Cubic critical, narrow range

Percentage Accurate: 55.7% → 92.2%

Time: 17.9s

Alternatives: 16

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + c \cdot \left(a \cdot -3\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -50:\\ \;\;\;\;\frac{\frac{t\_0}{3 \cdot a} - \frac{b \cdot b}{3 \cdot a}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + c \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(\frac{-3.1640625 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{t\_1 \cdot \left(t\_1 \cdot t\_1\right)} + \frac{\left(a \cdot a\right) \cdot -1.0546875}{{b}^{7}}\right) + \frac{a \cdot -0.5625}{\left(b \cdot b\right) \cdot t\_1}\right) + \frac{-0.375}{t\_1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* c (* a -3.0)))) (t_1 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -50.0)
     (/ (- (/ t_0 (* 3.0 a)) (/ (* b b) (* 3.0 a))) (+ b (sqrt t_0)))
     (+
      (/ (* c -0.5) b)
      (*
       c
       (*
        a
        (*
         c
         (+
          (*
           c
           (+
            (*
             c
             (+
              (/ (* -3.1640625 (* c (* a (* a a)))) (* t_1 (* t_1 t_1)))
              (/ (* (* a a) -1.0546875) (pow b 7.0))))
            (/ (* a -0.5625) (* (* b b) t_1))))
          (/ -0.375 t_1)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) + (c * (a * -3.0));
	double t_1 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0) {
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + sqrt(t_0));
	} else {
		tmp = ((c * -0.5) / b) + (c * (a * (c * ((c * ((c * (((-3.1640625 * (c * (a * (a * a)))) / (t_1 * (t_1 * t_1))) + (((a * a) * -1.0546875) / pow(b, 7.0)))) + ((a * -0.5625) / ((b * b) * t_1)))) + (-0.375 / t_1)))));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (b * b) + (c * (a * (-3.0d0)))
    t_1 = b * (b * b)
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-50.0d0)) then
        tmp = ((t_0 / (3.0d0 * a)) - ((b * b) / (3.0d0 * a))) / (b + sqrt(t_0))
    else
        tmp = ((c * (-0.5d0)) / b) + (c * (a * (c * ((c * ((c * ((((-3.1640625d0) * (c * (a * (a * a)))) / (t_1 * (t_1 * t_1))) + (((a * a) * (-1.0546875d0)) / (b ** 7.0d0)))) + ((a * (-0.5625d0)) / ((b * b) * t_1)))) + ((-0.375d0) / t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) + (c * (a * -3.0));
	double t_1 = b * (b * b);
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0) {
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + Math.sqrt(t_0));
	} else {
		tmp = ((c * -0.5) / b) + (c * (a * (c * ((c * ((c * (((-3.1640625 * (c * (a * (a * a)))) / (t_1 * (t_1 * t_1))) + (((a * a) * -1.0546875) / Math.pow(b, 7.0)))) + ((a * -0.5625) / ((b * b) * t_1)))) + (-0.375 / t_1)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) + (c * (a * -3.0))
	t_1 = b * (b * b)
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0:
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + math.sqrt(t_0))
	else:
		tmp = ((c * -0.5) / b) + (c * (a * (c * ((c * ((c * (((-3.1640625 * (c * (a * (a * a)))) / (t_1 * (t_1 * t_1))) + (((a * a) * -1.0546875) / math.pow(b, 7.0)))) + ((a * -0.5625) / ((b * b) * t_1)))) + (-0.375 / t_1)))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -50.0)
		tmp = Float64(Float64(Float64(t_0 / Float64(3.0 * a)) - Float64(Float64(b * b) / Float64(3.0 * a))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(c * Float64(a * Float64(c * Float64(Float64(c * Float64(Float64(c * Float64(Float64(Float64(-3.1640625 * Float64(c * Float64(a * Float64(a * a)))) / Float64(t_1 * Float64(t_1 * t_1))) + Float64(Float64(Float64(a * a) * -1.0546875) / (b ^ 7.0)))) + Float64(Float64(a * -0.5625) / Float64(Float64(b * b) * t_1)))) + Float64(-0.375 / t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) + (c * (a * -3.0));
	t_1 = b * (b * b);
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0)
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + sqrt(t_0));
	else
		tmp = ((c * -0.5) / b) + (c * (a * (c * ((c * ((c * (((-3.1640625 * (c * (a * (a * a)))) / (t_1 * (t_1 * t_1))) + (((a * a) * -1.0546875) / (b ^ 7.0)))) + ((a * -0.5625) / ((b * b) * t_1)))) + (-0.375 / t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $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], -50.0], N[(N[(N[(t$95$0 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(c * N[(a * N[(c * N[(N[(c * N[(N[(c * N[(N[(N[(-3.1640625 * N[(c * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -1.0546875), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.5625), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 / t$95$1), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -50

