Cubic critical, narrow range

Percentage Accurate: 55.3% → 91.9%

Time: 17.1s

Alternatives: 15

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{a}{0.3333333333333333}}\\ t_1 := 0 \cdot \frac{{c}^{3}}{{b}^{4}}\\ t_2 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ t_3 := \sqrt[3]{{t_2}^{1.5} - {b}^{3}}\\ t_4 := \frac{{c}^{3}}{{b}^{3}}\\ t_5 := \frac{c \cdot c}{b \cdot b} \cdot 0\\ t_6 := \frac{{c}^{4}}{{b}^{6}}\\ t_7 := t_2 + b \cdot \left(b + \sqrt{t_2}\right)\\ t_8 := \mathsf{fma}\left(5.0625, t_6, {\left(-1.125 \cdot \frac{c \cdot c}{{b}^{3}}\right)}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{t_3 \cdot \left(t_3 \cdot t_3\right)}{t_7} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-1.6875, t_4, \mathsf{fma}\left(-1.5, \frac{t_5}{\frac{b}{c}}, \mathsf{fma}\left(t_1, b, t_4 \cdot 3.375\right)\right)\right), \mathsf{fma}\left(a, \left(b \cdot c\right) \cdot -4.5, \mathsf{fma}\left(\mathsf{fma}\left(t_5, b, 3.375 \cdot \frac{c \cdot c}{b}\right), a \cdot a, {a}^{4} \cdot \mathsf{fma}\left(-1.125, \frac{t_5}{\frac{{b}^{3}}{c \cdot c}}, \mathsf{fma}\left(\mathsf{fma}\left(-1, t_8, t_6 \cdot 6.328125\right), b, \mathsf{fma}\left(-0.5, b \cdot t_8, \mathsf{fma}\left(-1.5, \frac{t_1}{\frac{b}{c}}, \frac{5.0625 \cdot {c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)}{t_7}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ a 0.3333333333333333)))
        (t_1 (* 0.0 (/ (pow c 3.0) (pow b 4.0))))
        (t_2 (fma b b (* c (* a -3.0))))
        (t_3 (cbrt (- (pow t_2 1.5) (pow b 3.0))))
        (t_4 (/ (pow c 3.0) (pow b 3.0)))
        (t_5 (* (/ (* c c) (* b b)) 0.0))
        (t_6 (/ (pow c 4.0) (pow b 6.0)))
        (t_7 (+ t_2 (* b (+ b (sqrt t_2)))))
        (t_8 (fma 5.0625 t_6 (pow (* -1.125 (/ (* c c) (pow b 3.0))) 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5)
     (* (/ (* t_3 (* t_3 t_3)) t_7) t_0)
     (*
      t_0
      (/
       (fma
        (pow a 3.0)
        (fma -1.6875 t_4 (fma -1.5 (/ t_5 (/ b c)) (fma t_1 b (* t_4 3.375))))
        (fma
         a
         (* (* b c) -4.5)
         (fma
          (fma t_5 b (* 3.375 (/ (* c c) b)))
          (* a a)
          (*
           (pow a 4.0)
           (fma
            -1.125
            (/ t_5 (/ (pow b 3.0) (* c c)))
            (fma
             (fma -1.0 t_8 (* t_6 6.328125))
             b
             (fma
              -0.5
              (* b t_8)
              (fma
               -1.5
               (/ t_1 (/ b c))
               (/ (* 5.0625 (pow c 4.0)) (pow b 5.0))))))))))
       t_7)))))

C

double code(double a, double b, double c) {
	double t_0 = 1.0 / (a / 0.3333333333333333);
	double t_1 = 0.0 * (pow(c, 3.0) / pow(b, 4.0));
	double t_2 = fma(b, b, (c * (a * -3.0)));
	double t_3 = cbrt((pow(t_2, 1.5) - pow(b, 3.0)));
	double t_4 = pow(c, 3.0) / pow(b, 3.0);
	double t_5 = ((c * c) / (b * b)) * 0.0;
	double t_6 = pow(c, 4.0) / pow(b, 6.0);
	double t_7 = t_2 + (b * (b + sqrt(t_2)));
	double t_8 = fma(5.0625, t_6, pow((-1.125 * ((c * c) / pow(b, 3.0))), 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.5) {
		tmp = ((t_3 * (t_3 * t_3)) / t_7) * t_0;
	} else {
		tmp = t_0 * (fma(pow(a, 3.0), fma(-1.6875, t_4, fma(-1.5, (t_5 / (b / c)), fma(t_1, b, (t_4 * 3.375)))), fma(a, ((b * c) * -4.5), fma(fma(t_5, b, (3.375 * ((c * c) / b))), (a * a), (pow(a, 4.0) * fma(-1.125, (t_5 / (pow(b, 3.0) / (c * c))), fma(fma(-1.0, t_8, (t_6 * 6.328125)), b, fma(-0.5, (b * t_8), fma(-1.5, (t_1 / (b / c)), ((5.0625 * pow(c, 4.0)) / pow(b, 5.0)))))))))) / t_7);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(1.0 / Float64(a / 0.3333333333333333))
	t_1 = Float64(0.0 * Float64((c ^ 3.0) / (b ^ 4.0)))
	t_2 = fma(b, b, Float64(c * Float64(a * -3.0)))
	t_3 = cbrt(Float64((t_2 ^ 1.5) - (b ^ 3.0)))
	t_4 = Float64((c ^ 3.0) / (b ^ 3.0))
	t_5 = Float64(Float64(Float64(c * c) / Float64(b * b)) * 0.0)
	t_6 = Float64((c ^ 4.0) / (b ^ 6.0))
	t_7 = Float64(t_2 + Float64(b * Float64(b + sqrt(t_2))))
	t_8 = fma(5.0625, t_6, (Float64(-1.125 * Float64(Float64(c * c) / (b ^ 3.0))) ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(t_3 * Float64(t_3 * t_3)) / t_7) * t_0);
	else
		tmp = Float64(t_0 * Float64(fma((a ^ 3.0), fma(-1.6875, t_4, fma(-1.5, Float64(t_5 / Float64(b / c)), fma(t_1, b, Float64(t_4 * 3.375)))), fma(a, Float64(Float64(b * c) * -4.5), fma(fma(t_5, b, Float64(3.375 * Float64(Float64(c * c) / b))), Float64(a * a), Float64((a ^ 4.0) * fma(-1.125, Float64(t_5 / Float64((b ^ 3.0) / Float64(c * c))), fma(fma(-1.0, t_8, Float64(t_6 * 6.328125)), b, fma(-0.5, Float64(b * t_8), fma(-1.5, Float64(t_1 / Float64(b / c)), Float64(Float64(5.0625 * (c ^ 4.0)) / (b ^ 5.0)))))))))) / t_7));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Power[t$95$2, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 + N[(b * N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(5.0625 * t$95$6 + N[Power[N[(-1.125 * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[(t$95$3 * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(-1.6875 * t$95$4 + N[(-1.5 * N[(t$95$5 / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * b + N[(t$95$4 * 3.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(b * c), $MachinePrecision] * -4.5), $MachinePrecision] + N[(N[(t$95$5 * b + N[(3.375 * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] * N[(-1.125 * N[(t$95$5 / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * t$95$8 + N[(t$95$6 * 6.328125), $MachinePrecision]), $MachinePrecision] * b + N[(-0.5 * N[(b * t$95$8), $MachinePrecision] + N[(-1.5 * N[(t$95$1 / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(N[(5.0625 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$7), $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)) < -1.5

