t_0 - {b}^{2}\right) \cdot \frac{1}{b + \sqrt[3]{{t_0}^{1.5}}}\right) \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (*
      (/ 1.0 a)
      (*
       (* (- t_0 (pow b 2.0)) (/ 1.0 (+ b (cbrt (pow t_0 1.5)))))
       0.3333333333333333))
     (+
      (* -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) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (1.0 / a) * (((t_0 - pow(b, 2.0)) * (1.0 / (b + cbrt(pow(t_0, 1.5))))) * 0.3333333333333333);
	} 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(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(Float64(t_0 - (b ^ 2.0)) * Float64(1.0 / Float64(b + cbrt((t_0 ^ 1.5))))) * 0.3333333333333333));
	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[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b + N[Power[N[Power[t$95$0, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.2%

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

      1. add-cbrt-cube 81.8%

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

      2. unpow3 81.9%

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

      3. fma-udef 81.9%

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

      4. +-commutative 81.9%

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

      5. fma-udef 81.9%

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

      6. sqrt-pow2 82.0%

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

      7. fma-udef 82.0%

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

      8. +-commutative 82.0%

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

      9. associate-*r* 82.0%

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

      10. fma-def 82.0%

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

      11. pow2 82.0%

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

      12. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. metadata-eval 82.0%

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

      3. *-commutative 82.0%

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

      4. times-frac 82.0%

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

      5. metadata-eval 82.0%

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

      6. div-inv 82.0%

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

      7. metadata-eval 82.0%

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

   7. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(\sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}} - b\right) \cdot 0.3333333333333333\right)} \]
   8. Step-by-step derivation

      1. flip-- 82.4%

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

      2. div-inv 82.3%

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

      3. cbrt-unprod 83.4%

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

      4. pow-sqr 83.2%

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

      5. metadata-eval 83.2%

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

      6. unpow2 83.2%

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

      7. +-commutative 83.2%

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

   9. Applied egg-rr 83.2%

      \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{3}} - {b}^{2}\right) \cdot \frac{1}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}\right)} \cdot 0.3333333333333333\right) \]
   10. Step-by-step derivation

       1. rem-cbrt-cube 84.8%

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

   11. Simplified 84.8%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 92.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}\right) \cdot \frac{1}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}\right) \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 3: 89.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \frac{t_0 - {b}^{2}}{b + \sqrt[3]{{t_0}^{1.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (*
      (/ 1.0 a)
      (*
       0.3333333333333333
       (/ (- t_0 (pow b 2.0)) (+ b (cbrt (pow t_0 1.5))))))
     (+
      (* -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) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (1.0 / a) * (0.3333333333333333 * ((t_0 - pow(b, 2.0)) / (b + cbrt(pow(t_0, 1.5)))));
	} 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(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(1.0 / a) * Float64(0.3333333333333333 * Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + cbrt((t_0 ^ 1.5))))));
	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[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(1.0 / a), $MachinePrecision] * N[(0.3333333333333333 * N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Power[N[Power[t$95$0, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $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)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.2%

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

      1. add-cbrt-cube 81.8%

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

      2. unpow3 81.9%

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

      3. fma-udef 81.9%

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

      4. +-commutative 81.9%

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

      5. fma-udef 81.9%

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

      6. sqrt-pow2 82.0%

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

      7. fma-udef 82.0%

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

      8. +-commutative 82.0%

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

      9. associate-*r* 82.0%

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

      10. fma-def 82.0%

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

      11. pow2 82.0%

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

      12. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. metadata-eval 82.0%

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

      3. *-commutative 82.0%

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

      4. times-frac 82.0%

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

      5. metadata-eval 82.0%

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

      6. div-inv 82.0%

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

      7. metadata-eval 82.0%

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

   7. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(\sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}} - b\right) \cdot 0.3333333333333333\right)} \]
   8. Step-by-step derivation

