Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.1%

Time: 17.3s

Alternatives: 11

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: 55.2% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left({b}^{2} - {\left(-b\right)}^{2}\right) - 3 \cdot \left(a \cdot c\right)}{b + \sqrt{{b}^{2} - a \cdot \left(3 \cdot c\right)}}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (- (- (pow b 2.0) (pow (- b) 2.0)) (* 3.0 (* a c)))
   (+ b (sqrt (- (pow b 2.0) (* a (* 3.0 c))))))
  (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (((pow(b, 2.0) - pow(-b, 2.0)) - (3.0 * (a * c))) / (b + sqrt((pow(b, 2.0) - (a * (3.0 * 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 ** 2.0d0) - (-b ** 2.0d0)) - (3.0d0 * (a * c))) / (b + sqrt(((b ** 2.0d0) - (a * (3.0d0 * c)))))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (((Math.pow(b, 2.0) - Math.pow(-b, 2.0)) - (3.0 * (a * c))) / (b + Math.sqrt((Math.pow(b, 2.0) - (a * (3.0 * c)))))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (((math.pow(b, 2.0) - math.pow(-b, 2.0)) - (3.0 * (a * c))) / (b + math.sqrt((math.pow(b, 2.0) - (a * (3.0 * c)))))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((b ^ 2.0) - (Float64(-b) ^ 2.0)) - Float64(3.0 * Float64(a * c))) / Float64(b + sqrt(Float64((b ^ 2.0) - Float64(a * Float64(3.0 * c)))))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((b ^ 2.0) - (-b ^ 2.0)) - (3.0 * (a * c))) / (b + sqrt(((b ^ 2.0) - (a * (3.0 * c)))))) / (3.0 * a);
end

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. sqr-neg 54.4%

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

   2. sqr-neg 54.4%

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

   3. associate-*l* 54.4%

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

3. Simplified 54.4%

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

   1. add-cube-cbrt 54.4%

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

   2. pow3 54.4%

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

6. Applied egg-rr 54.4%

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

   1. flip-+ 54.3%

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

   2. pow2 54.3%

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

   3. add-sqr-sqrt 56.0%

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

   4. pow2 56.0%

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

   5. unpow3 56.0%

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

   6. add-cube-cbrt 56.1%

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

   7. associate-*r* 56.1%

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

   8. *-commutative 56.1%

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

   9. pow2 56.1%

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

   10. unpow3 56.1%

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

   11. add-cube-cbrt 56.1%

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

   12. associate-*r* 56.1%

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

8. Applied egg-rr 56.1%

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

   1. associate--r- 99.3%

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

   2. associate-*l* 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-*l* 99.0%

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

   5. *-commutative 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in a around 0 99.1%

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

12. Final simplification 99.1%

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

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.08)
   (* (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 0.3333333333333333 (/ 1.0 a)))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) * (0.3333333333333333 * (1.0 / a));
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) * Float64(0.3333333333333333 * Float64(1.0 / a)));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0800000000000000017

   1. Initial program 81.2%

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

      1. sqr-neg 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-*l* 81.2%

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

   3. Simplified 81.2%

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

      1. add-cube-cbrt 81.1%

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

      2. pow3 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. div-inv 81.2%

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

      2. neg-mul-1 81.2%

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

      3. fma-define 81.2%

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

      4. pow2 81.2%

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

      5. unpow3 81.2%

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

      6. add-cube-cbrt 81.2%

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

      7. associate-*r* 81.3%

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

      8. *-commutative 81.3%

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

      9. *-commutative 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. *-commutative 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.2%

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

      4. metadata-eval 81.2%

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

      5. fma-undefine 81.2%

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

      6. neg-mul-1 81.2%

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

      7. +-commutative 81.2%

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

      8. unsub-neg 81.2%

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

      9. unpow2 81.2%

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

      10. fma-neg 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. distribute-rgt-neg-in 81.6%

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

      14. metadata-eval 81.6%

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

   10. Simplified 81.6%

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

       1. div-inv 81.6%

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

   12. Applied egg-rr 81.6%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.2%

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

      1. sqr-neg 48.2%

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

      2. sqr-neg 48.2%

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

      3. associate-*l* 48.2%

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

   3. Simplified 48.2%

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

      1. add-cube-cbrt 48.3%

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

      2. pow3 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in b around inf 87.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right) + -1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right) + -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      2. fma-define 87.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, {1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}}{3 \cdot a} \]