   1. Initial program 87.4%

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

      1. neg-sub0 87.4%

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

      2. associate-+l- 87.4%

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

      3. sub0-neg 87.4%

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

      4. neg-mul-1 87.4%

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

      5. associate-*r/ 87.4%

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

      6. *-commutative 87.4%

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

      7. metadata-eval 87.4%

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

      8. metadata-eval 87.4%

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

      9. times-frac 87.4%

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

      10. *-commutative 87.4%

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

      11. times-frac 87.4%

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

   3. Simplified 87.5%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   7. Simplified 87.4%

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

      1. flip3-- 87.9%

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

      2. associate-*r* 88.0%

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

      3. *-commutative 88.0%

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

      4. add-sqr-sqrt 88.0%

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

      5. associate-*r* 88.1%

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

      6. *-commutative 88.1%

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

      7. associate-*r* 88.0%

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

      8. *-commutative 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. associate-*l* 88.0%

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

       3. distribute-rgt-out 88.1%

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

       4. associate-*l* 88.1%

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

   11. Simplified 88.1%

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

       1. add-cube-cbrt 88.2%

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

       2. sqrt-pow2 88.4%

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

       3. metadata-eval 88.4%

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

       4. sqrt-pow2 89.1%

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

       5. metadata-eval 89.1%

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

       6. sqrt-pow2 89.4%

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

       7. metadata-eval 89.4%

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

   13. Applied egg-rr 89.4%

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

   if -1.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*r/ 51.4%

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

      6. *-commutative 51.4%

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

      7. metadata-eval 51.4%

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

      8. metadata-eval 51.4%

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

      9. times-frac 51.4%

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

      10. *-commutative 51.4%

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

      11. times-frac 51.4%

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

   3. Simplified 51.5%

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

      1. clear-num 51.5%

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

      2. inv-pow 51.5%

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

   5. Applied egg-rr 51.5%

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

      1. unpow-1 51.5%

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

   7. Simplified 51.5%

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

      1. flip3-- 51.5%

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

      2. associate-*r* 51.5%

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

      3. *-commutative 51.5%

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

      4. add-sqr-sqrt 51.5%

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

      5. associate-*r* 51.5%

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

      6. *-commutative 51.5%

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

      7. associate-*r* 51.5%

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

      8. *-commutative 51.5%

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

   9. Applied egg-rr 51.5%

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

       1. associate-*l* 51.5%

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

       2. associate-*l* 51.5%

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

       3. distribute-rgt-out 51.5%

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

       4. associate-*l* 51.5%

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

   11. Simplified 51.5%

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

   12. Taylor expanded in a around 0 94.7%

       \[\leadsto \frac{\color{blue}{{a}^{3} \cdot \left(-1.6875 \cdot \frac{{c}^{3}}{{b}^{3}} + \left(-1.5 \cdot \frac{\left(-2.25 \cdot \frac{{c}^{2}}{{b}^{2}} + 2.25 \cdot \frac{{c}^{2}}{{b}^{2}}\right) \cdot c}{b} + \left(\left(-3.375 \cdot \frac{{c}^{3}}{{b}^{4}} + 3.375 \cdot \frac{{c}^{3}}{{b}^{4}}\right) \cdot b + 3.375 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)\right) + \left(a \cdot \left(-3 \cdot \left(c \cdot b\right) + -1.5 \cdot \left(c \cdot b\right)\right) + \left(\left(\left(-2.25 \cdot \frac{{c}^{2}}{{b}^{2}} + 2.25 \cdot \frac{{c}^{2}}{{b}^{2}}\right) \cdot b + \left(-1.125 \cdot \frac{{c}^{2}}{b} + 4.5 \cdot \frac{{c}^{2}}{b}\right)\right) \cdot {a}^{2} + {a}^{4} \cdot \left(-1.125 \cdot \frac{\left(-2.25 \cdot \frac{{c}^{2}}{{b}^{2}} + 2.25 \cdot \frac{{c}^{2}}{{b}^{2}}\right) \cdot {c}^{2}}{{b}^{3}} + \left(\left(-1 \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) + \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right) \cdot b + \left(-0.5 \cdot \left(b \cdot \left({\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right) + \left(-1.5 \cdot \frac{\left(-3.375 \cdot \frac{{c}^{3}}{{b}^{4}} + 3.375 \cdot \frac{{c}^{3}}{{b}^{4}}\right) \cdot c}{b} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]

