Cubic critical, medium range

Percentage Accurate: 31.2% → 99.7%

Time: 14.5s

Alternatives: 7

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.2% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c / (-b - sqrt((pow(b, 2.0) - (c * (a * 3.0)))));
}

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 - sqrt(((b ** 2.0d0) - (c * (a * 3.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. add-cbrt-cube 38.4%

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

   2. pow3 38.4%

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

4. Applied egg-rr 38.4%

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

   1. flip-+ 38.6%

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

   2. pow2 38.6%

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

   3. add-sqr-sqrt 39.6%

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

   4. pow2 39.6%

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

   5. *-commutative 39.6%

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

   6. *-commutative 39.6%

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

   7. pow2 39.6%

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

   8. *-commutative 39.6%

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

   9. *-commutative 39.6%

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

6. Applied egg-rr 39.6%

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

   1. associate--r- 98.9%

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

8. Simplified 98.9%

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

   1. rem-cbrt-cube 99.4%

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

   2. div-inv 99.3%

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

   3. +-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. fma-define 99.3%

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

   6. *-commutative 99.3%

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

   7. neg-mul-1 99.3%

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

   8. unpow-prod-down 99.3%

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

   9. metadata-eval 99.3%

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

   10. *-un-lft-identity 99.3%

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

   11. *-commutative 99.3%

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

10. Applied egg-rr 99.3%

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

    1. associate-*l/ 99.4%

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

    2. +-inverses 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in c around 0 99.7%

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

Alternative 2: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0035:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0035)
     t_0
     (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* c a) b))) c)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0035) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.0035d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0035) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.0035:
		tmp = t_0
	else:
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.0035)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(c * a) / b))) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.0035)
		tmp = t_0;
	else
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -0.0035], t$95$0, N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.00350000000000000007

   1. Initial program 74.6%

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

   if -0.00350000000000000007 < (/.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 28.0%

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

      1. add-cbrt-cube 28.0%

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

      2. pow3 28.0%

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

   4. Applied egg-rr 28.0%

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

      1. flip-+ 28.2%

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

      2. pow2 28.2%

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

      3. add-sqr-sqrt 29.0%

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

      4. pow2 29.0%

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

      5. *-commutative 29.0%

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

      6. *-commutative 29.0%

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

      7. pow2 29.0%

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

      8. *-commutative 29.0%

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

      9. *-commutative 29.0%

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

   6. Applied egg-rr 29.0%

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

      1. associate--r- 98.9%

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

   8. Simplified 98.9%

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

      1. rem-cbrt-cube 99.4%

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

      2. clear-num 99.3%

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

      3. inv-pow 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. unpow-1 99.3%

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

       2. associate-/r/ 99.3%

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

       3. fma-undefine 99.3%

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

       4. +-inverses 99.3%

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

       5. +-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

   13. Taylor expanded in c around 0 92.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (+
   (* -2.0 (/ b c))
   (* a (+ (* 1.125 (/ (* c a) (pow b 3.0))) (* 1.5 (/ 1.0 b)))))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / pow(b, 3.0))) + (1.5 * (1.0 / 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)) + (a * ((1.125d0 * ((c * a) / (b ** 3.0d0))) + (1.5d0 * (1.0d0 / b)))))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / Math.pow(b, 3.0))) + (1.5 * (1.0 / b)))));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / math.pow(b, 3.0))) + (1.5 * (1.0 / b)))))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(a * Float64(Float64(1.125 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(1.5 * Float64(1.0 / b))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / (b ^ 3.0))) + (1.5 * (1.0 / b)))));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(1.125 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. add-cbrt-cube 38.4%

