Cubic critical, narrow range

Percentage Accurate: 55.1% → 92.0%

Time: 16.2s

Alternatives: 13

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)\\ \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{2}, t\_0, -{b}^{2}\right)}{b + \sqrt{{b}^{2} \cdot t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a (* c (pow b -2.0))) 1.0)))
   (if (<= b 0.095)
     (/
      (/
       (fma (pow b 2.0) t_0 (- (pow b 2.0)))
       (+ b (sqrt (* (pow b 2.0) t_0))))
      (* a 3.0))
     (/
      (+
       (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
       (+
        (* c -0.5)
        (+
         (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
         (/
          (* -0.16666666666666666 (* (pow (* a c) 4.0) 6.328125))
          (* a (pow b 6.0))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (a * (c * pow(b, -2.0))), 1.0);
	double tmp;
	if (b <= 0.095) {
		tmp = (fma(pow(b, 2.0), t_0, -pow(b, 2.0)) / (b + sqrt((pow(b, 2.0) * t_0)))) / (a * 3.0);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + ((-0.16666666666666666 * (pow((a * c), 4.0) * 6.328125)) / (a * pow(b, 6.0)))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)
	tmp = 0.0
	if (b <= 0.095)
		tmp = Float64(Float64(fma((b ^ 2.0), t_0, Float64(-(b ^ 2.0))) / Float64(b + sqrt(Float64((b ^ 2.0) * t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * 6.328125)) / Float64(a * (b ^ 6.0)))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, 0.095], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.095000000000000001

   1. Initial program 84.3%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in b around inf 84.2%

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

      1. flip-- 85.0%

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

      2. add-sqr-sqrt 86.4%

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

      3. +-commutative 86.4%

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

      4. fma-define 86.2%

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

      5. div-inv 86.4%

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

      6. pow-flip 86.3%

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

      7. metadata-eval 86.3%

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

      8. unpow2 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. fma-neg 86.3%

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

      2. associate-*l* 86.3%

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

      3. +-commutative 86.3%

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

      4. associate-*l* 86.3%

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

   9. Simplified 86.3%

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

   if 0.095000000000000001 < b

   1. Initial program 48.8%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 94.6%

      \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]

      2. distribute-rgt-out 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{6}}\right)\right)}{b} \]

      3. pow-prod-down 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

      4. metadata-eval 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   7. Applied egg-rr 94.6%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{2}, \mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right), -{b}^{2}\right)}{b + \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} \cdot \mathsf{fma}\left(-3, {b}^{-2} \cdot \left(a \cdot c\right), 1\right)\\ \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow b 2.0) (fma -3.0 (* (pow b -2.0) (* a c)) 1.0))))
   (if (<= b 0.095)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (/
      (+
       (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
       (+
        (* c -0.5)
        (+
         (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
         (/
          (* -0.16666666666666666 (* (pow (* a c) 4.0) 6.328125))
          (* a (pow b 6.0))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) * fma(-3.0, (pow(b, -2.0) * (a * c)), 1.0);
	double tmp;
	if (b <= 0.095) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + ((-0.16666666666666666 * (pow((a * c), 4.0) * 6.328125)) / (a * pow(b, 6.0)))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) * fma(-3.0, Float64((b ^ -2.0) * Float64(a * c)), 1.0))
	tmp = 0.0
	if (b <= 0.095)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * 6.328125)) / Float64(a * (b ^ 6.0)))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] * N[(-3.0 * N[(N[Power[b, -2.0], $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.095], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.095000000000000001

   1. Initial program 84.3%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in b around inf 84.2%

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

      1. flip-- 85.0%

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

      2. add-sqr-sqrt 86.4%

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

      3. +-commutative 86.4%

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

      4. fma-define 86.2%

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

      5. div-inv 86.4%

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

      6. pow-flip 86.3%

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

      7. metadata-eval 86.3%

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

      8. unpow2 86.3%

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

   7. Applied egg-rr 86.3%

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

   if 0.095000000000000001 < b

   1. Initial program 48.8%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 94.6%

      \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]

      2. distribute-rgt-out 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{6}}\right)\right)}{b} \]