   14. Simplified 94.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{3}}, \mathsf{fma}\left(-1.5, \frac{\frac{c \cdot c}{b \cdot b} \cdot 0}{\frac{b}{c}}, \mathsf{fma}\left(\frac{{c}^{3}}{{b}^{4}} \cdot 0, b, 3.375 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)\right), \mathsf{fma}\left(a, \left(c \cdot b\right) \cdot -4.5, \mathsf{fma}\left(\mathsf{fma}\left(\frac{c \cdot c}{b \cdot b} \cdot 0, b, \frac{c \cdot c}{b} \cdot 3.375\right), a \cdot a, {a}^{4} \cdot \mathsf{fma}\left(-1.125, \frac{\frac{c \cdot c}{b \cdot b} \cdot 0}{\frac{{b}^{3}}{c \cdot c}}, \mathsf{fma}\left(\mathsf{fma}\left(-1, \mathsf{fma}\left(5.0625, \frac{{c}^{4}}{{b}^{6}}, {\left(-1.125 \cdot \frac{c \cdot c}{{b}^{3}}\right)}^{2}\right), \frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), b, \mathsf{fma}\left(-0.5, b \cdot \mathsf{fma}\left(5.0625, \frac{{c}^{4}}{{b}^{6}}, {\left(-1.125 \cdot \frac{c \cdot c}{{b}^{3}}\right)}^{2}\right), \mathsf{fma}\left(-1.5, \frac{\frac{{c}^{3}}{{b}^{4}} \cdot 0}{\frac{b}{c}}, \frac{{c}^{4} \cdot 5.0625}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}} \cdot \left(\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}}\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{\mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{3}}, \mathsf{fma}\left(-1.5, \frac{\frac{c \cdot c}{b \cdot b} \cdot 0}{\frac{b}{c}}, \mathsf{fma}\left(0 \cdot \frac{{c}^{3}}{{b}^{4}}, b, \frac{{c}^{3}}{{b}^{3}} \cdot 3.375\right)\right)\right), \mathsf{fma}\left(a, \left(b \cdot c\right) \cdot -4.5, \mathsf{fma}\left(\mathsf{fma}\left(\frac{c \cdot c}{b \cdot b} \cdot 0, b, 3.375 \cdot \frac{c \cdot c}{b}\right), a \cdot a, {a}^{4} \cdot \mathsf{fma}\left(-1.125, \frac{\frac{c \cdot c}{b \cdot b} \cdot 0}{\frac{{b}^{3}}{c \cdot c}}, \mathsf{fma}\left(\mathsf{fma}\left(-1, \mathsf{fma}\left(5.0625, \frac{{c}^{4}}{{b}^{6}}, {\left(-1.125 \cdot \frac{c \cdot c}{{b}^{3}}\right)}^{2}\right), \frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), b, \mathsf{fma}\left(-0.5, b \cdot \mathsf{fma}\left(5.0625, \frac{{c}^{4}}{{b}^{6}}, {\left(-1.125 \cdot \frac{c \cdot c}{{b}^{3}}\right)}^{2}\right), \mathsf{fma}\left(-1.5, \frac{0 \cdot \frac{{c}^{3}}{{b}^{4}}}{\frac{b}{c}}, \frac{5.0625 \cdot {c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}\\ \end{array} \]

Alternative 2: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{a}{0.3333333333333333}}\\ t_1 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ t_2 := \sqrt[3]{{t_1}^{1.5} - {b}^{3}}\\ t_3 := \frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\\ t_4 := \frac{{c}^{3}}{{b}^{3}}\\ t_5 := t_1 + b \cdot \left(b + \sqrt{t_1}\right)\\ t_6 := \frac{c}{\frac{b}{0}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{t_2 \cdot \left(t_2 \cdot t_2\right)}{t_5} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-1.6875, t_4, \mathsf{fma}\left(-1.5, t_6, \mathsf{fma}\left(0, b, t_4 \cdot 3.375\right)\right)\right), \mathsf{fma}\left(a, b \cdot \left(c \cdot -4.5\right), \mathsf{fma}\left(\mathsf{fma}\left(0, b, 3.375 \cdot \frac{c \cdot c}{b}\right), a \cdot a, {a}^{4} \cdot \mathsf{fma}\left(-1.125, \frac{\left(c \cdot c\right) \cdot 0}{{b}^{3}}, \mathsf{fma}\left(t_3 - t_3, b, \mathsf{fma}\left(-0.5, b \cdot t_3, \mathsf{fma}\left(-1.5, t_6, 5.0625 \cdot \frac{{c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)}{t_5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ a 0.3333333333333333)))
        (t_1 (fma b b (* c (* a -3.0))))
        (t_2 (cbrt (- (pow t_1 1.5) (pow b 3.0))))
        (t_3 (* (/ (pow c 4.0) (pow b 6.0)) 6.328125))
        (t_4 (/ (pow c 3.0) (pow b 3.0)))
        (t_5 (+ t_1 (* b (+ b (sqrt t_1)))))
        (t_6 (/ c (/ b 0.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5)
     (* (/ (* t_2 (* t_2 t_2)) t_5) t_0)
     (*
      t_0
      (/
       (fma
        (pow a 3.0)
        (fma -1.6875 t_4 (fma -1.5 t_6 (fma 0.0 b (* t_4 3.375))))
        (fma
         a
         (* b (* c -4.5))
         (fma
          (fma 0.0 b (* 3.375 (/ (* c c) b)))
          (* a a)
          (*
           (pow a 4.0)
           (fma
            -1.125
            (/ (* (* c c) 0.0) (pow b 3.0))
            (fma
             (- t_3 t_3)
             b
             (fma
              -0.5
              (* b t_3)
              (fma -1.5 t_6 (* 5.0625 (/ (pow c 4.0) (pow b 5.0)))))))))))
       t_5)))))

