Cubic critical, narrow range

Percentage Accurate: 55.8% → 91.8%

Time: 23.8s

Alternatives: 12

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(3 \cdot c\right)\\ t_1 := \frac{{b}^{6} - {t\_0}^{3}}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(b, b, t\_0\right), {b}^{4}\right)}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\frac{{t\_1}^{1.5} - {b}^{3}}{{b}^{2} + \left(t\_1 + b \cdot \sqrt{t\_1}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* 3.0 c)))
        (t_1
         (/
          (- (pow b 6.0) (pow t_0 3.0))
          (fma t_0 (fma b b t_0) (pow b 4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.6)
     (/
      (/
       (- (pow t_1 1.5) (pow b 3.0))
       (+ (pow b 2.0) (+ t_1 (* b (sqrt t_1)))))
      (* 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) (/ 6.328125 (* a (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (3.0 * c);
	double t_1 = (pow(b, 6.0) - pow(t_0, 3.0)) / fma(t_0, fma(b, b, t_0), pow(b, 4.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.6) {
		tmp = ((pow(t_1, 1.5) - pow(b, 3.0)) / (pow(b, 2.0) + (t_1 + (b * sqrt(t_1))))) / (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) * (6.328125 / (a * pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(3.0 * c))
	t_1 = Float64(Float64((b ^ 6.0) - (t_0 ^ 3.0)) / fma(t_0, fma(b, b, t_0), (b ^ 4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.6)
		tmp = Float64(Float64(Float64((t_1 ^ 1.5) - (b ^ 3.0)) / Float64((b ^ 2.0) + Float64(t_1 + Float64(b * sqrt(t_1))))) / 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(a * c) ^ 4.0) * Float64(6.328125 / Float64(a * (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[b, 6.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(b * b + t$95$0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.6], N[(N[(N[(N[Power[t$95$1, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$1 + N[(b * N[Sqrt[t$95$1], $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[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.6000000000000001

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. sqr-neg 84.1%

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

      3. associate-*l* 84.2%

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

   3. Simplified 84.2%

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

      1. flip3-- 83.7%

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

      2. pow2 83.7%

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

      3. pow-pow 83.3%

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

      4. metadata-eval 83.3%

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

      5. pow2 83.3%

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

      6. pow2 83.3%

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

      7. pow-prod-up 83.8%

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

      8. metadata-eval 83.8%

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

      9. distribute-rgt-out 83.8%

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

      10. +-commutative 83.8%

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

      11. fma-define 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. flip3-+ 83.4%

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

   8. Applied egg-rr 84.5%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\mathsf{fma}\left(\left(a \cdot 3\right) \cdot c, \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right), {b}^{4}\right)}\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\mathsf{fma}\left(\left(a \cdot 3\right) \cdot c, \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right), {b}^{4}\right)} - \left(-b\right) \cdot \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\mathsf{fma}\left(\left(a \cdot 3\right) \cdot c, \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right), {b}^{4}\right)}}\right)}}}{3 \cdot a} \]

   10. Simplified 84.5%

       \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{{b}^{6} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{3}}{\mathsf{fma}\left(a \cdot \left(c \cdot 3\right), \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right), {b}^{4}\right)}\right)}^{1.5} - {b}^{3}}{{b}^{2} + \left(\frac{{b}^{6} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{3}}{\mathsf{fma}\left(a \cdot \left(c \cdot 3\right), \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right), {b}^{4}\right)} + b \cdot \sqrt{\frac{{b}^{6} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{3}}{\mathsf{fma}\left(a \cdot \left(c \cdot 3\right), \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right), {b}^{4}\right)}}\right)}}}{3 \cdot a} \]

   if -1.6000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.0%

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

      1. sqr-neg 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in b around inf 94.0%