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

   2. pow3 38.4%

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

4. Applied egg-rr 38.4%

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

   1. flip-+ 38.6%

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

   2. pow2 38.6%

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

   3. add-sqr-sqrt 39.6%

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

   4. pow2 39.6%

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

   5. *-commutative 39.6%

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

   6. *-commutative 39.6%

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

   7. pow2 39.6%

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

   8. *-commutative 39.6%

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

   9. *-commutative 39.6%

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

6. Applied egg-rr 39.6%

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

   1. associate--r- 98.9%

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

8. Simplified 98.9%

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

   1. rem-cbrt-cube 99.4%

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

   2. clear-num 99.3%

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

   3. inv-pow 99.3%

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

10. Applied egg-rr 99.3%

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

    1. unpow-1 99.3%

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

    2. associate-/r/ 99.2%

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

    3. fma-undefine 99.2%

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

    4. +-inverses 99.2%

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

    5. +-rgt-identity 99.2%

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

12. Simplified 99.2%

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

13. Taylor expanded in a around 0 91.2%

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

14. Final simplification 91.2%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)} \]
15. Add Preprocessing

Alternative 4: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7e-6)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-6) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} 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 <= 3.7d-6) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    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 <= 3.7e-6) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.7e-6:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	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 <= 3.7e-6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	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 <= 3.7e-6)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	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, 3.7e-6], 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[(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 < 3.7000000000000002e-6

   1. Initial program 80.5%

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

      1. 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 3.7000000000000002e-6 < b

   1. Initial program 35.0%

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

      1. add-cbrt-cube 35.0%

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

      2. pow3 35.0%

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

   4. Applied egg-rr 35.0%

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

      1. flip-+ 35.2%

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

      2. pow2 35.2%

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

      3. add-sqr-sqrt 36.2%

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

      4. pow2 36.2%

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

      5. *-commutative 36.2%

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

      6. *-commutative 36.2%

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

      7. pow2 36.2%

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

      8. *-commutative 36.2%

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

      9. *-commutative 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. associate--r- 98.9%

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

   8. Simplified 98.9%

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

      1. rem-cbrt-cube 99.4%

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

      2. clear-num 99.2%

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

      3. inv-pow 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. unpow-1 99.2%

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

       2. associate-/r/ 99.2%

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

       3. fma-undefine 99.2%

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

       4. +-inverses 99.2%

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

       5. +-rgt-identity 99.2%

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

   12. Simplified 99.2%

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

   13. Taylor expanded in a around 0 89.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% 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 38.4%

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

   1. add-cbrt-cube 38.4%

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

   2. pow3 38.4%

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

4. Applied egg-rr 38.4%

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

   1. flip-+ 38.6%

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

   2. pow2 38.6%

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

   3. add-sqr-sqrt 39.6%

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

   4. pow2 39.6%

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

   5. *-commutative 39.6%

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

   6. *-commutative 39.6%

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

   7. pow2 39.6%

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

   8. *-commutative 39.6%

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

   9. *-commutative 39.6%

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

6. Applied egg-rr 39.6%

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

   1. associate--r- 98.9%

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

8. Simplified 98.9%

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

   1. rem-cbrt-cube 99.4%

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

   2. clear-num 99.3%

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

   3. inv-pow 99.3%

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

10. Applied egg-rr 99.3%

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

    1. unpow-1 99.3%

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

    2. associate-/r/ 99.2%

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

    3. fma-undefine 99.2%

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

    4. +-inverses 99.2%

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

    5. +-rgt-identity 99.2%

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

12. Simplified 99.2%

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

13. Taylor expanded in a around 0 86.8%

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

Alternative 6: 81.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

3. Simplified 38.4%

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

5. Taylor expanded in b around inf 75.8%

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

Alternative 7: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 38.4%

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

3. Simplified 38.4%

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

5. Taylor expanded in c around inf 38.4%

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

   1. *-un-lft-identity 38.4%

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

   2. add-sqr-sqrt 38.2%

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

   3. prod-diff 39.1%

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

   4. add-sqr-sqrt 39.5%

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

   5. fmm-def 39.5%

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

   6. *-un-lft-identity 39.5%

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

   7. *-commutative 39.5%

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

   8. fma-define 39.5%

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

   9. add-sqr-sqrt 39.1%

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

7. Applied egg-rr 39.1%

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

8. Taylor expanded in c around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

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

11. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))