C

double code(double a, double b, double c) {
	double t_0 = 1.0 / (a / 0.3333333333333333);
	double t_1 = fma(b, b, (c * (a * -3.0)));
	double t_2 = cbrt((pow(t_1, 1.5) - pow(b, 3.0)));
	double t_3 = (pow(c, 4.0) / pow(b, 6.0)) * 6.328125;
	double t_4 = pow(c, 3.0) / pow(b, 3.0);
	double t_5 = t_1 + (b * (b + sqrt(t_1)));
	double t_6 = c / (b / 0.0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.5) {
		tmp = ((t_2 * (t_2 * t_2)) / t_5) * t_0;
	} else {
		tmp = t_0 * (fma(pow(a, 3.0), fma(-1.6875, t_4, fma(-1.5, t_6, fma(0.0, b, (t_4 * 3.375)))), fma(a, (b * (c * -4.5)), fma(fma(0.0, b, (3.375 * ((c * c) / b))), (a * a), (pow(a, 4.0) * fma(-1.125, (((c * c) * 0.0) / pow(b, 3.0)), fma((t_3 - t_3), b, fma(-0.5, (b * t_3), fma(-1.5, t_6, (5.0625 * (pow(c, 4.0) / pow(b, 5.0))))))))))) / t_5);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(1.0 / Float64(a / 0.3333333333333333))
	t_1 = fma(b, b, Float64(c * Float64(a * -3.0)))
	t_2 = cbrt(Float64((t_1 ^ 1.5) - (b ^ 3.0)))
	t_3 = Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125)
	t_4 = Float64((c ^ 3.0) / (b ^ 3.0))
	t_5 = Float64(t_1 + Float64(b * Float64(b + sqrt(t_1))))
	t_6 = Float64(c / Float64(b / 0.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(t_2 * Float64(t_2 * t_2)) / t_5) * t_0);
	else
		tmp = Float64(t_0 * Float64(fma((a ^ 3.0), fma(-1.6875, t_4, fma(-1.5, t_6, fma(0.0, b, Float64(t_4 * 3.375)))), fma(a, Float64(b * Float64(c * -4.5)), fma(fma(0.0, b, Float64(3.375 * Float64(Float64(c * c) / b))), Float64(a * a), Float64((a ^ 4.0) * fma(-1.125, Float64(Float64(Float64(c * c) * 0.0) / (b ^ 3.0)), fma(Float64(t_3 - t_3), b, fma(-0.5, Float64(b * t_3), fma(-1.5, t_6, Float64(5.0625 * Float64((c ^ 4.0) / (b ^ 5.0))))))))))) / t_5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Power[t$95$1, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 + N[(b * N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(c / N[(b / 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[(t$95$2 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(-1.6875 * t$95$4 + N[(-1.5 * t$95$6 + N[(0.0 * b + N[(t$95$4 * 3.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(b * N[(c * -4.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0 * b + N[(3.375 * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] * N[(-1.125 * N[(N[(N[(c * c), $MachinePrecision] * 0.0), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - t$95$3), $MachinePrecision] * b + N[(-0.5 * N[(b * t$95$3), $MachinePrecision] + N[(-1.5 * t$95$6 + N[(5.0625 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.5

   1. Initial program 87.4%

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

      1. neg-sub0 87.4%

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

      2. associate-+l- 87.4%

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

      3. sub0-neg 87.4%

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

      4. neg-mul-1 87.4%

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

      5. associate-*r/ 87.4%

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

      6. *-commutative 87.4%

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

      7. metadata-eval 87.4%

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

      8. metadata-eval 87.4%

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

      9. times-frac 87.4%

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

      10. *-commutative 87.4%

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

      11. times-frac 87.4%

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

   3. Simplified 87.5%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   7. Simplified 87.4%

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

      1. flip3-- 87.9%

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

      2. associate-*r* 88.0%

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

      3. *-commutative 88.0%

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

      4. add-sqr-sqrt 88.0%

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

      5. associate-*r* 88.1%

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

      6. *-commutative 88.1%

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

      7. associate-*r* 88.0%

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

      8. *-commutative 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. associate-*l* 88.0%

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

       3. distribute-rgt-out 88.1%

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

       4. associate-*l* 88.1%

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

   11. Simplified 88.1%

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

       1. add-cube-cbrt 88.2%

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

       2. sqrt-pow2 88.4%

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

       3. metadata-eval 88.4%

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

       4. sqrt-pow2 89.1%

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

       5. metadata-eval 89.1%

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

       6. sqrt-pow2 89.4%

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

       7. metadata-eval 89.4%

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

   13. Applied egg-rr 89.4%

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

   if -1.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*r/ 51.4%

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

      6. *-commutative 51.4%

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

      7. metadata-eval 51.4%

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

      8. metadata-eval 51.4%

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

      9. times-frac 51.4%

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

      10. *-commutative 51.4%

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

      11. times-frac 51.4%

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

   3. Simplified 51.5%

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

      1. clear-num 51.5%

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

      2. inv-pow 51.5%

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

   5. Applied egg-rr 51.5%

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

      1. unpow-1 51.5%

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

   7. Simplified 51.5%

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

      1. flip3-- 51.5%

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

      2. associate-*r* 51.5%

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

      3. *-commutative 51.5%

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

      4. add-sqr-sqrt 51.5%

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

      5. associate-*r* 51.5%

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

      6. *-commutative 51.5%

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

      7. associate-*r* 51.5%

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

      8. *-commutative 51.5%

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

   9. Applied egg-rr 51.5%

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

       1. associate-*l* 51.5%

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

       2. associate-*l* 51.5%

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

       3. distribute-rgt-out 51.5%

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

       4. associate-*l* 51.5%

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

   11. Simplified 51.5%

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

   12. Taylor expanded in a around 0 94.7%

       \[\leadsto \frac{\color{blue}{{a}^{3} \cdot \left(-1.6875 \cdot \frac{{c}^{3}}{{b}^{3}} + \left(-1.5 \cdot \frac{\left(-2.25 \cdot \frac{{c}^{2}}{{b}^{2}} + 2.25 \cdot \frac{{c}^{2}}{{b}^{2}}\right) \cdot c}{b} + \left(\left(-3.375 \cdot \frac{{c}^{3}}{{b}^{4}} + 3.375 \cdot \frac{{c}^{3}}{{b}^{4}}\right) \cdot b + 3.375 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)\right) + \left(a \cdot \left(-3 \cdot \left(c \cdot b\right) + -1.5 \cdot \left(c \cdot b\right)\right) + \left(\left(\left(-2.25 \cdot \frac{{c}^{2}}{{b}^{2}} + 2.25 \cdot \frac{{c}^{2}}{{b}^{2}}\right) \cdot b + \left(-1.125 \cdot \frac{{c}^{2}}{b} + 4.5 \cdot \frac{{c}^{2}}{b}\right)\right) \cdot {a}^{2} + {a}^{4} \cdot \left(-1.125 \cdot \frac{\left(-2.25 \cdot \frac{{c}^{2}}{{b}^{2}} + 2.25 \cdot \frac{{c}^{2}}{{b}^{2}}\right) \cdot {c}^{2}}{{b}^{3}} + \left(\left(-1 \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) + \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right) \cdot b + \left(-0.5 \cdot \left(b \cdot \left({\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right) + \left(-1.5 \cdot \frac{\left(-3.375 \cdot \frac{{c}^{3}}{{b}^{4}} + 3.375 \cdot \frac{{c}^{3}}{{b}^{4}}\right) \cdot c}{b} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]

