Cubic critical, narrow range

Percentage Accurate: 55.3% → 92.2%

Time: 19.8s

Alternatives: 13

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + \mathsf{fma}\left(b, b, b \cdot \sqrt{t_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -16.0)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (fma b b (* b (sqrt t_0)))))
      (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (/
          (+
           (* 5.0625 (* (pow a 4.0) (pow c 4.0)))
           (pow (* -1.125 (* (pow a 2.0) (pow c 2.0))) 2.0))
          (* a (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -16.0) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + fma(b, b, (b * sqrt(t_0))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * (((5.0625 * (pow(a, 4.0) * pow(c, 4.0))) + pow((-1.125 * (pow(a, 2.0) * pow(c, 2.0))), 2.0)) / (a * pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -16.0)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + fma(b, b, Float64(b * sqrt(t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64(Float64(5.0625 * Float64((a ^ 4.0) * (c ^ 4.0))) + (Float64(-1.125 * Float64((a ^ 2.0) * (c ^ 2.0))) ^ 2.0)) / Float64(a * (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -16.0], N[(N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(5.0625 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(-1.125 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -16

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. sqr-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. div-sub 91.5%

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

      5. --rgt-identity 91.5%

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

      6. div-sub 92.2%

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

   3. Simplified 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. metadata-eval 92.3%

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

      4. distribute-lft-neg-in 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. pow1/2 92.3%

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

      2. distribute-lft-neg-in 92.3%

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

      3. metadata-eval 92.3%

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

   8. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}} - b}{3 \cdot a} \]
   9. Step-by-step derivation

      1. unpow1/2 92.3%

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

      2. fma-udef 92.2%

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

      3. unpow2 92.2%

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

      4. +-commutative 92.2%

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

      5. associate-*r* 92.2%

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

      6. fma-def 92.1%

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

      7. *-commutative 92.1%

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

   10. Simplified 92.1%

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

       1. flip3-- 92.0%

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

       2. pow1/2 92.0%

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

       3. pow-pow 93.5%

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

       4. metadata-eval 93.5%

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

       5. add-sqr-sqrt 93.5%

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

       6. fma-def 93.6%

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

   12. Applied egg-rr 93.6%

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

       1. fma-def 93.7%

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

       2. *-commutative 93.7%

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

       3. associate-*r* 93.7%

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

       4. fma-def 93.7%

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

       5. fma-def 93.7%

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

       6. *-commutative 93.7%

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

       7. associate-*r* 93.7%

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

       8. fma-def 93.7%

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

       9. *-commutative 93.7%

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

       10. fma-def 93.7%

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

       11. *-commutative 93.7%

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

       12. associate-*r* 93.7%

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

       13. fma-def 93.7%

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

   14. Simplified 93.7%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.4%

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

   3. Taylor expanded in b around inf 94.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right) + \mathsf{fma}\left(b, b, b \cdot \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + \mathsf{fma}\left(b, b, b \cdot \sqrt{t_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -16.0)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (fma b b (* b (sqrt t_0)))))
      (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * c), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -16.0) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + fma(b, b, (b * sqrt(t_0))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * c), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -16.0)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + fma(b, b, Float64(b * sqrt(t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -16.0], N[(N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -16

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. sqr-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. div-sub 91.5%

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

      5. --rgt-identity 91.5%

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

      6. div-sub 92.2%

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

   3. Simplified 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. metadata-eval 92.3%

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

      4. distribute-lft-neg-in 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. pow1/2 92.3%

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

      2. distribute-lft-neg-in 92.3%

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

      3. metadata-eval 92.3%

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

   8. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}} - b}{3 \cdot a} \]
   9. Step-by-step derivation