      3. pow-base-1 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{1} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      4. *-lft-identity 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      5. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      6. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      7. swap-sqr 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      8. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      9. pow-base-1 87.8%

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

      10. associate-*r* 87.8%

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

      11. metadata-eval 87.8%

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

      12. associate-*r/ 87.8%

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

      13. associate-*r* 87.9%

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

      14. *-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. clear-num 87.8%

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

       2. inv-pow 87.8%

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

       3. *-commutative 87.8%

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

       4. div-inv 87.8%

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

       5. pow-flip 87.8%

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

       6. metadata-eval 87.8%

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

       7. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. unpow-1 87.9%

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

       2. *-commutative 87.9%

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

       3. associate-*l* 87.7%

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

   13. Simplified 87.7%

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

   14. Taylor expanded in a around 0 88.5%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.08)
   (* (/ 1.0 (/ a 0.3333333333333333)) (- (sqrt (fma b b (* c (* a -3.0)))) b))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08) {
		tmp = (1.0 / (a / 0.3333333333333333)) * (sqrt(fma(b, b, (c * (a * -3.0)))) - b);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(1.0 / Float64(a / 0.3333333333333333)) * Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0800000000000000017

   1. Initial program 81.2%

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

      1. sqr-neg 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-*l* 81.2%

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

   3. Simplified 81.2%

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

      1. add-cube-cbrt 81.1%

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

      2. pow3 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. div-inv 81.2%

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

      2. neg-mul-1 81.2%

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

      3. fma-define 81.2%

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

      4. pow2 81.2%

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

      5. unpow3 81.2%

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

      6. add-cube-cbrt 81.2%

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

      7. associate-*r* 81.3%

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

      8. *-commutative 81.3%

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

      9. *-commutative 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. *-commutative 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.2%

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

      4. metadata-eval 81.2%

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

      5. fma-undefine 81.2%

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

      6. neg-mul-1 81.2%

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

      7. +-commutative 81.2%

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

      8. unsub-neg 81.2%

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

      9. unpow2 81.2%

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

      10. fma-neg 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. distribute-rgt-neg-in 81.6%

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

      14. metadata-eval 81.6%

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

   10. Simplified 81.6%

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

       1. clear-num 81.7%

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

       2. inv-pow 81.7%

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

   12. Applied egg-rr 81.7%

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

       1. unpow-1 81.7%

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

   14. Simplified 81.7%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.2%

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

      1. sqr-neg 48.2%

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

      2. sqr-neg 48.2%

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

      3. associate-*l* 48.2%

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

   3. Simplified 48.2%

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

      1. add-cube-cbrt 48.3%

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

      2. pow3 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in b around inf 87.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right) + -1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right) + -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      2. fma-define 87.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, {1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}}{3 \cdot a} \]

      3. pow-base-1 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{1} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      4. *-lft-identity 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      5. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      6. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      7. swap-sqr 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      8. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      9. pow-base-1 87.8%

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

      10. associate-*r* 87.8%

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

      11. metadata-eval 87.8%

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

      12. associate-*r/ 87.8%

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

      13. associate-*r* 87.9%

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

      14. *-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. clear-num 87.8%

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

       2. inv-pow 87.8%

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

       3. *-commutative 87.8%

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

       4. div-inv 87.8%

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

       5. pow-flip 87.8%

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

       6. metadata-eval 87.8%

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

       7. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. unpow-1 87.9%

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

       2. *-commutative 87.9%

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

       3. associate-*l* 87.7%

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

   13. Simplified 87.7%

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

   14. Taylor expanded in a around 0 88.5%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333}} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.08)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma b b (* c (* a -3.0)))) b))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma(b, b, (c * (a * -3.0)))) - b);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0800000000000000017

   1. Initial program 81.2%

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

      1. sqr-neg 81.2%

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

      2. sqr-neg 81.2%

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

      3. associate-*l* 81.2%

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

   3. Simplified 81.2%

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

      1. add-cube-cbrt 81.1%

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

      2. pow3 81.1%

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

   6. Applied egg-rr 81.1%

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

      1. div-inv 81.2%

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

      2. neg-mul-1 81.2%

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

      3. fma-define 81.2%

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

      4. pow2 81.2%

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

      5. unpow3 81.2%

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

      6. add-cube-cbrt 81.2%

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

      7. associate-*r* 81.3%

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

      8. *-commutative 81.3%

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

      9. *-commutative 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. *-commutative 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.2%