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

   6. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-in 94.0%

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

      2. associate-*r* 94.0%

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

      3. associate-*r* 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\frac{{\left(\frac{{b}^{6} - {\left(a \cdot \left(3 \cdot c\right)\right)}^{3}}{\mathsf{fma}\left(a \cdot \left(3 \cdot c\right), \mathsf{fma}\left(b, b, a \cdot \left(3 \cdot c\right)\right), {b}^{4}\right)}\right)}^{1.5} - {b}^{3}}{{b}^{2} + \left(\frac{{b}^{6} - {\left(a \cdot \left(3 \cdot c\right)\right)}^{3}}{\mathsf{fma}\left(a \cdot \left(3 \cdot c\right), \mathsf{fma}\left(b, b, a \cdot \left(3 \cdot c\right)\right), {b}^{4}\right)} + b \cdot \sqrt{\frac{{b}^{6} - {\left(a \cdot \left(3 \cdot c\right)\right)}^{3}}{\mathsf{fma}\left(a \cdot \left(3 \cdot c\right), \mathsf{fma}\left(b, b, a \cdot \left(3 \cdot c\right)\right), {b}^{4}\right)}}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{7}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.6)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+
     (* -0.5 (/ c b))
     (+
      (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
      (*
       -0.16666666666666666
       (* (pow (* a c) 4.0) (/ 6.328125 (* a (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * (pow((a * c), 4.0) * (6.328125 / (a * pow(b, 7.0)))))));
	}
	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)) <= -1.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * Float64(6.328125 / Float64(a * (b ^ 7.0))))))));
	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], -1.6], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * 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[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.6000000000000001

   1. Initial program 84.1%

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

      1. /-rgt-identity 84.1%

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

      2. metadata-eval 84.1%

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

   3. Simplified 84.2%

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

   if -1.6000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.0%

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

      1. sqr-neg 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in b around inf 94.0%

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

   6. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-in 94.0%

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

      2. associate-*r* 94.0%

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

      3. associate-*r* 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{7}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2}}{a \cdot {b}^{3}} + \frac{1.5 \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{3}\right)}{a \cdot {b}^{5}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.6)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (fma
    -0.16666666666666666
    (+
     (/ (pow (* c (* a -1.5)) 2.0) (* a (pow b 3.0)))
     (/ (* 1.5 (* 2.25 (pow (* a c) 3.0))) (* a (pow b 5.0))))
    (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.16666666666666666, ((pow((c * (a * -1.5)), 2.0) / (a * pow(b, 3.0))) + ((1.5 * (2.25 * pow((a * c), 3.0))) / (a * pow(b, 5.0)))), ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.16666666666666666, Float64(Float64((Float64(c * Float64(a * -1.5)) ^ 2.0) / Float64(a * (b ^ 3.0))) + Float64(Float64(1.5 * Float64(2.25 * (Float64(a * c) ^ 3.0))) / Float64(a * (b ^ 5.0)))), Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.6], 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[(-0.16666666666666666 * N[(N[(N[Power[N[(c * N[(a * -1.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5 * N[(2.25 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.6000000000000001

   1. Initial program 84.1%

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

      1. /-rgt-identity 84.1%

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

      2. metadata-eval 84.1%

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

   3. Simplified 84.2%

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

   if -1.6000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.0%

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

      1. sqr-neg 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

      1. flip3-- 52.0%

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

      2. pow2 52.0%

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

      3. pow-pow 51.6%

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

      4. metadata-eval 51.6%

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

      5. pow2 51.6%

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

      6. pow2 51.6%

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

      7. pow-prod-up 51.8%

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

      8. metadata-eval 51.8%

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

      9. distribute-rgt-out 51.8%

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

      10. +-commutative 51.8%

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

      11. fma-define 51.8%

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

   6. Applied egg-rr 51.8%

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

   7. Taylor expanded in b around inf 90.9%

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

   9. Simplified 90.7%

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

       1. div-inv 90.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \color{blue}{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0\right), \left(c \cdot 0\right) \cdot \left(a \cdot -3\right)\right) \cdot \frac{1}{a \cdot {b}^{5}}}, c \cdot \frac{-0.5}{b}\right) \]

   11. Applied egg-rr 90.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \color{blue}{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right) \cdot \frac{1}{a \cdot {b}^{5}}}, c \cdot \frac{-0.5}{b}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 90.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \color{blue}{\frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right) \cdot 1}{a \cdot {b}^{5}}}, c \cdot \frac{-0.5}{b}\right) \]