      1. unpow1/2 92.3%

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

      2. fma-udef 92.2%

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

      3. unpow2 92.2%

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

      4. +-commutative 92.2%

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

      5. associate-*r* 92.2%

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

      6. fma-def 92.1%

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

      7. *-commutative 92.1%

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

   10. Simplified 92.1%

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

       1. flip3-- 92.0%

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

       2. pow1/2 92.0%

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

       3. pow-pow 93.5%

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

       4. metadata-eval 93.5%

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

       5. add-sqr-sqrt 93.5%

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

       6. fma-def 93.6%

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

   12. Applied egg-rr 93.6%

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

       1. fma-def 93.7%

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

       2. *-commutative 93.7%

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

       3. associate-*r* 93.7%

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

       4. fma-def 93.7%

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

       5. fma-def 93.7%

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

       6. *-commutative 93.7%

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

       7. associate-*r* 93.7%

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

       8. fma-def 93.7%

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

       9. *-commutative 93.7%

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

       10. fma-def 93.7%

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

       11. *-commutative 93.7%

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

       12. associate-*r* 93.7%

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

       13. fma-def 93.7%

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

   14. Simplified 93.7%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.4%

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

   3. Taylor expanded in b around inf 94.0%

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

   4. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-out 94.0%

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

      2. associate-*r* 94.0%

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

      3. *-commutative 94.0%

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

      4. times-frac 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right) + \mathsf{fma}\left(b, b, b \cdot \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -16.0)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0)))))
      (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -16.0) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -16.0)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -16.0], N[(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] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -16

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. sqr-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. div-sub 91.5%

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

      5. --rgt-identity 91.5%

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

      6. div-sub 92.2%

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

   3. Simplified 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. metadata-eval 92.3%

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

      4. distribute-lft-neg-in 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. pow1/2 92.3%

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

      2. distribute-lft-neg-in 92.3%

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

      3. metadata-eval 92.3%

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

   8. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}} - b}{3 \cdot a} \]
   9. Step-by-step derivation

      1. unpow1/2 92.3%

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

      2. fma-udef 92.2%

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

      3. unpow2 92.2%

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

      4. +-commutative 92.2%

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

      5. associate-*r* 92.2%

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

      6. fma-def 92.1%

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

      7. *-commutative 92.1%

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

   10. Simplified 92.1%

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

       1. flip3-- 92.0%

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

       2. pow1/2 92.0%

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

       3. pow-pow 93.5%

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

       4. metadata-eval 93.5%

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

       5. add-sqr-sqrt 93.5%

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

       6. fma-def 93.6%

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

   12. Applied egg-rr 93.6%

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

       1. fma-def 93.7%

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

       2. associate-*l* 93.7%

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

       3. fma-def 93.6%

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

       4. fma-def 93.6%

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

       5. associate-*l* 93.6%

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

       6. fma-def 93.6%

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

       7. fma-udef 93.5%

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

       8. distribute-rgt-out 93.6%

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

       9. fma-def 93.6%

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

       10. associate-*l* 93.8%

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

       11. fma-def 93.6%

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

   14. Simplified 93.6%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.4%

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

   3. Taylor expanded in b around inf 94.0%

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

   4. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-out 94.0%

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

      2. associate-*r* 94.0%

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

      3. *-commutative 94.0%

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

      4. times-frac 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot -3, c, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a -3.0) c (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -16.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * -3.0), c, pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -16.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * -3.0), c, (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -16.0)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -3.0), $MachinePrecision] * c + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -16.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -16

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. sqr-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. div-sub 91.5%

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

      5. --rgt-identity 91.5%

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

      6. div-sub 92.2%

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

   3. Simplified 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. metadata-eval 92.3%

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

      4. distribute-lft-neg-in 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. pow1/2 92.3%

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

      2. distribute-lft-neg-in 92.3%

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

      3. metadata-eval 92.3%

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

   8. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}} - b}{3 \cdot a} \]
   9. Step-by-step derivation

      1. unpow1/2 92.3%

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

      2. fma-udef 92.2%

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

      3. unpow2 92.2%

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

      4. +-commutative 92.2%

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

      5. associate-*r* 92.2%

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

      6. fma-def 92.1%

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

      7. *-commutative 92.1%

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

   10. Simplified 92.1%

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

       1. flip-- 91.5%

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

       2. add-sqr-sqrt 93.0%

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

       3. unpow2 93.0%

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

   12. Applied egg-rr 93.0%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.4%

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

   3. Taylor expanded in b around inf 94.0%

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

   4. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-out 94.0%

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

      2. associate-*r* 94.0%

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

      3. *-commutative 94.0%

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

      4. times-frac 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot -3, c, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a \cdot -3, c, {b}^{2}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a -3.0) c (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (fma
       -0.375
       (/ a (/ (pow b 3.0) (pow c 2.0)))
       (/ (* -0.5625 (pow a 2.0)) (/ (pow b 5.0) (pow c 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * -3.0), c, pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.375, (a / (pow(b, 3.0) / pow(c, 2.0))), ((-0.5625 * pow(a, 2.0)) / (pow(b, 5.0) / pow(c, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * -3.0), c, (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7.0)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.375, Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))), Float64(Float64(-0.5625 * (a ^ 2.0)) / Float64((b ^ 5.0) / (c ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -3.0), $MachinePrecision] * c + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -7.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