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

      4. metadata-eval 81.2%

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

      5. fma-undefine 81.2%

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

      6. neg-mul-1 81.2%

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

      7. +-commutative 81.2%

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

      8. unsub-neg 81.2%

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

      9. unpow2 81.2%

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

      10. fma-neg 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. distribute-rgt-neg-in 81.6%

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

      14. metadata-eval 81.6%

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

   10. Simplified 81.6%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.2%

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

      1. sqr-neg 48.2%

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

      2. sqr-neg 48.2%

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

      3. associate-*l* 48.2%

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

   3. Simplified 48.2%

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

      1. add-cube-cbrt 48.3%

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

      2. pow3 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in b around inf 87.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right) + -1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right) + -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      2. fma-define 87.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, {1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}}{3 \cdot a} \]

      3. pow-base-1 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{1} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      4. *-lft-identity 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      5. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      6. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      7. swap-sqr 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      8. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      9. pow-base-1 87.8%

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

      10. associate-*r* 87.8%

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

      11. metadata-eval 87.8%

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

      12. associate-*r/ 87.8%

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

      13. associate-*r* 87.9%

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

      14. *-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. clear-num 87.8%

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

       2. inv-pow 87.8%

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

       3. *-commutative 87.8%

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

       4. div-inv 87.8%

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

       5. pow-flip 87.8%

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

       6. metadata-eval 87.8%

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

       7. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. unpow-1 87.9%

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

       2. *-commutative 87.9%

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

       3. associate-*l* 87.7%

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

   13. Simplified 87.7%

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

   14. Taylor expanded in a around 0 88.5%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{3 \cdot a}{\mathsf{fma}\left(3, a \cdot c, 0\right)} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -1.0
  (*
   (/ (* 3.0 a) (fma 3.0 (* a c) 0.0))
   (+ b (sqrt (fma b b (* c (* a -3.0))))))))

C

double code(double a, double b, double c) {
	return -1.0 / (((3.0 * a) / fma(3.0, (a * c), 0.0)) * (b + sqrt(fma(b, b, (c * (a * -3.0))))));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / Float64(Float64(Float64(3.0 * a) / fma(3.0, Float64(a * c), 0.0)) * Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(N[(N[(3.0 * a), $MachinePrecision] / N[(3.0 * N[(a * c), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

   1. sqr-neg 54.4%

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

   2. sqr-neg 54.4%

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

   3. associate-*l* 54.4%

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

3. Simplified 54.4%

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

   1. add-cube-cbrt 54.4%

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

   2. pow3 54.4%

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

6. Applied egg-rr 54.4%

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

   1. flip-+ 54.3%

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

   2. pow2 54.3%

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

   3. add-sqr-sqrt 56.0%

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

   4. pow2 56.0%

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

   5. unpow3 56.0%

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

   6. add-cube-cbrt 56.1%

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

   7. associate-*r* 56.1%

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

   8. *-commutative 56.1%

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

   9. pow2 56.1%

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

   10. unpow3 56.1%

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

   11. add-cube-cbrt 56.1%

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

   12. associate-*r* 56.1%

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

8. Applied egg-rr 56.1%

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

   1. associate--r- 99.3%

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

   2. associate-*l* 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-*l* 99.0%

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

   5. *-commutative 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in a around 0 99.1%

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

    1. clear-num 99.0%

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

    2. inv-pow 99.0%

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

13. Applied egg-rr 99.0%

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

    1. unpow-1 99.0%

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

    2. associate-/r/ 99.0%

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

    3. *-commutative 99.0%

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

    4. *-commutative 99.0%

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

    5. +-inverses 99.0%

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

    6. cancel-sign-sub-inv 99.0%

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

15. Simplified 99.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(3, c \cdot a, 0\right)} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}} \]

16. Final simplification 99.0%

    \[\leadsto \frac{-1}{\frac{3 \cdot a}{\mathsf{fma}\left(3, a \cdot c, 0\right)} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)} \]
17. Add Preprocessing

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(3, a \cdot c, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (fma 3.0 (* a c) 0.0) (/ 0.3333333333333333 a))
  (- (- b) (sqrt (fma b b (* c (* a -3.0)))))))

C

double code(double a, double b, double c) {
	return (fma(3.0, (a * c), 0.0) * (0.3333333333333333 / a)) / (-b - sqrt(fma(b, b, (c * (a * -3.0)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(3.0, Float64(a * c), 0.0) * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(3.0 * N[(a * c), $MachinePrecision] + 0.0), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