   1. Initial program 92.6%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right), \color{blue}{\left(\frac{b}{3 \cdot a}\right)}\right) \]

   4. Applied egg-rr 91.0%

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

      1. sub-div N/A

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

      2. flip-- N/A

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

      3. rem-square-sqrt N/A

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

      4. +-commutative N/A

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

      5. sub-div N/A

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

      6. div-sub N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{b \cdot b + a \cdot \left(-3 \cdot c\right)}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{3 \cdot a}\right), \color{blue}{\left(\frac{\frac{b \cdot b}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{3 \cdot a}\right)}\right) \]

   6. Applied egg-rr 92.0%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

   8. Applied egg-rr 93.4%

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

   if -50 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 52.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   7. Applied egg-rr 93.5%

      \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\frac{\frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot 0.140625 - \left(\frac{\left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right) \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{{b}^{10}} + \frac{\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}}{\frac{b}{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}}\right) \cdot \left(a \cdot a\right)}{\left(-0.375 \cdot c\right) \cdot \frac{\frac{c}{b \cdot b}}{b} - a \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot \frac{-0.5625}{{b}^{5}}\right) + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right)}} \]

   8. Taylor expanded in c around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(c \cdot \left(\frac{-405}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{9}} - \frac{135}{128} \cdot \frac{{a}^{2}}{{b}^{7}}\right) - \frac{9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right) \]

   10. Simplified 93.6%

       \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \left(c \cdot \left(\frac{-3.1640625 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{9}} + \frac{-1.0546875 \cdot \left(a \cdot a\right)}{{b}^{7}}\right) + -0.5625 \cdot \frac{a}{{b}^{5}}\right) - \frac{0.375}{b \cdot \left(b \cdot b\right)}\right)\right)} \]

   12. Applied egg-rr 93.6%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + c \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot \left(\frac{-3.1640625 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(a \cdot a\right) \cdot -1.0546875}{{b}^{7}}\right) + \frac{a \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) + \frac{-0.375}{b \cdot \left(b \cdot b\right)}\right)\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -50:\\ \;\;\;\;\frac{\frac{b \cdot b + c \cdot \left(a \cdot -3\right)}{3 \cdot a} - \frac{b \cdot b}{3 \cdot a}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + c \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(\frac{-3.1640625 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(a \cdot a\right) \cdot -1.0546875}{{b}^{7}}\right) + \frac{a \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) + \frac{-0.375}{b \cdot \left(b \cdot b\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -50:\\ \;\;\;\;\frac{\frac{t\_0}{3 \cdot a} - \frac{b \cdot b}{3 \cdot a}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\left(\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)\right) + \frac{\left(\left(a \cdot a\right) \cdot -1.0546875\right) \cdot {c}^{4}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* c (* a -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -50.0)
     (/ (- (/ t_0 (* 3.0 a)) (/ (* b b) (* 3.0 a))) (+ b (sqrt t_0)))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (/
        (+
         (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
         (/ (* (* (* a a) -1.0546875) (pow c 4.0)) (* (* b b) (* b b))))
        (* b (* b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) + (c * (a * -3.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0) {
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + sqrt(t_0));
	} else {
		tmp = ((c * -0.5) / b) + (a * ((((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) + ((((a * a) * -1.0546875) * pow(c, 4.0)) / ((b * b) * (b * b)))) / (b * (b * 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 = (b * b) + (c * (a * (-3.0d0)))
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-50.0d0)) then
        tmp = ((t_0 / (3.0d0 * a)) - ((b * b) / (3.0d0 * a))) / (b + sqrt(t_0))
    else
        tmp = ((c * (-0.5d0)) / b) + (a * ((((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) + ((((a * a) * (-1.0546875d0)) * (c ** 4.0d0)) / ((b * b) * (b * b)))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) + (c * (a * -3.0));
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0) {
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + Math.sqrt(t_0));
	} else {
		tmp = ((c * -0.5) / b) + (a * ((((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) + ((((a * a) * -1.0546875) * Math.pow(c, 4.0)) / ((b * b) * (b * b)))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) + (c * (a * -3.0))
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0:
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + math.sqrt(t_0))
	else:
		tmp = ((c * -0.5) / b) + (a * ((((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) + ((((a * a) * -1.0546875) * math.pow(c, 4.0)) / ((b * b) * (b * b)))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -50.0)
		tmp = Float64(Float64(Float64(t_0 / Float64(3.0 * a)) - Float64(Float64(b * b) / Float64(3.0 * a))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) + Float64(Float64(Float64(Float64(a * a) * -1.0546875) * (c ^ 4.0)) / Float64(Float64(b * b) * Float64(b * b)))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) + (c * (a * -3.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -50.0)
		tmp = ((t_0 / (3.0 * a)) - ((b * b) / (3.0 * a))) / (b + sqrt(t_0));
	else
		tmp = ((c * -0.5) / b) + (a * ((((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) + ((((a * a) * -1.0546875) * (c ^ 4.0)) / ((b * b) * (b * b)))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -50.0], N[(N[(N[(t$95$0 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * -1.0546875), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -50