      1. flip-- 82.4%

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

      2. cbrt-unprod 83.4%

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

      3. pow-sqr 83.1%

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

      4. metadata-eval 83.1%

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

      5. unpow2 83.1%

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

      6. +-commutative 83.1%

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

   9. Applied egg-rr 83.1%

      \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\frac{\sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{3}} - {b}^{2}}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}} \cdot 0.3333333333333333\right) \]
   10. Step-by-step derivation

       1. rem-cbrt-cube 84.8%

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

   11. Simplified 84.8%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 92.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \frac{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 4: 89.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt[3]{{t_0}^{1.5}}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (cbrt (pow t_0 1.5)))) (* a 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 t_0 = fma((a * c), -3.0, pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + cbrt(pow(t_0, 1.5)))) / (a * 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)
	t_0 = fma(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + cbrt((t_0 ^ 1.5)))) / Float64(a * 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_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Power[N[Power[t$95$0, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $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)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.2%

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

      1. add-cbrt-cube 81.8%

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

      2. unpow3 81.9%

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

      3. fma-udef 81.9%

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

      4. +-commutative 81.9%

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

      5. fma-udef 81.9%

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

      6. sqrt-pow2 82.0%

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

      7. fma-udef 82.0%

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

      8. +-commutative 82.0%

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

      9. associate-*r* 82.0%

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

      10. fma-def 82.0%

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

      11. pow2 82.0%

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

      12. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. flip-- 82.4%

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

      2. cbrt-unprod 83.4%

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

      3. pow-sqr 83.1%

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

      4. metadata-eval 83.1%

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

      5. unpow2 83.1%

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

      6. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

      1. rem-cbrt-cube 84.8%

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

   9. Simplified 84.8%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 92.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{\left(t_0 - {b}^{2}\right) \cdot \frac{1}{b + \sqrt[3]{{t_0}^{1.5}}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= b 0.009)
     (/ (* (- t_0 (pow b 2.0)) (/ 1.0 (+ b (cbrt (pow t_0 1.5))))) (* a 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)))
        (/ (/ (* -1.0546875 (pow (* a c) 4.0)) (pow b 7.0)) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -3.0, pow(b, 2.0));
	double tmp;
	if (b <= 0.009) {
		tmp = ((t_0 - pow(b, 2.0)) * (1.0 / (b + cbrt(pow(t_0, 1.5))))) / (a * 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))) + (((-1.0546875 * pow((a * c), 4.0)) / pow(b, 7.0)) / a)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.009)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) * Float64(1.0 / Float64(b + cbrt((t_0 ^ 1.5))))) / Float64(a * 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(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(Float64(Float64(-1.0546875 * (Float64(a * c) ^ 4.0)) / (b ^ 7.0)) / a))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.009], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b + N[Power[N[Power[t$95$0, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $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[(N[(N[(-1.0546875 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00899999999999999932

   1. Initial program 91.7%

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

   3. Simplified 91.8%

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

      1. add-cbrt-cube 89.5%

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

      2. unpow3 89.5%

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

      3. fma-udef 89.3%

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

      4. +-commutative 89.3%

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

      5. fma-udef 89.2%

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

      6. sqrt-pow2 90.2%

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

      7. fma-udef 90.0%

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

      8. +-commutative 90.0%

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

      9. associate-*r* 90.0%

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

      10. fma-def 90.0%

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

      11. pow2 90.0%

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

      12. metadata-eval 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. flip-- 90.2%

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

      2. div-inv 90.1%

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

      3. cbrt-unprod 91.8%

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

      4. pow-sqr 90.9%

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

      5. metadata-eval 90.9%

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

      6. unpow2 90.9%

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

      7. +-commutative 90.9%

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

   7. Applied egg-rr 90.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{3}} - {b}^{2}\right) \cdot \frac{1}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 92.7%

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

   9. Simplified 92.9%

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

   if 0.00899999999999999932 < b

   1. Initial program 53.7%

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

   2. Taylor expanded in b around inf 93.9%

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

      1. div-inv 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\left(5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)}\right)\right) \]

