Cubic critical, narrow range

Percentage Accurate: 55.7% → 99.6%

Time: 24.3s

Alternatives: 9

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c / (-b - pow(fma(b, b, (-3.0 * (c * a))), 0.5));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

   1. add-cbrt-cube 57.5%

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

   2. pow1/3 57.3%

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

   3. pow3 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 57.3%

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

8. Applied egg-rr 58.7%

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

   1. associate--r- 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

    8. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. *-lft-identity 99.1%

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

    4. associate-/r* 99.1%

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

    5. fma-undefine 99.1%

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

    6. +-inverses 99.1%

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

    7. +-rgt-identity 99.1%

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

14. Simplified 99.2%

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

15. Taylor expanded in a around 0 99.6%

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

    1. pow1/2 99.6%

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

17. Applied egg-rr 99.6%

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

Alternative 2: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.011:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.011)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (/ c (* b (- (* 1.5 (/ (* c a) (pow b 2.0))) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.011) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c / (b * ((1.5 * ((c * a) / pow(b, 2.0))) - 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.011d0)) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = c / (b * ((1.5d0 * ((c * a) / (b ** 2.0d0))) - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.011) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c / (b * ((1.5 * ((c * a) / Math.pow(b, 2.0))) - 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.011:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = c / (b * ((1.5 * ((c * a) / math.pow(b, 2.0))) - 2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.011)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c / Float64(b * Float64(Float64(1.5 * Float64(Float64(c * a) / (b ^ 2.0))) - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.011)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = c / (b * ((1.5 * ((c * a) / (b ^ 2.0))) - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.011], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c / N[(b * N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.010999999999999999

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.9%

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

   3. Simplified 78.9%

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

      1. add-cbrt-cube 78.9%

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

      2. pow1/3 78.5%

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

      3. pow3 78.5%

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

   6. Applied egg-rr 78.5%

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

   7. Taylor expanded in a around 0 78.9%

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

      1. *-commutative 78.9%

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

      2. associate-*l* 78.9%

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

   9. Simplified 78.9%

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

   if -0.010999999999999999 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.4%

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

      1. sqr-neg 47.4%

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

      2. sqr-neg 47.4%

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

      3. associate-*l* 47.4%

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

   3. Simplified 47.4%

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

      1. add-cbrt-cube 47.4%

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

      2. pow1/3 47.3%

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

      3. pow3 47.3%

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

   6. Applied egg-rr 47.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. flip-+ 47.2%

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

   8. Applied egg-rr 48.4%

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

      1. associate--r- 99.1%

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

      2. associate-*l* 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

       1. div-inv 99.1%

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

       2. +-commutative 99.1%

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

       3. fma-define 99.1%

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

       4. neg-mul-1 99.1%

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

       5. unpow-prod-down 99.1%

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

       6. metadata-eval 99.1%

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

       7. *-un-lft-identity 99.1%

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

       8. *-commutative 99.1%

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

   12. Applied egg-rr 99.1%

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

       1. *-commutative 99.1%

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

       2. times-frac 99.1%

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

       3. *-lft-identity 99.1%

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

       4. associate-/r* 99.2%

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

       5. fma-undefine 99.2%

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

       6. +-inverses 99.2%

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

       7. +-rgt-identity 99.2%

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

   14. Simplified 99.2%

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

   15. Taylor expanded in a around 0 99.7%

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

   16. Taylor expanded in b around inf 90.0%

       \[\leadsto \frac{c}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.011:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

   1. add-cbrt-cube 57.5%

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

   2. pow1/3 57.3%

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

   3. pow3 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 57.3%

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

8. Applied egg-rr 58.7%

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

   1. associate--r- 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

    8. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. *-lft-identity 99.1%

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

    4. associate-/r* 99.1%

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

    5. fma-undefine 99.1%

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

    6. +-inverses 99.1%

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

    7. +-rgt-identity 99.1%

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

14. Simplified 99.2%

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

15. Taylor expanded in a around 0 99.6%

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

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ c (* b (- (* 1.5 (/ (* c a) (pow b 2.0))) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = c / (b * ((1.5 * ((c * a) / pow(b, 2.0))) - 2.0));
	}
	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 <= 5.2d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = c / (b * ((1.5d0 * ((c * a) / (b ** 2.0d0))) - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = c / (b * ((1.5 * ((c * a) / Math.pow(b, 2.0))) - 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = c / (b * ((1.5 * ((c * a) / math.pow(b, 2.0))) - 2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c / Float64(b * Float64(Float64(1.5 * Float64(Float64(c * a) / (b ^ 2.0))) - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.2)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = c / (b * ((1.5 * ((c * a) / (b ^ 2.0))) - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c / N[(b * N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.20000000000000018