      2. sqr-neg 88.9%

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

      3. unsub-neg 88.9%

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

      4. div-sub 87.9%

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

      5. --rgt-identity 87.9%

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

      6. div-sub 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*r* 88.9%

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

      2. *-commutative 88.9%

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

      3. metadata-eval 88.9%

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

      4. distribute-lft-neg-in 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. pow1/2 88.9%

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

      2. distribute-lft-neg-in 88.9%

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

      3. metadata-eval 88.9%

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

   8. Applied egg-rr 88.9%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}} - b}{3 \cdot a} \]
   9. Step-by-step derivation

      1. unpow1/2 88.9%

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

      2. fma-udef 88.9%

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

      3. unpow2 88.9%

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

      4. +-commutative 88.9%

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

      5. associate-*r* 88.9%

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

      6. fma-def 88.8%

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

      7. *-commutative 88.8%

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

   10. Simplified 88.8%

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

       1. flip-- 88.6%

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

       2. add-sqr-sqrt 90.1%

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

       3. unpow2 90.1%

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

   12. Applied egg-rr 90.1%

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

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

   1. Initial program 51.5%

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

   3. Taylor expanded in b around inf 90.9%

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

      1. log1p-expm1-u 90.8%

         \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}\right)}{3 \cdot a} \]

      2. log1p-udef 81.2%

         \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}\right)}{3 \cdot a} \]

      3. div-inv 81.2%

         \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \log \left(1 + \mathsf{expm1}\left(\color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}\right)\right)\right)}{3 \cdot a} \]

      4. pow-prod-down 81.2%

         \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \log \left(1 + \mathsf{expm1}\left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}\right)\right)\right)}{3 \cdot a} \]

      5. pow-flip 81.2%

         \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \log \left(1 + \mathsf{expm1}\left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)\right)}{3 \cdot a} \]

      6. metadata-eval 81.2%

         \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \log \left(1 + \mathsf{expm1}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}\right)\right)\right)}{3 \cdot a} \]

   5. Applied egg-rr 81.2%

      \[\leadsto \frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)\right)}\right)}{3 \cdot a} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 77.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \log \left(1 + \mathsf{expm1}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)\right)\right)}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 59.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \log \left(1 + \mathsf{expm1}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)\right)\right)}{3 \cdot a}\right)} - 1} \]

   7. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.6875, {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}, \mathsf{fma}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.125, -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right)\right)}{a \cdot 3}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 86.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.6875, {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}, \mathsf{fma}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.125, -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right)\right)}{a \cdot 3}\right)\right)} \]

      2. expm1-log1p 90.9%

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

      3. *-lft-identity 90.9%

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

      4. *-commutative 90.9%

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

      5. times-frac 90.8%

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

      6. metadata-eval 90.8%

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

      7. *-commutative 90.8%

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

      8. associate-/r/ 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in a around 0 91.2%

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

       1. +-commutative 91.2%

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

       2. associate-+l+ 91.3%

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

       3. +-commutative 91.3%

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

       4. fma-def 91.3%

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

       5. +-commutative 91.3%

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

       6. fma-def 91.3%

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

       7. associate-/l* 91.3%

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

       8. associate-/l* 91.3%

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

       9. associate-*r/ 91.3%

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

   12. Simplified 91.3%

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

4. Final simplification 91.1%

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

Alternative 6: 89.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a -3.0) c (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -7.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * -3.0), c, pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -7.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * -3.0), c, (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -7.0)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -3.0), $MachinePrecision] * c + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -7.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

      2. sqr-neg 88.9%

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

      3. unsub-neg 88.9%

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

      4. div-sub 87.9%

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

      5. --rgt-identity 87.9%

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

      6. div-sub 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*r* 88.9%

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

      2. *-commutative 88.9%

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

      3. metadata-eval 88.9%

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

      4. distribute-lft-neg-in 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. pow1/2 88.9%

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

      2. distribute-lft-neg-in 88.9%

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

      3. metadata-eval 88.9%

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

   8. Applied egg-rr 88.9%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\right)}^{0.5}} - b}{3 \cdot a} \]
   9. Step-by-step derivation