   14. Simplified 94.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{3}}, \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(0, b, 3.375 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)\right), \mathsf{fma}\left(a, b \cdot \left(c \cdot -4.5\right), \mathsf{fma}\left(\mathsf{fma}\left(0, b, \frac{c \cdot c}{b} \cdot 3.375\right), a \cdot a, {a}^{4} \cdot \mathsf{fma}\left(-1.125, \frac{\left(c \cdot c\right) \cdot 0}{{b}^{3}}, \mathsf{fma}\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125 - \frac{{c}^{4}}{{b}^{6}} \cdot 6.328125, b, \mathsf{fma}\left(-0.5, b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{0}}, 5.0625 \cdot \frac{{c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}} \cdot \left(\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}}\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{\mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-1.6875, \frac{{c}^{3}}{{b}^{3}}, \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(0, b, \frac{{c}^{3}}{{b}^{3}} \cdot 3.375\right)\right)\right), \mathsf{fma}\left(a, b \cdot \left(c \cdot -4.5\right), \mathsf{fma}\left(\mathsf{fma}\left(0, b, 3.375 \cdot \frac{c \cdot c}{b}\right), a \cdot a, {a}^{4} \cdot \mathsf{fma}\left(-1.125, \frac{\left(c \cdot c\right) \cdot 0}{{b}^{3}}, \mathsf{fma}\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125 - \frac{{c}^{4}}{{b}^{6}} \cdot 6.328125, b, \mathsf{fma}\left(-0.5, b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{0}}, 5.0625 \cdot \frac{{c}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}\\ \end{array} \]

Alternative 3: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ t_1 := \sqrt[3]{{t_0}^{1.5} - {b}^{3}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{t_1 \cdot \left(t_1 \cdot t_1\right)}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -3.0))))
        (t_1 (cbrt (- (pow t_0 1.5) (pow b 3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5)
     (*
      (/ (* t_1 (* t_1 t_1)) (+ t_0 (* b (+ b (sqrt t_0)))))
      (/ 1.0 (/ a 0.3333333333333333)))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma
       -0.5
       (/ c b)
       (fma
        -0.16666666666666666
        (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0)))
        (* -0.375 (* a (/ (* c c) (pow b 3.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -3.0)));
	double t_1 = cbrt((pow(t_0, 1.5) - pow(b, 3.0)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.5) {
		tmp = ((t_1 * (t_1 * t_1)) / (t_0 + (b * (b + sqrt(t_0))))) * (1.0 / (a / 0.3333333333333333));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), fma(-0.16666666666666666, ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))), (-0.375 * (a * ((c * c) / pow(b, 3.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -3.0)))
	t_1 = cbrt(Float64((t_0 ^ 1.5) - (b ^ 3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(t_1 * Float64(t_1 * t_1)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))) * Float64(1.0 / Float64(a / 0.3333333333333333)));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0))), Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.5

   1. Initial program 87.4%

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

      1. neg-sub0 87.4%

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

      2. associate-+l- 87.4%

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

      3. sub0-neg 87.4%

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

      4. neg-mul-1 87.4%

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

      5. associate-*r/ 87.4%

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

      6. *-commutative 87.4%

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

      7. metadata-eval 87.4%

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

      8. metadata-eval 87.4%

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

      9. times-frac 87.4%

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

      10. *-commutative 87.4%

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

      11. times-frac 87.4%

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

   3. Simplified 87.5%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   7. Simplified 87.4%

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

      1. flip3-- 87.9%

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

      2. associate-*r* 88.0%

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

      3. *-commutative 88.0%

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

      4. add-sqr-sqrt 88.0%

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

      5. associate-*r* 88.1%

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

      6. *-commutative 88.1%

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

      7. associate-*r* 88.0%

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

      8. *-commutative 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. associate-*l* 88.0%

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

       3. distribute-rgt-out 88.1%

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

       4. associate-*l* 88.1%

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

   11. Simplified 88.1%

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

       1. add-cube-cbrt 88.2%

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

       2. sqrt-pow2 88.4%

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

       3. metadata-eval 88.4%

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

       4. sqrt-pow2 89.1%

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

       5. metadata-eval 89.1%

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

       6. sqrt-pow2 89.4%

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

       7. metadata-eval 89.4%

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

   13. Applied egg-rr 89.4%

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

   if -1.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.4%

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

      1. /-rgt-identity 51.4%

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

      2. metadata-eval 51.4%

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

      3. associate-/l* 51.4%

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

      4. associate-*r/ 51.4%

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

      5. *-commutative 51.4%

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

      6. associate-*l/ 51.4%

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

      7. associate-*r/ 51.4%

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

      8. metadata-eval 51.4%

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

      9. metadata-eval 51.4%

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

      10. times-frac 51.4%

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

      11. neg-mul-1 51.4%

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

      12. distribute-rgt-neg-in 51.4%

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

      13. times-frac 51.4%

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

      14. metadata-eval 51.4%

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

      15. neg-mul-1 51.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 94.4%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 94.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]

      2. associate-/l* 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      3. unpow2 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. fma-def 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right) \]

      5. associate-/l* 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      6. unpow2 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      7. fma-def 94.5%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right)\right) \]

   6. Simplified 94.5%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]

   7. Taylor expanded in b around 0 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}} \cdot \left(\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}}\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{{t_0}^{1.5} - {b}^{3}}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5)
     (*
      (/ 1.0 (/ a 0.3333333333333333))
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0))))))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma
       -0.5
       (/ c b)
       (fma
        -0.16666666666666666
        (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0)))
        (* -0.375 (* a (/ (* c c) (pow b 3.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -3.0)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.5) {
		tmp = (1.0 / (a / 0.3333333333333333)) * ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0)))));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), fma(-0.16666666666666666, ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))), (-0.375 * (a * ((c * c) / pow(b, 3.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(1.0 / Float64(a / 0.3333333333333333)) * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0))), Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.5