   1. Initial program 80.6%

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

      1. sqr-neg 80.6%

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

      2. sqr-neg 80.6%

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

      3. associate-*l* 80.6%

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

   3. Simplified 80.6%

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

   if 5.20000000000000018 < b

   1. Initial program 51.9%

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

      1. sqr-neg 51.9%

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

      2. sqr-neg 51.9%

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

      3. associate-*l* 51.9%

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

   3. Simplified 51.9%

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

      1. add-cbrt-cube 51.8%

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

      2. pow1/3 51.8%

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

      3. pow3 51.8%

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

   6. Applied egg-rr 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. flip-+ 51.7%

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

   8. Applied egg-rr 53.0%

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

      1. associate--r- 99.1%

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

      2. associate-*l* 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

       1. div-inv 99.1%

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

       2. +-commutative 99.1%

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

       3. fma-define 99.1%

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

       4. neg-mul-1 99.1%

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

       5. unpow-prod-down 99.1%

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

       6. metadata-eval 99.1%

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

       7. *-un-lft-identity 99.1%

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

       8. *-commutative 99.1%

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

   12. Applied egg-rr 99.1%

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

       1. *-commutative 99.1%

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

       2. times-frac 99.1%

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

       3. *-lft-identity 99.1%

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

       4. associate-/r* 99.2%

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

       5. fma-undefine 99.2%

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

       6. +-inverses 99.2%

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

       7. +-rgt-identity 99.2%

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

   14. Simplified 99.2%

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

   15. Taylor expanded in a around 0 99.6%

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

   16. Taylor expanded in b around inf 86.1%

       \[\leadsto \frac{c}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return c / (b * ((1.5 * ((c * a) / pow(b, 2.0))) - 2.0));
}

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 / (b * ((1.5d0 * ((c * a) / (b ** 2.0d0))) - 2.0d0))
end function

Java

public static double code(double a, double b, double c) {
	return c / (b * ((1.5 * ((c * a) / Math.pow(b, 2.0))) - 2.0));
}

Python

def code(a, b, c):
	return c / (b * ((1.5 * ((c * a) / math.pow(b, 2.0))) - 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

   1. add-cbrt-cube 57.5%

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

   2. pow1/3 57.3%

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

   3. pow3 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 57.3%

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

8. Applied egg-rr 58.7%

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

   1. associate--r- 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

    8. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. *-lft-identity 99.1%

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

    4. associate-/r* 99.1%

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

    5. fma-undefine 99.1%

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

    6. +-inverses 99.1%

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

    7. +-rgt-identity 99.1%

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

14. Simplified 99.2%

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

15. Taylor expanded in a around 0 99.6%

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

16. Taylor expanded in b around inf 81.6%

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

17. Final simplification 81.6%

    \[\leadsto \frac{c}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)} \]
18. Add Preprocessing

Alternative 6: 82.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

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 / ((1.5d0 * ((c * a) / b)) - (b * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

   1. add-cbrt-cube 57.5%

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

   2. pow1/3 57.3%

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

   3. pow3 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 57.3%

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

8. Applied egg-rr 58.7%

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

   1. associate--r- 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

    8. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. *-lft-identity 99.1%

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

    4. associate-/r* 99.1%

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

    5. fma-undefine 99.1%

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

    6. +-inverses 99.1%

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

    7. +-rgt-identity 99.1%

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

14. Simplified 99.2%

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

15. Taylor expanded in a around 0 99.6%

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

16. Taylor expanded in c around 0 81.6%

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

17. Final simplification 81.6%

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

Alternative 7: 64.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in b around inf 62.9%

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

   1. associate-*r/ 62.9%

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

   2. *-commutative 62.9%

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

7. Simplified 62.9%

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

Alternative 8: 64.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in b around inf 62.7%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate-/l* 62.7%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

   2. associate-/l* 62.8%

      \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

   3. *-commutative 62.8%

      \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

7. Applied egg-rr 62.8%

   \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
8. Step-by-step derivation

   1. times-frac 62.8%

      \[\leadsto -1.5 \cdot \color{blue}{\left(\frac{a}{a} \cdot \frac{\frac{c}{b}}{3}\right)} \]

   2. *-inverses 62.8%

      \[\leadsto -1.5 \cdot \left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right) \]

   3. associate-*r* 62.8%

      \[\leadsto \color{blue}{\left(-1.5 \cdot 1\right) \cdot \frac{\frac{c}{b}}{3}} \]

   4. metadata-eval 62.8%

      \[\leadsto \color{blue}{-1.5} \cdot \frac{\frac{c}{b}}{3} \]

9. Simplified 62.8%

   \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{c}{b}}{3}} \]
10. Step-by-step derivation

    1. associate-*r/ 62.8%

       \[\leadsto \color{blue}{\frac{-1.5 \cdot \frac{c}{b}}{3}} \]

11. Applied egg-rr 62.8%

    \[\leadsto \color{blue}{\frac{-1.5 \cdot \frac{c}{b}}{3}} \]
12. Step-by-step derivation

    1. associate-/l* 62.8%

       \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{c}{b}}{3}} \]