       2. *-rgt-identity 90.7%

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

       3. fma-undefine 90.7%

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

       4. +-rgt-identity 90.7%

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

       5. associate-*r* 90.7%

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

       6. associate-*r* 90.7%

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

       7. unpow2 90.7%

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

       8. cube-mult 90.7%

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

       9. cube-prod 90.7%

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

       10. *-commutative 90.7%

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

       11. cube-prod 90.7%

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

   13. Simplified 90.7%

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

       1. associate-*r/ 90.9%

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

   15. Applied egg-rr 90.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2}}{a \cdot {b}^{3}} + \frac{1.5 \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{3}\right)}{a \cdot {b}^{5}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.6)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) 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)) <= -1.6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - 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)) <= -1.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - 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], -1.6], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * 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)) < -1.6000000000000001

   1. Initial program 84.1%

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

      1. /-rgt-identity 84.1%

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

      2. metadata-eval 84.1%

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

   3. Simplified 84.2%

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

   if -1.6000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.0%

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

      1. sqr-neg 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

   5. 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 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -1.5\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{1.5 \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{3}\right)}{a \cdot {b}^{5}} + \frac{t\_0 \cdot t\_0}{a \cdot {b}^{3}}, c \cdot \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -1.5))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.6)
     (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
     (fma
      -0.16666666666666666
      (+
       (/ (* 1.5 (* 2.25 (pow (* a c) 3.0))) (* a (pow b 5.0)))
       (/ (* t_0 t_0) (* a (pow b 3.0))))
      (* c (/ -0.5 b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -1.5);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.16666666666666666, (((1.5 * (2.25 * pow((a * c), 3.0))) / (a * pow(b, 5.0))) + ((t_0 * t_0) / (a * pow(b, 3.0)))), (c * (-0.5 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -1.5))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.16666666666666666, Float64(Float64(Float64(1.5 * Float64(2.25 * (Float64(a * c) ^ 3.0))) / Float64(a * (b ^ 5.0))) + Float64(Float64(t_0 * t_0) / Float64(a * (b ^ 3.0)))), Float64(c * Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.6], 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[(-0.16666666666666666 * N[(N[(N[(1.5 * N[(2.25 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.6000000000000001

   1. Initial program 84.1%

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

      1. /-rgt-identity 84.1%

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

      2. metadata-eval 84.1%

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

   3. Simplified 84.2%

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

   if -1.6000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.0%

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

      1. sqr-neg 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

      1. flip3-- 52.0%

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

      2. pow2 52.0%

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

      3. pow-pow 51.6%

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

      4. metadata-eval 51.6%

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

      5. pow2 51.6%

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

      6. pow2 51.6%

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

      7. pow-prod-up 51.8%

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

      8. metadata-eval 51.8%

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

      9. distribute-rgt-out 51.8%

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

      10. +-commutative 51.8%

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

      11. fma-define 51.8%

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

   6. Applied egg-rr 51.8%

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

   7. Taylor expanded in b around inf 90.9%

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

   9. Simplified 90.7%

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

       1. div-inv 90.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \color{blue}{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0\right), \left(c \cdot 0\right) \cdot \left(a \cdot -3\right)\right) \cdot \frac{1}{a \cdot {b}^{5}}}, c \cdot \frac{-0.5}{b}\right) \]

   11. Applied egg-rr 90.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \color{blue}{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right) \cdot \frac{1}{a \cdot {b}^{5}}}, c \cdot \frac{-0.5}{b}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 90.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \color{blue}{\frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right) \cdot 1}{a \cdot {b}^{5}}}, c \cdot \frac{-0.5}{b}\right) \]

       2. *-rgt-identity 90.7%

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

       3. fma-undefine 90.7%

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

       4. +-rgt-identity 90.7%

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

       5. associate-*r* 90.7%

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

       6. associate-*r* 90.7%

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

       7. unpow2 90.7%

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

       8. cube-mult 90.7%

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

       9. cube-prod 90.7%

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

       10. *-commutative 90.7%

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

       11. cube-prod 90.7%

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

   13. Simplified 90.7%

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

       1. unpow2 90.7%

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

   15. Applied egg-rr 90.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{1.5 \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{3}\right)}{a \cdot {b}^{5}} + \frac{\left(c \cdot \left(a \cdot -1.5\right)\right) \cdot \left(c \cdot \left(a \cdot -1.5\right)\right)}{a \cdot {b}^{3}}, c \cdot \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -5e-7)
   (/ (- (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)) <= -5e-7) {
		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)) <= -5e-7)
		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], -5e-7], 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)) < -4.99999999999999977e-7