      2. fma-def 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\color{blue}{\mathsf{fma}\left(5.0625, {a}^{4} \cdot {c}^{4}, {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right)} \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      3. pow-prod-down 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, \color{blue}{{\left(a \cdot c\right)}^{4}}, {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      4. unpow2 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      5. *-commutative 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -1.125\right)}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      6. *-commutative 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -1.125\right)} \cdot \left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      7. swap-sqr 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-1.125 \cdot -1.125\right)}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      8. pow-prod-down 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      9. pow-prod-down 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      10. pow-sqr 93.9%

          \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      11. metadata-eval 93.9%

          \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot \left(-1.125 \cdot -1.125\right)\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

      12. metadata-eval 93.9%

          \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot \color{blue}{1.265625}\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)\right)\right) \]

   4. Applied egg-rr 93.9%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right) \cdot \frac{1}{a \cdot {b}^{7}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right) \cdot 1}{a \cdot {b}^{7}}}\right)\right) \]

      2. associate-*l/ 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}{a \cdot {b}^{7}} \cdot 1\right)}\right)\right) \]

      3. *-rgt-identity 93.9%

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

      4. fma-udef 93.9%

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

      5. *-commutative 93.9%

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

      6. distribute-rgt-out 93.9%

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

      7. metadata-eval 93.9%

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

   6. Simplified 93.9%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-/r* 93.9%

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

      4. *-commutative 93.9%

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

      5. associate-*r* 93.9%

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

      6. metadata-eval 93.9%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{\color{blue}{-1.0546875} \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}\right)\right) \]

   8. Applied egg-rr 93.9%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}\right) \cdot \frac{1}{b + \sqrt[3]{{\left(\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\right)}^{1.5}}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a}\right)\right)\\ \end{array} \]

Alternative 6: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot 3}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\left(b + t_0\right) \cdot \left(b - t_0\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (/ (- (sqrt (* (+ b t_0) (- b t_0))) b) (* a 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 t_0 = sqrt(((a * c) * 3.0));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 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;
}

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(((a * c) * 3.0d0))
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.25d0)) then
        tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * 3.0));
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (Math.sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * 3.0))
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25:
		tmp = (math.sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + t_0) * Float64(b - t_0))) - b) / Float64(a * 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

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((a * c) * 3.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25)
		tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[Sqrt[N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -0.25

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. difference-of-squares 83.2%

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

      3. associate-*l* 83.3%

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

      4. associate-*l* 83.3%

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

   3. Applied egg-rr 83.3%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 92.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\left(b + \sqrt{\left(a \cdot c\right) \cdot 3}\right) \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot 3}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 7: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot 3}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\left(b + t_0\right) \cdot \left(b - t_0\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(\frac{1}{b} \cdot {\left(c \cdot \frac{a}{b}\right)}^{2}\right)\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (/ (- (sqrt (* (+ b t_0) (- b t_0))) b) (* a 3.0))
     (/
      (+
       (* -1.6875 (/ (* (pow c 3.0) (pow a 3.0)) (pow b 5.0)))
       (+
        (* -1.5 (/ (* a c) b))
        (* -1.125 (* (/ 1.0 b) (pow (* c (/ a b)) 2.0)))))
      (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * 3.0));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	} else {
		tmp = ((-1.6875 * ((pow(c, 3.0) * pow(a, 3.0)) / pow(b, 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * pow((c * (a / b)), 2.0))))) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((a * c) * 3.0d0))
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.25d0)) then
        tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0d0)
    else
        tmp = (((-1.6875d0) * (((c ** 3.0d0) * (a ** 3.0d0)) / (b ** 5.0d0))) + (((-1.5d0) * ((a * c) / b)) + ((-1.125d0) * ((1.0d0 / b) * ((c * (a / b)) ** 2.0d0))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * 3.0));
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (Math.sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	} else {
		tmp = ((-1.6875 * ((Math.pow(c, 3.0) * Math.pow(a, 3.0)) / Math.pow(b, 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * Math.pow((c * (a / b)), 2.0))))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * 3.0))
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25:
		tmp = (math.sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0)
	else:
		tmp = ((-1.6875 * ((math.pow(c, 3.0) * math.pow(a, 3.0)) / math.pow(b, 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * math.pow((c * (a / b)), 2.0))))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + t_0) * Float64(b - t_0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-1.6875 * Float64(Float64((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-1.5 * Float64(Float64(a * c) / b)) + Float64(-1.125 * Float64(Float64(1.0 / b) * (Float64(c * Float64(a / b)) ^ 2.0))))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((a * c) * 3.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25)
		tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	else
		tmp = ((-1.6875 * (((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * ((c * (a / b)) ^ 2.0))))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[Sqrt[N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.6875 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[(1.0 / b), $MachinePrecision] * N[Power[N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $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)) < -0.25