   1. Initial program 92.6%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right), \color{blue}{\left(\frac{b}{3 \cdot a}\right)}\right) \]

   4. Applied egg-rr 91.0%

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

      1. sub-div N/A

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

      2. flip-- N/A

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

      3. rem-square-sqrt N/A

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

      4. +-commutative N/A

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

      5. sub-div N/A

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

      6. div-sub N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{b \cdot b + a \cdot \left(-3 \cdot c\right)}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{3 \cdot a}\right), \color{blue}{\left(\frac{\frac{b \cdot b}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{3 \cdot a}\right)}\right) \]

   6. Applied egg-rr 92.0%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

   8. Applied egg-rr 93.4%

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

   if -50 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 52.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{4}} + \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{4}} + \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

   8. Simplified 93.1%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -50:\\ \;\;\;\;\frac{\frac{b \cdot b + c \cdot \left(a \cdot -3\right)}{3 \cdot a} - \frac{b \cdot b}{3 \cdot a}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\left(\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)\right) + \frac{\left(\left(a \cdot a\right) \cdot -1.0546875\right) \cdot {c}^{4}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;b \leq 0.32:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* c (* a -3.0)))))
   (if (<= b 0.32)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (/
        (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
        (* b (* b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) + (c * (a * -3.0));
	double tmp;
	if (b <= 0.32) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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 = (b * b) + (c * (a * (-3.0d0)))
    if (b <= 0.32d0) then
        tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0d0 * a)
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) + (c * (a * -3.0));
	double tmp;
	if (b <= 0.32) {
		tmp = ((t_0 - (b * b)) / (b + Math.sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) + (c * (a * -3.0))
	tmp = 0
	if b <= 0.32:
		tmp = ((t_0 - (b * b)) / (b + math.sqrt(t_0))) / (3.0 * a)
	else:
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (b <= 0.32)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) + (c * (a * -3.0));
	tmp = 0.0;
	if (b <= 0.32)
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	else
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.32], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.320000000000000007

   1. Initial program 83.4%

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

   4. Applied egg-rr 83.8%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right)\right)}{\mathsf{neg}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)} - \frac{b \cdot b}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right)\right)}{\mathsf{neg}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)} - \frac{\mathsf{neg}\left(b \cdot b\right)}{\mathsf{neg}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)}\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      3. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right)\right)\right) - \left(\mathsf{neg}\left(b \cdot b\right)\right)}{\mathsf{neg}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)}\right), \mathsf{*.f64}\left(\color{blue}{3}, a\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right)\right)\right) - \left(\mathsf{neg}\left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{3}, a\right)\right) \]