      1. unpow1/2 88.9%

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

      2. fma-udef 88.9%

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

      3. unpow2 88.9%

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

      4. +-commutative 88.9%

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

      5. associate-*r* 88.9%

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

      6. fma-def 88.8%

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

      7. *-commutative 88.8%

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

   10. Simplified 88.8%

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

       1. flip-- 88.6%

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

       2. add-sqr-sqrt 90.1%

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

       3. unpow2 90.1%

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

   12. Applied egg-rr 90.1%

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

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

   1. Initial program 51.5%

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

   3. Taylor expanded in b around inf 91.2%

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

4. Final simplification 91.1%

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

Alternative 7: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -16.0)
   (/ (- (sqrt (fma b b (* 3.0 (* a (- c))))) b) (* 3.0 a))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -16.0) {
		tmp = (sqrt(fma(b, b, (3.0 * (a * -c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

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)) <= -16.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(3.0 * Float64(a * Float64(-c))))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

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], -16.0], N[(N[(N[Sqrt[N[(b * b + N[(3.0 * N[(a * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -16

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. sqr-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. div-sub 91.5%

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

      5. --rgt-identity 91.5%

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

      6. div-sub 92.2%

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

   3. Simplified 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. metadata-eval 92.3%

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

      4. distribute-lft-neg-in 92.3%

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

   6. Applied egg-rr 92.3%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.4%

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

   3. Taylor expanded in b around inf 90.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.18:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.18)
   (/ (- (sqrt (fma b b (* 3.0 (* a (- c))))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.18) {
		tmp = (sqrt(fma(b, b, (3.0 * (a * -c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.18)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(3.0 * Float64(a * Float64(-c))))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.18], N[(N[(N[Sqrt[N[(b * b + N[(3.0 * N[(a * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.17999999999999999

   1. Initial program 82.5%

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

      1. +-commutative 82.5%

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

      2. sqr-neg 82.5%

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

      3. unsub-neg 82.5%

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

      4. div-sub 82.5%

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

      5. --rgt-identity 82.5%

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

      6. div-sub 82.5%

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

   3. Simplified 82.6%

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

      1. associate-*r* 82.6%

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

      2. *-commutative 82.6%

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

      3. metadata-eval 82.6%

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

      4. distribute-lft-neg-in 82.6%

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

   6. Applied egg-rr 82.6%

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

   if -0.17999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf 86.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.18:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -3.02e-5)
   (/ (- (sqrt (fma b b (* 3.0 (* a (- c))))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.02e-5) {
		tmp = (sqrt(fma(b, b, (3.0 * (a * -c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -3.02e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(3.0 * Float64(a * Float64(-c))))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.02e-5], N[(N[(N[Sqrt[N[(b * b + N[(3.0 * N[(a * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.01999999999999988e-5

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. sqr-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. div-sub 74.1%

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

      5. --rgt-identity 74.1%

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

      6. div-sub 74.5%

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

   3. Simplified 74.6%

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

      1. associate-*r* 74.6%

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

      2. *-commutative 74.6%

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

      3. metadata-eval 74.6%

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

      4. distribute-lft-neg-in 74.6%

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

   6. Applied egg-rr 74.6%

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

   if -3.01999999999999988e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -3.02e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -3.02e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -3.02e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.02e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.01999999999999988e-5

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. sqr-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. div-sub 74.1%

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

      5. --rgt-identity 74.1%

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

      6. div-sub 74.5%

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

   3. Simplified 74.6%

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

   if -3.01999999999999988e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -3.02e-5) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -3.02e-5) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-3.02d-5)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -3.02e-5) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -3.02e-5:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -3.02e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -3.02e-5)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3.02e-5], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.01999999999999988e-5

   1. Initial program 74.5%

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

   if -3.01999999999999988e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 37.0%

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

   3. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 256:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 256.0)
   (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 256.0) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 256.0d0) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 256.0) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 256.0:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 256.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 256.0)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 256.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 256

   1. Initial program 75.9%

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

   3. Taylor expanded in a around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*l* 75.9%

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

   5. Simplified 75.9%

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

   if 256 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf 73.4%

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

      1. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 256:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

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

5. Simplified 63.8%

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

6. Final simplification 63.8%

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

Reproduce

herbie shell --seed 2024024 
(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)))