   1. Initial program 87.4%

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

      1. neg-sub0 87.4%

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

      2. associate-+l- 87.4%

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

      3. sub0-neg 87.4%

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

      4. neg-mul-1 87.4%

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

      5. associate-*r/ 87.4%

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

      6. *-commutative 87.4%

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

      7. metadata-eval 87.4%

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

      8. metadata-eval 87.4%

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

      9. times-frac 87.4%

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

      10. *-commutative 87.4%

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

      11. times-frac 87.4%

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

   3. Simplified 87.5%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   7. Simplified 87.4%

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

      1. flip3-- 87.9%

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

      2. associate-*r* 88.0%

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

      3. *-commutative 88.0%

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

      4. add-sqr-sqrt 88.0%

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

      5. associate-*r* 88.1%

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

      6. *-commutative 88.1%

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

      7. associate-*r* 88.0%

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

      8. *-commutative 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. associate-*l* 88.0%

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

       3. distribute-rgt-out 88.1%

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

       4. associate-*l* 88.1%

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

   11. Simplified 88.1%

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

       1. sub-neg 88.1%

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

       2. sqrt-pow2 89.2%

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

       3. metadata-eval 89.2%

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

   13. Applied egg-rr 89.2%

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

       1. sub-neg 89.2%

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

   15. Simplified 89.2%

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

   if -1.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.4%

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

      1. /-rgt-identity 51.4%

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

      2. metadata-eval 51.4%

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

      3. associate-/l* 51.4%

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

      4. associate-*r/ 51.4%

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

      5. *-commutative 51.4%

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

      6. associate-*l/ 51.4%

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

      7. associate-*r/ 51.4%

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

      8. metadata-eval 51.4%

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

      9. metadata-eval 51.4%

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

      10. times-frac 51.4%

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

      11. neg-mul-1 51.4%

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

      12. distribute-rgt-neg-in 51.4%

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

      13. times-frac 51.4%

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

      14. metadata-eval 51.4%

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

      15. neg-mul-1 51.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 94.4%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 94.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]

      2. associate-/l* 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      3. unpow2 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. fma-def 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right) \]

      5. associate-/l* 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      6. unpow2 94.4%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

      7. fma-def 94.5%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right)\right) \]

   6. Simplified 94.5%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]

   7. Taylor expanded in b around 0 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 89.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5)
     (*
      (/ 1.0 (/ a 0.3333333333333333))
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0))))))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -3.0)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.5) {
		tmp = (1.0 / (a / 0.3333333333333333)) * ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0)))));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(1.0 / Float64(a / 0.3333333333333333)) * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $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)) < -1.5

   1. Initial program 87.4%

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

      1. neg-sub0 87.4%

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

      2. associate-+l- 87.4%

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

      3. sub0-neg 87.4%

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

      4. neg-mul-1 87.4%

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

      5. associate-*r/ 87.4%

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

      6. *-commutative 87.4%

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

      7. metadata-eval 87.4%

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

      8. metadata-eval 87.4%

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

      9. times-frac 87.4%

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

      10. *-commutative 87.4%

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

      11. times-frac 87.4%

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

   3. Simplified 87.5%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   7. Simplified 87.4%

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

      1. flip3-- 87.9%

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

      2. associate-*r* 88.0%

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

      3. *-commutative 88.0%

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

      4. add-sqr-sqrt 88.0%

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

      5. associate-*r* 88.1%

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

      6. *-commutative 88.1%

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

      7. associate-*r* 88.0%

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

      8. *-commutative 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. associate-*l* 88.0%

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

       2. associate-*l* 88.0%

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

       3. distribute-rgt-out 88.1%

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

       4. associate-*l* 88.1%

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

   11. Simplified 88.1%

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

       1. sub-neg 88.1%

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

       2. sqrt-pow2 89.2%

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

       3. metadata-eval 89.2%

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

   13. Applied egg-rr 89.2%

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

       1. sub-neg 89.2%

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

   15. Simplified 89.2%

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

   if -1.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*r/ 51.4%

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

      6. metadata-eval 51.4%

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

      7. metadata-eval 51.4%

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

      8. times-frac 51.4%

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

      9. *-commutative 51.4%

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

      10. times-frac 51.4%

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

      11. associate-*l/ 51.4%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 91.5%

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

      1. fma-def 91.5%

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

      2. associate-/l* 91.5%

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

      3. unpow2 91.5%

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

      4. fma-def 91.5%

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

      5. associate-*r/ 91.5%

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

      6. *-commutative 91.5%

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

      7. unpow2 91.5%

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

   6. Simplified 91.5%

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

4. Final simplification 91.3%

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

Alternative 6: 89.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.5)
     (* (/ 1.0 (/ a 0.3333333333333333)) (/ (- t_0 (* b b)) (+ b (sqrt t_0))))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (-3.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.5) {
		tmp = (1.0 / (a / 0.3333333333333333)) * ((t_0 - (b * b)) / (b + sqrt(t_0)));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(-3.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(1.0 / Float64(a / 0.3333333333333333)) * Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $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)) < -1.5

   1. Initial program 87.4%

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

      1. neg-sub0 87.4%

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

      2. associate-+l- 87.4%

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

      3. sub0-neg 87.4%

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

      4. neg-mul-1 87.4%

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

      5. associate-*r/ 87.4%

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

      6. *-commutative 87.4%

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

      7. metadata-eval 87.4%

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

      8. metadata-eval 87.4%

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

      9. times-frac 87.4%

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

      10. *-commutative 87.4%

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

      11. times-frac 87.4%

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

   3. Simplified 87.5%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   7. Simplified 87.4%

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

      1. flip-- 87.2%

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

      2. add-sqr-sqrt 88.4%

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

      3. associate-*r* 88.5%

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

      4. *-commutative 88.5%

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

      5. associate-*r* 88.5%

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

      6. *-commutative 88.5%

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

   9. Applied egg-rr 88.5%

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

   if -1.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*r/ 51.4%

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

      6. metadata-eval 51.4%

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

      7. metadata-eval 51.4%

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

      8. times-frac 51.4%

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

      9. *-commutative 51.4%

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

      10. times-frac 51.4%

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

      11. associate-*l/ 51.4%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 91.5%