   1. sqr-neg 54.4%

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

   2. sqr-neg 54.4%

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

   3. associate-*l* 54.4%

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

3. Simplified 54.4%

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

   1. add-cube-cbrt 54.4%

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

   2. pow3 54.4%

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

6. Applied egg-rr 54.4%

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

   1. flip-+ 54.3%

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

   2. pow2 54.3%

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

   3. add-sqr-sqrt 56.0%

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

   4. pow2 56.0%

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

   5. unpow3 56.0%

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

   6. add-cube-cbrt 56.1%

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

   7. associate-*r* 56.1%

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

   8. *-commutative 56.1%

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

   9. pow2 56.1%

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

   10. unpow3 56.1%

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

   11. add-cube-cbrt 56.1%

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

   12. associate-*r* 56.1%

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

8. Applied egg-rr 56.1%

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

   1. associate--r- 99.3%

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

   2. associate-*l* 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-*l* 99.0%

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

   5. *-commutative 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in a around 0 99.1%

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

    1. div-inv 99.0%

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

    2. +-commutative 99.0%

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

    3. fma-define 99.0%

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

    4. neg-mul-1 99.0%

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

    5. unpow-prod-down 99.0%

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

    6. metadata-eval 99.0%

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

    7. *-un-lft-identity 99.0%

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

    8. associate-*r* 99.0%

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

    9. *-commutative 99.0%

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

13. Applied egg-rr 99.0%

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

    1. associate-*l/ 99.0%

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

    2. *-commutative 99.0%

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

    3. +-inverses 99.0%

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

    4. associate-/r* 99.0%

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

    5. metadata-eval 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}} \]

    6. cancel-sign-sub-inv 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-3\right) \cdot \left(a \cdot c\right)}}} \]

    7. unpow2 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} + \left(-3\right) \cdot \left(a \cdot c\right)}} \]

    8. metadata-eval 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{b \cdot b + \color{blue}{-3} \cdot \left(a \cdot c\right)}} \]

    9. *-commutative 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot -3}}} \]

    10. *-commutative 99.0%

        \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{b \cdot b + \color{blue}{\left(c \cdot a\right)} \cdot -3}} \]

    11. associate-*r* 99.0%

        \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -3\right)}}} \]

    12. fma-undefine 99.0%

        \[\leadsto \frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}} \]

15. Simplified 99.0%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(3, c \cdot a, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}} \]

16. Final simplification 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(3, a \cdot c, 0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \]
17. Add Preprocessing

Alternative 7: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.08)
   (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (* 3.0 a))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-0.08d0)) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / (3.0d0 * a)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / 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 * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.08)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.08)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.0800000000000000017

   1. Initial program 81.2%

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

   3. Taylor expanded in a around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

      3. associate-*l* 81.3%

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

   5. Simplified 81.3%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.2%

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

      1. sqr-neg 48.2%

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

      2. sqr-neg 48.2%

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

      3. associate-*l* 48.2%

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

   3. Simplified 48.2%

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

      1. add-cube-cbrt 48.3%

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

      2. pow3 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in b around inf 87.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right) + -1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right) + -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      2. fma-define 87.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, {1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}}{3 \cdot a} \]

      3. pow-base-1 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{1} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      4. *-lft-identity 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      5. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      6. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      7. swap-sqr 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      8. unpow2 87.8%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      9. pow-base-1 87.8%

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

      10. associate-*r* 87.8%

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

      11. metadata-eval 87.8%

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

      12. associate-*r/ 87.8%

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

      13. associate-*r* 87.9%

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

      14. *-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. clear-num 87.8%

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

       2. inv-pow 87.8%

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

       3. *-commutative 87.8%

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

       4. div-inv 87.8%

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

       5. pow-flip 87.8%

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

       6. metadata-eval 87.8%

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

       7. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. unpow-1 87.9%

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

       2. *-commutative 87.9%

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

       3. associate-*l* 87.7%

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

   13. Simplified 87.7%

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

   14. Taylor expanded in a around 0 88.5%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 12.6)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.6) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 <= 12.6d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.6) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 12.6:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 12.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 12.6)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 12.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 12.5999999999999996