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. difference-of-squares 83.2%

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

      3. associate-*l* 83.3%

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

      4. associate-*l* 83.3%

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

   3. Applied egg-rr 83.3%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 92.4%

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

      1. *-un-lft-identity 92.4%

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

      2. cube-mult 92.4%

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

      3. unpow2 92.4%

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

      4. times-frac 92.4%

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

      5. unpow2 92.4%

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

      6. unpow2 92.4%

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

      7. swap-sqr 92.4%

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

      8. unpow2 92.4%

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

      9. frac-times 92.4%

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

      10. pow1 92.4%

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

      11. pow1 92.4%

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

      12. pow-sqr 92.4%

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

      13. div-inv 92.4%

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

      14. *-commutative 92.4%

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

      15. associate-*l* 92.4%

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

      16. div-inv 92.4%

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

      17. metadata-eval 92.4%

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

   4. Applied egg-rr 92.4%

      \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left(\frac{1}{b} \cdot {\left(c \cdot \frac{a}{b}\right)}^{2}\right)}\right)}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\left(b + \sqrt{\left(a \cdot c\right) \cdot 3}\right) \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot 3}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(\frac{1}{b} \cdot {\left(c \cdot \frac{a}{b}\right)}^{2}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 8: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot 3}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\left(b + t_0\right) \cdot \left(b - t_0\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (/ (- (sqrt (* (+ b t_0) (- b t_0))) b) (* a 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 t_0 = sqrt(((a * c) * 3.0));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((a * c) * 3.0d0))
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.25d0)) then
        tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * 3.0));
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (Math.sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * 3.0))
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25:
		tmp = (math.sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + t_0) * Float64(b - t_0))) - b) / Float64(a * 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

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((a * c) * 3.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25)
		tmp = (sqrt(((b + t_0) * (b - t_0))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[Sqrt[N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -0.25

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. difference-of-squares 83.2%

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

      3. associate-*l* 83.3%

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

      4. associate-*l* 83.3%

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

   3. Applied egg-rr 83.3%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 87.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\left(b + \sqrt{\left(a \cdot c\right) \cdot 3}\right) \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot 3}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 9: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 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) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 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(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 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[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.3%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 87.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 10: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -7.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -7.3e-5)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -7.3e-5) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (a * 3.0);
	} 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(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -7.3e-5)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(a * 3.0));
	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[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -7.3e-5], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -7.2999999999999999e-5

   1. Initial program 74.0%

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

   3. Simplified 74.1%

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

   if -7.2999999999999999e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 38.4%

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

   2. Taylor expanded in b around inf 79.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -7.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -7.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -7.3e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -7.3e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} 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(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -7.3e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	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[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -7.3e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -7.2999999999999999e-5

   1. Initial program 74.0%

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

   3. Simplified 74.1%

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

   if -7.2999999999999999e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 38.4%

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

   2. Taylor expanded in b around inf 79.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -7.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -7.3 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -7.3e-5) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -7.3e-5) {
		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) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-7.3d-5)) 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) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -7.3e-5) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -7.3e-5:
		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(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -7.3e-5)
		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) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -7.3e-5)
		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[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -7.3e-5], 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)) < -7.2999999999999999e-5

   1. Initial program 74.0%

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

   if -7.2999999999999999e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 38.4%

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

   2. Taylor expanded in b around inf 79.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -7.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13: 64.6% 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 56.0%

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

2. Taylor expanded in b around inf 64.1%

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

3. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2023305 
(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)))