   6. Applied egg-rr 84.8%

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

   if 0.320000000000000007 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 92.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.32:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right) - b \cdot b}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{\frac{b \cdot b + \left(t\_0 - b \cdot b\right)}{b + \sqrt{b \cdot b + t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (if (<= b 0.28)
     (/ (/ (+ (* b b) (- t_0 (* b b))) (+ b (sqrt (+ (* b b) t_0)))) (* 3.0 a))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (/
        (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
        (* b (* b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	double tmp;
	if (b <= 0.28) {
		tmp = (((b * b) + (t_0 - (b * b))) / (b + sqrt(((b * b) + t_0)))) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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 = c * (a * (-3.0d0))
    if (b <= 0.28d0) then
        tmp = (((b * b) + (t_0 - (b * b))) / (b + sqrt(((b * b) + t_0)))) / (3.0d0 * a)
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	double tmp;
	if (b <= 0.28) {
		tmp = (((b * b) + (t_0 - (b * b))) / (b + Math.sqrt(((b * b) + t_0)))) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * -3.0)
	tmp = 0
	if b <= 0.28:
		tmp = (((b * b) + (t_0 - (b * b))) / (b + math.sqrt(((b * b) + t_0)))) / (3.0 * a)
	else:
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	tmp = 0.0
	if (b <= 0.28)
		tmp = Float64(Float64(Float64(Float64(b * b) + Float64(t_0 - Float64(b * b))) / Float64(b + sqrt(Float64(Float64(b * b) + t_0)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * -3.0);
	tmp = 0.0;
	if (b <= 0.28)
		tmp = (((b * b) + (t_0 - (b * b))) / (b + sqrt(((b * b) + t_0)))) / (3.0 * a);
	else
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.28], N[(N[(N[(N[(b * b), $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28000000000000003

   1. Initial program 83.4%

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

   4. Applied egg-rr 83.8%

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

      1. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right) - b \cdot b}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}\right), \mathsf{*.f64}\left(\color{blue}{3}, a\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right) - b \cdot b\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(\color{blue}{3}, a\right)\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b + \left(a \cdot \left(-3 \cdot c\right) - b \cdot b\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(-3 \cdot c\right) - b \cdot b\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(-3 \cdot c\right) - b \cdot b\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(a \cdot \left(-3 \cdot c\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\left(-3 \cdot c\right) \cdot a\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(\left(c \cdot -3\right) \cdot a\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(c \cdot \left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \left(b \cdot b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(b, \left(\sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

   6. Applied egg-rr 84.6%

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

   if 0.28000000000000003 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 92.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{\frac{b \cdot b + \left(c \cdot \left(a \cdot -3\right) - b \cdot b\right)}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{b \cdot b + \left(t\_0 - b \cdot b\right)}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{b \cdot b + t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (if (<= b 0.28)
     (/ (+ (* b b) (- t_0 (* b b))) (* (* 3.0 a) (+ b (sqrt (+ (* b b) t_0)))))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (/
        (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
        (* b (* b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	double tmp;
	if (b <= 0.28) {
		tmp = ((b * b) + (t_0 - (b * b))) / ((3.0 * a) * (b + sqrt(((b * b) + t_0))));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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 = c * (a * (-3.0d0))
    if (b <= 0.28d0) then
        tmp = ((b * b) + (t_0 - (b * b))) / ((3.0d0 * a) * (b + sqrt(((b * b) + t_0))))
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	double tmp;
	if (b <= 0.28) {
		tmp = ((b * b) + (t_0 - (b * b))) / ((3.0 * a) * (b + Math.sqrt(((b * b) + t_0))));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * -3.0)
	tmp = 0
	if b <= 0.28:
		tmp = ((b * b) + (t_0 - (b * b))) / ((3.0 * a) * (b + math.sqrt(((b * b) + t_0))))
	else:
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	tmp = 0.0
	if (b <= 0.28)
		tmp = Float64(Float64(Float64(b * b) + Float64(t_0 - Float64(b * b))) / Float64(Float64(3.0 * a) * Float64(b + sqrt(Float64(Float64(b * b) + t_0)))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * -3.0);
	tmp = 0.0;
	if (b <= 0.28)
		tmp = ((b * b) + (t_0 - (b * b))) / ((3.0 * a) * (b + sqrt(((b * b) + t_0))));
	else
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.28], N[(N[(N[(b * b), $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28000000000000003

   1. Initial program 83.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right), \color{blue}{\left(\frac{b}{3 \cdot a}\right)}\right) \]

   4. Applied egg-rr 82.3%

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

      1. sub-div N/A

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

      2. flip-- N/A

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

      3. rem-square-sqrt N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right) - b \cdot b\right), \color{blue}{\left(\left(3 \cdot a\right) \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)}\right) \]