   1. Initial program 71.5%

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

      1. /-rgt-identity 71.5%

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

      2. metadata-eval 71.5%

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

   3. Simplified 71.6%

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

   if -4.99999999999999977e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.1%

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

      1. sqr-neg 29.1%

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

      2. sqr-neg 29.1%

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

      3. associate-*l* 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b around inf 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*l/ 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -5e-7) 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 <= -5e-7) {
		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 <= (-5d-7)) 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 <= -5e-7) {
		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 <= -5e-7:
		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 <= -5e-7)
		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 <= -5e-7)
		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, -5e-7], 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)) < -4.99999999999999977e-7

   1. Initial program 71.5%

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

   if -4.99999999999999977e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.1%

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

      1. sqr-neg 29.1%

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

      2. sqr-neg 29.1%

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

      3. associate-*l* 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b around inf 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*l/ 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.6)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.6], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5999999999999996

   1. Initial program 81.8%

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

      1. /-rgt-identity 81.8%

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

      2. metadata-eval 81.8%

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

   3. Simplified 81.9%

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

   if 4.5999999999999996 < b

   1. Initial program 49.3%

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

      1. sqr-neg 49.3%

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

      2. sqr-neg 49.3%

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

      3. associate-*l* 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in b around inf 86.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2650:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2650.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2650.0) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2650.0d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2650.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2650.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2650.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2650.0)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2650.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2650

   1. Initial program 70.5%

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

      1. sqr-neg 70.5%

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

      2. sqr-neg 70.5%

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

      3. associate-*l* 70.5%

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

   3. Simplified 70.5%

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

   if 2650 < b

   1. Initial program 42.1%

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

      1. sqr-neg 42.1%

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

      2. sqr-neg 42.1%

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

      3. associate-*l* 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in b around inf 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*l/ 75.6%

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

   7. Simplified 75.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2650:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 23.2× speedup

Math

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

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(c * Float64(-0.5 / b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. sqr-neg 55.4%

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

   2. sqr-neg 55.4%

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

   3. associate-*l* 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in b around inf 81.6%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

6. Taylor expanded in c around 0 64.3%

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

   1. associate-*r/ 64.3%

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

   2. *-commutative 64.3%

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

   3. associate-/l* 64.2%

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

8. Simplified 64.2%

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

9. Final simplification 64.2%

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

Alternative 11: 64.1% 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 55.4%

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

   1. sqr-neg 55.4%

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

   2. sqr-neg 55.4%

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

   3. associate-*l* 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in b around inf 64.3%

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

   1. *-commutative 64.3%

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

   2. associate-*l/ 64.3%

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

7. Simplified 64.3%

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

8. Final simplification 64.3%

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

Alternative 12: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. sqr-neg 55.4%

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

   2. sqr-neg 55.4%

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

   3. associate-*l* 55.4%

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

3. Simplified 55.4%

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

   1. flip3-- 55.3%

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

   2. pow2 55.3%

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

   3. pow-pow 55.0%

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

   4. metadata-eval 55.0%

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

   5. pow2 55.0%

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

   6. pow2 55.0%

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

   7. pow-prod-up 55.2%

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

   8. metadata-eval 55.2%

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

   9. distribute-rgt-out 55.2%

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

   10. +-commutative 55.2%

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

   11. fma-define 55.1%

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

6. Applied egg-rr 55.1%

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

   1. log1p-expm1-u 51.0%

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

8. Applied egg-rr 51.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\mathsf{fma}\left(\left(a \cdot 3\right) \cdot c, \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right), {b}^{4}\right)}}\right)}{a \cdot 3}\right)\right)} \]

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024043 
(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)))