13. Applied egg-rr 62.8%

    \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{c}{b}}{3}} \]
14. Step-by-step derivation

    1. associate-/l/ 62.8%

       \[\leadsto -1.5 \cdot \color{blue}{\frac{c}{3 \cdot b}} \]

15. Simplified 62.8%

    \[\leadsto \color{blue}{-1.5 \cdot \frac{c}{3 \cdot b}} \]
16. Step-by-step derivation

    1. clear-num 62.8%

       \[\leadsto -1.5 \cdot \color{blue}{\frac{1}{\frac{3 \cdot b}{c}}} \]

    2. un-div-inv 62.8%

       \[\leadsto \color{blue}{\frac{-1.5}{\frac{3 \cdot b}{c}}} \]

    3. *-un-lft-identity 62.8%

       \[\leadsto \frac{-1.5}{\frac{3 \cdot b}{\color{blue}{1 \cdot c}}} \]

    4. times-frac 62.8%

       \[\leadsto \frac{-1.5}{\color{blue}{\frac{3}{1} \cdot \frac{b}{c}}} \]

    5. metadata-eval 62.8%

       \[\leadsto \frac{-1.5}{\color{blue}{3} \cdot \frac{b}{c}} \]

17. Applied egg-rr 62.8%

    \[\leadsto \color{blue}{\frac{-1.5}{3 \cdot \frac{b}{c}}} \]
18. Step-by-step derivation

    1. associate-/r* 62.8%

       \[\leadsto \color{blue}{\frac{\frac{-1.5}{3}}{\frac{b}{c}}} \]

    2. metadata-eval 62.8%

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

19. Simplified 62.8%

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

Alternative 9: 64.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. sqr-neg 57.5%

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

   2. sqr-neg 57.5%

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

   3. associate-*l* 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in b around inf 62.7%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate-/l* 62.7%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

   2. associate-/l* 62.8%

      \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

   3. *-commutative 62.8%

      \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

7. Applied egg-rr 62.8%

   \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
8. Step-by-step derivation

   1. times-frac 62.8%

      \[\leadsto -1.5 \cdot \color{blue}{\left(\frac{a}{a} \cdot \frac{\frac{c}{b}}{3}\right)} \]

   2. *-inverses 62.8%

      \[\leadsto -1.5 \cdot \left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right) \]

   3. associate-*r* 62.8%

      \[\leadsto \color{blue}{\left(-1.5 \cdot 1\right) \cdot \frac{\frac{c}{b}}{3}} \]

   4. metadata-eval 62.8%

      \[\leadsto \color{blue}{-1.5} \cdot \frac{\frac{c}{b}}{3} \]

9. Simplified 62.8%

   \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{c}{b}}{3}} \]
10. Step-by-step derivation

    1. associate-*r/ 62.8%

       \[\leadsto \color{blue}{\frac{-1.5 \cdot \frac{c}{b}}{3}} \]

11. Applied egg-rr 62.8%

    \[\leadsto \color{blue}{\frac{-1.5 \cdot \frac{c}{b}}{3}} \]
12. Step-by-step derivation

    1. associate-/l* 62.8%

       \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{c}{b}}{3}} \]

13. Applied egg-rr 62.8%

    \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{c}{b}}{3}} \]
14. Step-by-step derivation

    1. associate-/l/ 62.8%

       \[\leadsto -1.5 \cdot \color{blue}{\frac{c}{3 \cdot b}} \]

15. Simplified 62.8%

    \[\leadsto \color{blue}{-1.5 \cdot \frac{c}{3 \cdot b}} \]
16. Step-by-step derivation

    1. clear-num 62.8%

       \[\leadsto -1.5 \cdot \color{blue}{\frac{1}{\frac{3 \cdot b}{c}}} \]

    2. un-div-inv 62.8%

       \[\leadsto \color{blue}{\frac{-1.5}{\frac{3 \cdot b}{c}}} \]

    3. *-un-lft-identity 62.8%

       \[\leadsto \frac{-1.5}{\frac{3 \cdot b}{\color{blue}{1 \cdot c}}} \]

    4. times-frac 62.8%

       \[\leadsto \frac{-1.5}{\color{blue}{\frac{3}{1} \cdot \frac{b}{c}}} \]

    5. metadata-eval 62.8%

       \[\leadsto \frac{-1.5}{\color{blue}{3} \cdot \frac{b}{c}} \]

17. Applied egg-rr 62.8%

    \[\leadsto \color{blue}{\frac{-1.5}{3 \cdot \frac{b}{c}}} \]
18. Step-by-step derivation

    1. associate-/r* 62.8%

       \[\leadsto \color{blue}{\frac{\frac{-1.5}{3}}{\frac{b}{c}}} \]

    2. metadata-eval 62.8%

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

19. Simplified 62.8%

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

    1. associate-/r/ 62.8%

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

21. Applied egg-rr 62.8%

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

22. Final simplification 62.8%

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

Reproduce

herbie shell --seed 2024105 
(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)))