   1. Initial program 79.8%

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

      1. sqr-neg 79.8%

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

      2. sqr-neg 79.8%

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

      3. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

   if 12.5999999999999996 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

      1. add-cube-cbrt 47.4%

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

      2. pow3 47.4%

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf 88.1%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right) + -1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. +-commutative 88.1%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right) + -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      2. fma-define 88.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, {1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}}{3 \cdot a} \]

      3. pow-base-1 88.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{1} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      4. *-lft-identity 88.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      5. unpow2 88.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      6. unpow2 88.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      7. swap-sqr 88.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      8. unpow2 88.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

      9. pow-base-1 88.1%

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

      10. associate-*r* 88.1%

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

      11. metadata-eval 88.1%

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

      12. associate-*r/ 88.2%

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

      13. associate-*r* 88.2%

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

      14. *-commutative 88.2%

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

   9. Simplified 88.2%

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

       1. clear-num 88.1%

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

       2. inv-pow 88.1%

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

       3. *-commutative 88.1%

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

       4. div-inv 88.1%

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

       5. pow-flip 88.1%

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

       6. metadata-eval 88.1%

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

       7. associate-/l* 88.2%

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

   11. Applied egg-rr 88.2%

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

       1. unpow-1 88.2%

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

       2. *-commutative 88.2%

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

       3. associate-*l* 88.0%

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

   13. Simplified 88.0%

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

   14. Taylor expanded in a around 0 88.8%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 12.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

   1. sqr-neg 54.4%

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

   2. sqr-neg 54.4%

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

   3. associate-*l* 54.4%

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

3. Simplified 54.4%

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

   1. add-cube-cbrt 54.4%

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

   2. pow3 54.4%

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

6. Applied egg-rr 54.4%

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

7. Taylor expanded in b around inf 82.6%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right) + -1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
8. Step-by-step derivation

   1. +-commutative 82.6%

      \[\leadsto \frac{\color{blue}{-1.125 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right) + -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   2. fma-define 82.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, {1}^{0.3333333333333333} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}}{3 \cdot a} \]

   3. pow-base-1 82.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{1} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

   4. *-lft-identity 82.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

   5. unpow2 82.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

   6. unpow2 82.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

   7. swap-sqr 82.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

   8. unpow2 82.6%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, -1.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{a \cdot c}{b}\right)\right)}{3 \cdot a} \]

   9. pow-base-1 82.6%

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

   10. associate-*r* 82.6%

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

   11. metadata-eval 82.6%

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

   12. associate-*r/ 82.6%

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

   13. associate-*r* 82.7%

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

   14. *-commutative 82.7%

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

9. Simplified 82.7%

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

    1. clear-num 82.6%

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

    2. inv-pow 82.6%

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

    3. *-commutative 82.6%

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

    4. div-inv 82.6%

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

    5. pow-flip 82.6%

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

    6. metadata-eval 82.6%

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

    7. associate-/l* 82.7%

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

11. Applied egg-rr 82.7%

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

    1. unpow-1 82.7%

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

    2. *-commutative 82.7%

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

    3. associate-*l* 82.5%

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

13. Simplified 82.5%

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

14. Taylor expanded in a around 0 83.4%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]

15. Final simplification 83.4%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]
16. Add Preprocessing

Alternative 10: 64.4% 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 54.4%

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

   1. sqr-neg 54.4%

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

   2. sqr-neg 54.4%

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

   3. associate-*l* 54.4%

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

3. Simplified 54.4%

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

5. Taylor expanded in b around inf 65.4%

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

   1. *-commutative 65.4%

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

   2. associate-*l/ 65.4%

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

7. Simplified 65.4%

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

8. Final simplification 65.4%

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

Alternative 11: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. sqr-neg 54.4%

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

   2. sqr-neg 54.4%

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

   3. associate-*l* 54.4%

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

3. Simplified 54.4%

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

   1. add-cube-cbrt 54.4%

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

   2. pow3 54.4%

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

6. Applied egg-rr 54.4%

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

   1. add-cube-cbrt 54.4%

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

   2. pow3 54.4%

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

8. Applied egg-rr 54.4%

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

9. Taylor expanded in c around 0 3.2%

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

    1. pow-base-1 3.2%

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

    2. associate-*r* 3.2%

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

    3. metadata-eval 3.2%

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

    4. associate-*r/ 3.2%

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

    5. distribute-rgt1-in 3.2%

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

    6. metadata-eval 3.2%

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

    7. mul0-lft 3.2%

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

    8. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024044 
(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)))