   6. Applied egg-rr 84.6%

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

   if 0.28000000000000003 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 92.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{b \cdot b + \left(c \cdot \left(a \cdot -3\right) - b \cdot b\right)}{\left(3 \cdot a\right) \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{\left(b \cdot b + \left(t\_0 - b \cdot b\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{b \cdot b + t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (if (<= b 0.28)
     (/
      (* (+ (* b b) (- t_0 (* b b))) (/ 0.3333333333333333 a))
      (+ b (sqrt (+ (* b b) t_0))))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (/
        (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
        (* b (* b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	double tmp;
	if (b <= 0.28) {
		tmp = (((b * b) + (t_0 - (b * b))) * (0.3333333333333333 / a)) / (b + sqrt(((b * b) + t_0)));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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 = c * (a * (-3.0d0))
    if (b <= 0.28d0) then
        tmp = (((b * b) + (t_0 - (b * b))) * (0.3333333333333333d0 / a)) / (b + sqrt(((b * b) + t_0)))
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	double tmp;
	if (b <= 0.28) {
		tmp = (((b * b) + (t_0 - (b * b))) * (0.3333333333333333 / a)) / (b + Math.sqrt(((b * b) + t_0)));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * -3.0)
	tmp = 0
	if b <= 0.28:
		tmp = (((b * b) + (t_0 - (b * b))) * (0.3333333333333333 / a)) / (b + math.sqrt(((b * b) + t_0)))
	else:
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	tmp = 0.0
	if (b <= 0.28)
		tmp = Float64(Float64(Float64(Float64(b * b) + Float64(t_0 - Float64(b * b))) * Float64(0.3333333333333333 / a)) / Float64(b + sqrt(Float64(Float64(b * b) + t_0))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * -3.0);
	tmp = 0.0;
	if (b <= 0.28)
		tmp = (((b * b) + (t_0 - (b * b))) * (0.3333333333333333 / a)) / (b + sqrt(((b * b) + t_0)));
	else
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.28], N[(N[(N[(N[(b * b), $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28000000000000003

   1. Initial program 83.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right), \color{blue}{\left(\frac{b}{3 \cdot a}\right)}\right) \]

   4. Applied egg-rr 82.3%

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

      1. sub-div N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. rem-square-sqrt N/A

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

      5. +-commutative N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right) - b \cdot b\right) \cdot \frac{1}{3 \cdot a}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied egg-rr 84.5%

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

   if 0.28000000000000003 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 92.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{\left(b \cdot b + \left(c \cdot \left(a \cdot -3\right) - b \cdot b\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.28)
   (/ (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) b) 3.0) a)
   (+
    (/ (* c -0.5) b)
    (*
     a
     (/
      (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
      (* b (* b b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.28) {
		tmp = ((sqrt(((b * b) + (c * (a * -3.0)))) - b) / 3.0) / a;
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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 <= 0.28d0) then
        tmp = ((sqrt(((b * b) + (c * (a * (-3.0d0))))) - b) / 3.0d0) / a
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.28) {
		tmp = ((Math.sqrt(((b * b) + (c * (a * -3.0)))) - b) / 3.0) / a;
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.28:
		tmp = ((math.sqrt(((b * b) + (c * (a * -3.0)))) - b) / 3.0) / a
	else:
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.28)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) / 3.0) / a);
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.28)
		tmp = ((sqrt(((b * b) + (c * (a * -3.0)))) - b) / 3.0) / a;
	else
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.28], N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28000000000000003

   1. Initial program 83.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right), \color{blue}{\left(\frac{b}{3 \cdot a}\right)}\right) \]

   4. Applied egg-rr 82.3%

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

      1. sub-div N/A

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

      2. flip-- N/A

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

      3. rem-square-sqrt N/A

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

      4. +-commutative N/A

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

      5. sub-div N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{b \cdot b + a \cdot \left(-3 \cdot c\right)}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}} - \frac{b \cdot b}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{3}\right), \color{blue}{a}\right) \]

   6. Applied egg-rr 83.5%

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

   if 0.28000000000000003 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 92.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.28:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.32:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.32)
   (* (/ -0.3333333333333333 a) (- b (sqrt (+ (* b b) (* a (* c -3.0))))))
   (+
    (/ (* c -0.5) b)
    (*
     a
     (/
      (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
      (* b (* b b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.32) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(((b * b) + (a * (c * -3.0)))));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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 <= 0.32d0) then
        tmp = ((-0.3333333333333333d0) / a) * (b - sqrt(((b * b) + (a * (c * (-3.0d0))))))
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.32) {
		tmp = (-0.3333333333333333 / a) * (b - Math.sqrt(((b * b) + (a * (c * -3.0)))));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.32:
		tmp = (-0.3333333333333333 / a) * (b - math.sqrt(((b * b) + (a * (c * -3.0)))))
	else:
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.32)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0))))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.32)
		tmp = (-0.3333333333333333 / a) * (b - sqrt(((b * b) + (a * (c * -3.0)))));
	else
		tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.32], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.320000000000000007