      3. pow-prod-down 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

      4. metadata-eval 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   7. Applied egg-rr 94.6%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\frac{{b}^{2} \cdot \mathsf{fma}\left(-3, {b}^{-2} \cdot \left(a \cdot c\right), 1\right) - {b}^{2}}{b + \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-3, {b}^{-2} \cdot \left(a \cdot c\right), 1\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.095)
   (/
    (fma (fabs b) (sqrt (fma -3.0 (* a (* c (pow b -2.0))) 1.0)) (- b))
    (* a 3.0))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+
      (* c -0.5)
      (+
       (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
       (/
        (* -0.16666666666666666 (* (pow (* a c) 4.0) 6.328125))
        (* a (pow b 6.0))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.095) {
		tmp = fma(fabs(b), sqrt(fma(-3.0, (a * (c * pow(b, -2.0))), 1.0)), -b) / (a * 3.0);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + ((-0.16666666666666666 * (pow((a * c), 4.0) * 6.328125)) / (a * pow(b, 6.0)))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.095)
		tmp = Float64(fma(abs(b), sqrt(fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)), Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * 6.328125)) / Float64(a * (b ^ 6.0)))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.095], N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.095000000000000001

   1. Initial program 84.3%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in b around inf 84.2%

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

      1. sqrt-prod 84.3%

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

      2. fma-neg 85.1%

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

      3. +-commutative 85.1%

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

      4. fma-define 85.1%

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

      5. div-inv 85.1%

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

      6. pow-flip 85.1%

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

      7. metadata-eval 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. unpow2 85.1%

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

      2. rem-sqrt-square 85.1%

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

      3. associate-*l* 85.1%

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

   9. Simplified 85.1%

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

   if 0.095000000000000001 < b

   1. Initial program 48.8%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 94.6%

      \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]

      2. distribute-rgt-out 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{6}}\right)\right)}{b} \]

      3. pow-prod-down 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

      4. metadata-eval 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   7. Applied egg-rr 94.6%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}} + -0.375 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.095)
   (/
    (fma (fabs b) (sqrt (fma -3.0 (* a (* c (pow b -2.0))) 1.0)) (- b))
    (* a 3.0))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+
      (* c -0.5)
      (+
       (/
        (* -0.16666666666666666 (* (pow (* a c) 4.0) 6.328125))
        (* a (pow b 6.0)))
       (* -0.375 (* a (* (/ c b) (/ c b)))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.095) {
		tmp = fma(fabs(b), sqrt(fma(-3.0, (a * (c * pow(b, -2.0))), 1.0)), -b) / (a * 3.0);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + (((-0.16666666666666666 * (pow((a * c), 4.0) * 6.328125)) / (a * pow(b, 6.0))) + (-0.375 * (a * ((c / b) * (c / b))))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.095)
		tmp = Float64(fma(abs(b), sqrt(fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)), Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * 6.328125)) / Float64(a * (b ^ 6.0))) + Float64(-0.375 * Float64(a * Float64(Float64(c / b) * Float64(c / b))))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.095], N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.095000000000000001

   1. Initial program 84.3%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in b around inf 84.2%

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

      1. sqrt-prod 84.3%

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

      2. fma-neg 85.1%

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

      3. +-commutative 85.1%

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

      4. fma-define 85.1%

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

      5. div-inv 85.1%

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

      6. pow-flip 85.1%

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

      7. metadata-eval 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. unpow2 85.1%

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

      2. rem-sqrt-square 85.1%

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

      3. associate-*l* 85.1%

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

   9. Simplified 85.1%

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

   if 0.095000000000000001 < b

   1. Initial program 48.8%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 94.6%

      \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]

      2. distribute-rgt-out 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{6}}\right)\right)}{b} \]

      3. pow-prod-down 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

      4. metadata-eval 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   7. Applied egg-rr 94.6%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]
   8. Step-by-step derivation

      1. associate-/l* 94.6%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   9. Applied egg-rr 94.6%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]
   10. Step-by-step derivation

       1. unpow2 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

       2. unpow2 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

       3. times-frac 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

       4. unpow1 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

       5. pow-plus 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

       6. metadata-eval 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   11. Simplified 94.6%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]
   12. Step-by-step derivation

       1. unpow2 94.6%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   13. Applied egg-rr 94.6%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}} + -0.375 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.135)
   (/
    (fma (fabs b) (sqrt (fma -3.0 (* a (* c (pow b -2.0))) 1.0)) (- b))
    (* a 3.0))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
      (* -0.375 (/ (pow c 2.0) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.135) {
		tmp = fma(fabs(b), sqrt(fma(-3.0, (a * (c * pow(b, -2.0))), 1.0)), -b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.135)
		tmp = Float64(fma(abs(b), sqrt(fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)), Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.135], N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.13500000000000001