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

      1. fma-def 91.5%

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

      2. associate-/l* 91.5%

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

      3. unpow2 91.5%

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

      4. fma-def 91.5%

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

      5. associate-*r/ 91.5%

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

      6. *-commutative 91.5%

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

      7. unpow2 91.5%

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

   6. Simplified 91.5%

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

4. Final simplification 91.2%

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

Alternative 7: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{1}{a}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ 1.0 a))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
     (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (cbrt (* t_0 (* t_0 t_0))))
     (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (1.0 / a);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * cbrt((t_0 * (t_0 * t_0)));
	} else {
		tmp = fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(0.3333333333333333 * Float64(1.0 / a))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * cbrt(Float64(t_0 * Float64(t_0 * t_0))));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. neg-sub0 79.5%

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

      2. associate-+l- 79.5%

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

      3. sub0-neg 79.5%

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

      4. neg-mul-1 79.5%

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

      5. associate-*r/ 79.5%

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

      6. *-commutative 79.5%

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

      7. metadata-eval 79.5%

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

      8. metadata-eval 79.5%

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

      9. times-frac 79.5%

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

      10. *-commutative 79.5%

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

      11. times-frac 79.5%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

      1. add-cbrt-cube 79.6%

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

      2. associate-/r/ 79.6%

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

      3. associate-/r/ 79.6%

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

      4. associate-/r/ 79.6%

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

   9. Applied egg-rr 79.6%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 92.1%

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

      1. fma-def 92.1%

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

      2. associate-*r/ 92.1%

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

      3. *-commutative 92.1%

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

      4. unpow2 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt[3]{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 8: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{t_0 - b \cdot b}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
     (* (/ 1.0 (/ a 0.3333333333333333)) (/ (- t_0 (* b b)) (+ b (sqrt t_0))))
     (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (-3.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (1.0 / (a / 0.3333333333333333)) * ((t_0 - (b * b)) / (b + sqrt(t_0)));
	} else {
		tmp = fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(-3.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(Float64(1.0 / Float64(a / 0.3333333333333333)) * Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. neg-sub0 79.5%

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

      2. associate-+l- 79.5%

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

      3. sub0-neg 79.5%

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

      4. neg-mul-1 79.5%

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

      5. associate-*r/ 79.5%

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

      6. *-commutative 79.5%

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

      7. metadata-eval 79.5%

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

      8. metadata-eval 79.5%

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

      9. times-frac 79.5%

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

      10. *-commutative 79.5%

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

      11. times-frac 79.5%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

      1. flip-- 79.3%

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

      2. add-sqr-sqrt 80.7%

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

      3. associate-*r* 80.8%

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

      4. *-commutative 80.8%

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

      5. associate-*r* 80.8%

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

      6. *-commutative 80.8%

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

   9. Applied egg-rr 80.8%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 92.1%

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

      1. fma-def 92.1%

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

      2. associate-*r/ 92.1%

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

      3. *-commutative 92.1%

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

      4. unpow2 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \left(\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (cbrt
     (*
      (/ 0.3333333333333333 a)
      (* (/ 0.3333333333333333 a) (/ 0.3333333333333333 a)))))
   (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * cbrt(((0.3333333333333333 / a) * ((0.3333333333333333 / a) * (0.3333333333333333 / a))));
	} else {
		tmp = fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * cbrt(Float64(Float64(0.3333333333333333 / a) * Float64(Float64(0.3333333333333333 / a) * Float64(0.3333333333333333 / a)))));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Power[N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. neg-sub0 79.5%

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

      2. associate-+l- 79.5%

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

      3. sub0-neg 79.5%

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

      4. neg-mul-1 79.5%

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

      5. associate-*r/ 79.5%

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

      6. *-commutative 79.5%

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

      7. metadata-eval 79.5%

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

      8. metadata-eval 79.5%

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

      9. times-frac 79.5%

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

      10. *-commutative 79.5%

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

      11. times-frac 79.5%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

      1. add-cbrt-cube 79.6%

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

      2. associate-/r/ 79.6%

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

      3. associate-/r/ 79.6%

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

      4. associate-/r/ 79.6%

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

   9. Applied egg-rr 79.6%

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

       1. associate-*l* 79.6%

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

       2. associate-*l/ 79.6%

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

       3. metadata-eval 79.6%

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

       4. associate-*l/ 79.6%

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

       5. metadata-eval 79.6%

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

       6. associate-*l/ 79.6%

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

       7. metadata-eval 79.6%

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

   11. Simplified 79.6%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 92.1%

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

      1. fma-def 92.1%

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

      2. associate-*r/ 92.1%

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

      3. *-commutative 92.1%

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

      4. unpow2 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \left(\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 10: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. neg-sub0 79.5%

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

      2. associate-+l- 79.5%

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

      3. sub0-neg 79.5%

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

      4. neg-mul-1 79.5%

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

      5. associate-*r/ 79.5%

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

      6. metadata-eval 79.5%

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

      7. metadata-eval 79.5%

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

      8. times-frac 79.5%

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

      9. *-commutative 79.5%

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

      10. times-frac 79.5%

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

      11. associate-*l/ 79.5%

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

   3. Simplified 79.6%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 92.1%

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

      1. fma-def 92.1%

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

      2. associate-*r/ 92.1%

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

      3. *-commutative 92.1%

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

      4. unpow2 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 11: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.125 \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (/
    (+ (/ (* -1.125 (* a (* c (* a c)))) (pow b 3.0)) (* -1.5 (/ (* a c) b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = (((-1.125 * (a * (c * (a * c)))) / pow(b, 3.0)) + (-1.5 * ((a * c) / b))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(Float64(Float64(Float64(-1.125 * Float64(a * Float64(c * Float64(a * c)))) / (b ^ 3.0)) + Float64(-1.5 * Float64(Float64(a * c) / b))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.125 * N[(a * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. /-rgt-identity 79.5%