   1. Initial program 83.4%

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

   4. Applied egg-rr 83.4%

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

   if 0.320000000000000007 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 92.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.32:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (*
   a
   (/
    (+ (/ (* (* a -0.5625) (* c (* c c))) (* b b)) (* -0.375 (* c c)))
    (* b (* b b))))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * 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) + (a * (((((a * (-0.5625d0)) * (c * (c * c))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(a * -0.5625) * Float64(c * Float64(c * c))) / Float64(b * b)) + Float64(-0.375 * Float64(c * c))) / Float64(b * Float64(b * b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * (((((a * -0.5625) * (c * (c * c))) / (b * b)) + (-0.375 * (c * c))) / (b * (b * b))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(a * -0.5625), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Simplified 91.4%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   18. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

   21. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

   22. *-lowering-*.f64 88.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

8. Simplified 88.6%

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

9. Final simplification 88.6%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(a \cdot -0.5625\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} \]
10. Add Preprocessing

Alternative 10: 87.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right) + \frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -0.5625\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)} + \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (/
    (+ (* -0.375 (* a c)) (/ (* (* c c) (* (* a a) -0.5625)) (* b b)))
    (* b (* b b)))
   (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((((-0.375 * (a * c)) + (((c * c) * ((a * a) * -0.5625)) / (b * b))) / (b * (b * b))) + (-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 * (((((-0.375d0) * (a * c)) + (((c * c) * ((a * a) * (-0.5625d0))) / (b * b))) / (b * (b * b))) + ((-0.5d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((((-0.375 * (a * c)) + (((c * c) * ((a * a) * -0.5625)) / (b * b))) / (b * (b * b))) + (-0.5 / b));
}

Python

def code(a, b, c):
	return c * ((((-0.375 * (a * c)) + (((c * c) * ((a * a) * -0.5625)) / (b * b))) / (b * (b * b))) + (-0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(Float64(-0.375 * Float64(a * c)) + Float64(Float64(Float64(c * c) * Float64(Float64(a * a) * -0.5625)) / Float64(b * b))) / Float64(b * Float64(b * b))) + Float64(-0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((((-0.375 * (a * c)) + (((c * c) * ((a * a) * -0.5625)) / (b * b))) / (b * (b * b))) + (-0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(N[(-0.375 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)\right) \]

5. Simplified 88.4%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}, \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot c\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{-1}{2}}, b\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot c\right) + \frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot c\right) \cdot \frac{-3}{8}\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), \frac{-3}{8}\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), \frac{-3}{8}\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot {c}^{2}\right)}{{b}^{2}}\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot {a}^{2}\right) \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot {a}^{2}\right), \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \left({a}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \left(a \cdot a\right)\right), \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \left(c \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{2}\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot b\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left({b}^{3}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   19. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(b \cdot \left(b \cdot b\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(b \cdot {b}^{2}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(b, \left({b}^{2}\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]

8. Simplified 88.4%

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

9. Final simplification 88.4%

   \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right) + \frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -0.5625\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)} + \frac{-0.5}{b}\right) \]
10. Add Preprocessing

Alternative 11: 81.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (/ (* c -0.5) b) (* a (/ (* -0.375 (* c (/ c (* b b)))) b))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((-0.375 * (c * (c / (b * b)))) / 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) + (a * (((-0.375d0) * (c * (c / (b * b)))) / b))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((-0.375 * (c * (c / (b * b)))) / b));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * ((-0.375 * (c * (c / (b * b)))) / b))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(-0.375 * Float64(c * Float64(c / Float64(b * b)))) / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * ((-0.375 * (c * (c / (b * b)))) / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \color{blue}{\frac{-3}{8}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + \left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right) \cdot \frac{-3}{8} \]

   3. associate-*r* N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + a \cdot \color{blue}{\left(\frac{{c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \color{blue}{\frac{{c}^{2}}{{b}^{3}}}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{c}{b}\right), \color{blue}{\left(a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot c}{b}\right), \left(\color{blue}{a} \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), b\right), \left(\color{blue}{a} \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right), \left(a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)}\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{-3}{8} \cdot {c}^{2}}{\color{blue}{{b}^{3}}}\right)\right)\right) \]