   1. Initial program 84.2%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in b around inf 84.1%

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

      1. sqrt-prod 84.2%

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

      2. fma-neg 84.9%

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

      3. +-commutative 84.9%

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

      4. fma-define 84.9%

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

      5. div-inv 84.9%

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

      6. pow-flip 84.9%

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

      7. metadata-eval 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. unpow2 84.9%

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

      2. rem-sqrt-square 84.9%

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

      3. associate-*l* 84.9%

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

   9. Simplified 84.9%

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

   if 0.13500000000000001 < b

   1. Initial program 48.5%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in a around 0 91.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)}, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.215)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
      (* -0.375 (/ (pow c 2.0) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.215) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.215)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.215], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.214999999999999997

   1. Initial program 84.2%

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

   3. Simplified 84.3%

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

   if 0.214999999999999997 < b

   1. Initial program 48.5%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in a around 0 91.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (/ (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.02d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = ((c * (-0.5d0)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0)))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.5) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0)))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = ((c * -0.5) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 2.0)))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = ((c * -0.5) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 2.0)))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 82.5%

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

      1. sqr-neg 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-*l* 82.5%

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

   3. Simplified 82.5%

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

   if -0.0200000000000000004 < (/.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 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in b around inf 88.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.135:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.135)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
       (* -0.375 (/ a (pow b 3.0)))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.135) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.135)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.135], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.13500000000000001

   1. Initial program 84.2%

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

   3. Simplified 84.3%

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

   if 0.13500000000000001 < b

   1. Initial program 48.5%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in c around 0 91.7%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.135:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot -0.375, {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (/ (fma (* a -0.375) (pow (/ c b) 2.0) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = fma((a * -0.375), pow((c / b), 2.0), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(Float64(a * -0.375), (Float64(c / b) ^ 2.0), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * -0.375), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 82.5%

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

      1. sqr-neg 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-*l* 82.5%

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

   3. Simplified 82.5%

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

   if -0.0200000000000000004 < (/.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 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in a around 0 88.5%

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

   6. Taylor expanded in b around inf 88.5%

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

      1. +-commutative 88.5%

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

      2. associate-*r/ 88.5%

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

      3. associate-*r* 88.5%

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

      4. fma-define 88.5%

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

      5. unpow2 88.5%

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

      6. unpow2 88.5%

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

      7. times-frac 88.5%

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

      8. unpow1 88.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375 \cdot a, \color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}, -0.5 \cdot c\right)}{b} \]

      9. pow-plus 88.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375 \cdot a, \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, -0.5 \cdot c\right)}{b} \]

      10. metadata-eval 88.5%

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

   8. Simplified 88.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot -0.375, {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.02d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 82.5%

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

      1. sqr-neg 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-*l* 82.5%

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

   3. Simplified 82.5%

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

   if -0.0200000000000000004 < (/.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 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in c around 0 88.4%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 88.4%

         \[\leadsto c \cdot \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

      2. associate-*r/ 88.4%

         \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

      3. metadata-eval 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (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 / (b ** 3.0d0)))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
end

Wolfram

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

Derivation

1. Initial program 53.2%

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

3. Simplified 53.3%

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

5. Taylor expanded in c around 0 82.3%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-/l* 82.3%

      \[\leadsto c \cdot \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r/ 82.3%

      \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   3. metadata-eval 82.3%

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

7. Simplified 82.3%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)} \]
8. Add Preprocessing

Alternative 12: 64.6% 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 53.2%

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

3. Simplified 53.3%

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

5. Taylor expanded in b around inf 65.9%

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

   1. associate-*r/ 65.9%

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

   2. *-commutative 65.9%

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

7. Simplified 65.9%

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

Alternative 13: 64.5% 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 53.2%

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

3. Simplified 53.3%

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

5. Taylor expanded in b around inf 65.9%

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

   1. associate-*r/ 65.9%

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

   2. *-commutative 65.9%

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

7. Simplified 65.9%

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

   1. associate-/l* 65.8%

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

9. Applied egg-rr 65.8%

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

Reproduce

herbie shell --seed 2024145 
(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)))