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

      2. metadata-eval 79.5%

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

      3. associate-/l* 79.5%

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

      4. associate-*r/ 79.5%

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

      5. *-commutative 79.5%

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

      6. associate-*l/ 79.5%

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

      7. associate-*r/ 79.5%

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

      8. metadata-eval 79.5%

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

      9. metadata-eval 79.5%

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

      10. times-frac 79.5%

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

      11. neg-mul-1 79.5%

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

      12. distribute-rgt-neg-in 79.5%

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

      13. times-frac 79.5%

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

      14. metadata-eval 79.5%

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

      15. neg-mul-1 79.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.6%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 91.7%

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

      1. pow-prod-down 91.7%

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

      2. pow2 91.7%

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

      3. associate-*r/ 91.7%

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

      4. pow2 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. pow2 91.7%

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

      2. associate-*r* 91.7%

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

   8. Applied egg-rr 91.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.125 \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \]

Alternative 12: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.125 \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (/
    (+ (/ (* -1.125 (* a (* c (* a c)))) (pow b 3.0)) (* -1.5 (/ (* a c) b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (((-1.125 * (a * (c * (a * c)))) / pow(b, 3.0)) + (-1.5 * ((a * c) / b))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(-1.125 * Float64(a * Float64(c * Float64(a * c)))) / (b ^ 3.0)) + Float64(-1.5 * Float64(Float64(a * c) / b))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.125 * N[(a * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. neg-sub0 79.5%

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

      2. associate-+l- 79.5%

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

      3. sub0-neg 79.5%

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

      4. neg-mul-1 79.5%

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

      5. associate-*r/ 79.5%

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

      6. metadata-eval 79.5%

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

      7. metadata-eval 79.5%

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

      8. times-frac 79.5%

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

      9. *-commutative 79.5%

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

      10. times-frac 79.5%

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

      11. associate-*l/ 79.5%

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

   3. Simplified 79.6%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 91.7%

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

      1. pow-prod-down 91.7%

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

      2. pow2 91.7%

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

      3. associate-*r/ 91.7%

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

      4. pow2 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. pow2 91.7%

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

      2. associate-*r* 91.7%

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

   8. Applied egg-rr 91.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.125 \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \]

Alternative 13: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.125 \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00025)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (/
    (+ (/ (* -1.125 (* a (* c (* a c)))) (pow b 3.0)) (* -1.5 (/ (* a c) b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (((-1.125 * (a * (c * (a * c)))) / pow(b, 3.0)) + (-1.5 * ((a * c) / b))) / (3.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-0.00025d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = ((((-1.125d0) * (a * (c * (a * c)))) / (b ** 3.0d0)) + ((-1.5d0) * ((a * c) / b))) / (3.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (((-1.125 * (a * (c * (a * c)))) / Math.pow(b, 3.0)) + (-1.5 * ((a * c) / b))) / (3.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (((-1.125 * (a * (c * (a * c)))) / math.pow(b, 3.0)) + (-1.5 * ((a * c) / b))) / (3.0 * a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00025)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(-1.125 * Float64(a * Float64(c * Float64(a * c)))) / (b ^ 3.0)) + Float64(-1.5 * Float64(Float64(a * c) / b))) / Float64(3.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00025)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (((-1.125 * (a * (c * (a * c)))) / (b ^ 3.0)) + (-1.5 * ((a * c) / b))) / (3.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.125 * N[(a * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5000000000000001e-4

   1. Initial program 79.5%

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

      1. neg-sub0 79.5%

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

      2. associate-+l- 79.5%

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

      3. sub0-neg 79.5%

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

      4. neg-mul-1 79.5%

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

      5. associate-*r/ 79.5%

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

      6. metadata-eval 79.5%

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

      7. metadata-eval 79.5%

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

      8. times-frac 79.5%

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

      9. *-commutative 79.5%

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

      10. times-frac 79.5%

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

      11. associate-*l/ 79.5%

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

   3. Simplified 79.5%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 40.8%

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

      1. neg-sub0 40.8%

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

      2. associate-+l- 40.8%

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

      3. sub0-neg 40.8%

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

      4. neg-mul-1 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. metadata-eval 40.8%

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

      8. times-frac 40.8%

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

      9. *-commutative 40.8%

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

      10. times-frac 40.8%

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

      11. associate-*l/ 40.8%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in b around inf 91.7%

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

      1. pow-prod-down 91.7%

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

      2. pow2 91.7%

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

      3. associate-*r/ 91.7%

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

      4. pow2 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. pow2 91.7%

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

      2. associate-*r* 91.7%

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

   8. Applied egg-rr 91.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.125 \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \]

Alternative 14: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7e-6)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7e-6) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} 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) :: tmp
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-7d-6)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7e-6) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7e-6:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7e-6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7e-6)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -7e-6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6.99999999999999989e-6

   1. Initial program 76.5%

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

      1. neg-sub0 76.5%

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

      2. associate-+l- 76.5%

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

      3. sub0-neg 76.5%

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

      4. neg-mul-1 76.5%

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

      5. associate-*r/ 76.5%

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

      6. metadata-eval 76.5%

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

      7. metadata-eval 76.5%

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

      8. times-frac 76.5%

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

      9. *-commutative 76.5%

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

      10. times-frac 76.5%

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

      11. associate-*l/ 76.5%

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

   3. Simplified 76.5%

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

   if -6.99999999999999989e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 35.4%

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

      1. neg-sub0 35.4%

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

      2. associate-+l- 35.4%

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

      3. sub0-neg 35.4%

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

      4. neg-mul-1 35.4%

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

      5. associate-*r/ 35.4%

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

      6. metadata-eval 35.4%

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

      7. metadata-eval 35.4%

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

      8. times-frac 35.4%

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

      9. *-commutative 35.4%

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

      10. times-frac 35.4%

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

      11. associate-*l/ 35.4%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in b around inf 80.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 15: 64.5% 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 55.8%

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

   1. neg-sub0 55.8%

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

   2. associate-+l- 55.8%

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

   3. sub0-neg 55.8%

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

   4. neg-mul-1 55.8%

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

   5. associate-*r/ 55.8%

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

   6. metadata-eval 55.8%

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

   7. metadata-eval 55.8%

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

   8. times-frac 55.8%

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

   9. *-commutative 55.8%

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

   10. times-frac 55.8%

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

   11. associate-*l/ 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in b around inf 63.7%

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

5. Final simplification 63.7%

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

Reproduce

herbie shell --seed 2023224 
(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)))