   12. unpow3 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{-3}{8} \cdot {c}^{2}}{\left(b \cdot b\right) \cdot \color{blue}{b}}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{2} \cdot b}\right)\right)\right) \]

   14. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \left(\frac{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}}{\color{blue}{b}}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}\right), \color{blue}{b}\right)\right)\right) \]

5. Simplified 82.4%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b}} \]
6. Add Preprocessing

Alternative 12: 81.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5 + a \cdot \left(-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (* c -0.5) (* a (* -0.375 (* c (/ c (* b b)))))) b))

C

double code(double a, double b, double c) {
	return ((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) / 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)) + (a * ((-0.375d0) * (c * (c / (b * b)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) / b;
}

Python

def code(a, b, c):
	return ((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) + Float64(a * Float64(-0.375 * Float64(c * Float64(c / Float64(b * b)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] + N[(a * N[(-0.375 * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right), \color{blue}{b}\right) \]

5. Simplified 82.4%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5 + a \cdot \left(-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)\right)}{b}} \]
6. Add Preprocessing

Alternative 13: 81.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-0.5}{b} + \frac{c \cdot \left(a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (+ (/ -0.5 b) (/ (* c (* a -0.375)) (* b (* b b))))))

C

double code(double a, double b, double c) {
	return c * ((-0.5 / b) + ((c * (a * -0.375)) / (b * (b * 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) + ((c * (a * (-0.375d0))) / (b * (b * b))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.5 / b) + ((c * (a * -0.375)) / (b * (b * b))));
}

Python

def code(a, b, c):
	return c * ((-0.5 / b) + ((c * (a * -0.375)) / (b * (b * b))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.5 / b) + Float64(Float64(c * Float64(a * -0.375)) / Float64(b * Float64(b * b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.5 / b) + ((c * (a * -0.375)) / (b * (b * b))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.5 / b), $MachinePrecision] + N[(N[(c * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

      \[\leadsto c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

   2. associate-*r/ N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto c \cdot \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{b}\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{b}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) + \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right), \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b}\right)\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b}\right)\right), \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{\color{blue}{{b}^{3}}}\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b}}^{3}}\right)\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

5. Simplified 82.2%

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

6. Final simplification 82.2%

   \[\leadsto c \cdot \left(\frac{-0.5}{b} + \frac{c \cdot \left(a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right) \]
7. Add Preprocessing

Alternative 14: 81.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5 + -0.375 \cdot \left(a \cdot \frac{c}{b \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (+ -0.5 (* -0.375 (* a (/ c (* b b))))) b)))

C

double code(double a, double b, double c) {
	return c * ((-0.5 + (-0.375 * (a * (c / (b * b))))) / 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) + ((-0.375d0) * (a * (c / (b * b))))) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.5 + (-0.375 * (a * (c / (b * b))))) / b);
}

Python

def code(a, b, c):
	return c * ((-0.5 + (-0.375 * (a * (c / (b * b))))) / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.5 + Float64(-0.375 * Float64(a * Float64(c / Float64(b * b))))) / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.5 + (-0.375 * (a * (c / (b * b))))) / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.5 + N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)\right) \]

5. Simplified 88.4%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}\right)}\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right), \color{blue}{b}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), b\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{-1}{2}\right), b\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}\right), \frac{-1}{2}\right), b\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \frac{c}{{b}^{2}}\right), \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{{b}^{2}}\right)\right), \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right), \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right), \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

   11. *-lowering-*.f64 82.2%

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \frac{-3}{8}\right), \frac{-1}{2}\right), b\right)\right) \]

8. Simplified 82.2%

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

9. Final simplification 82.2%

   \[\leadsto c \cdot \frac{-0.5 + -0.375 \cdot \left(a \cdot \frac{c}{b \cdot b}\right)}{b} \]
10. Add Preprocessing

Alternative 15: 64.2% 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 54.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 64.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

5. Simplified 64.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Add Preprocessing

Alternative 16: 64.1% 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 54.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 64.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

5. Simplified 64.9%

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

   1. associate-/l* N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-1}{2}}{b}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{2}}{b} \cdot \color{blue}{c} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \color{blue}{c}\right) \]

   4. /-lowering-/.f64 64.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

7. Applied egg-rr 64.8%

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

8. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2024185 
(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)))