Quadratic roots, narrow range

Percentage Accurate: 56.3% → 99.6%

Time: 13.5s

Alternatives: 10

Speedup: 29.0×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot 2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c 2.0) (- (- b) (sqrt (fma -4.0 (* c a) (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return (c * 2.0) / (-b - sqrt(fma(-4.0, (c * a), pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(fma(-4.0, Float64(c * a), (b ^ 2.0)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 56.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 54.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 56.6%

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

8. Simplified 56.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. flip-+ 56.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow2 56.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. add-sqr-sqrt 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. pow1/3 54.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. metadata-eval 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. associate-*l* 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. pow1/3 58.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   9. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   10. metadata-eval 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   11. associate-*l* 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

10. Applied egg-rr 58.3%

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

    1. *-un-lft-identity 58.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

    2. associate-/l/ 58.3%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

    3. neg-mul-1 58.3%

       \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    4. unpow-prod-down 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    5. metadata-eval 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    6. *-un-lft-identity 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    7. cancel-sign-sub-inv 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    8. unpow2 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    9. fma-define 57.8%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    10. metadata-eval 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    11. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    12. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

12. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

    2. times-frac 57.8%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

    3. *-commutative 57.8%

       \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

    4. associate-*l/ 57.8%

       \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

14. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]

15. Taylor expanded in c around 0 99.5%

    \[\leadsto \frac{\color{blue}{2 \cdot c}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b} \]
16. Step-by-step derivation

    1. *-commutative 99.5%

       \[\leadsto \frac{\color{blue}{c \cdot 2}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b} \]

17. Simplified 99.5%

    \[\leadsto \frac{\color{blue}{c \cdot 2}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b} \]

18. Final simplification 99.5%

    \[\leadsto \frac{c \cdot 2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}} \]
19. Add Preprocessing

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{2 \cdot a} \leq -0.021:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot t\_0}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* 2.0 a)) -0.021)
     (* (- (sqrt (fma b b (* -4.0 (* c a)))) b) (/ 1.0 (* 2.0 a)))
     (/ (* (/ 0.5 a) t_0) (* 2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (2.0 * a)) <= -0.021) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) - b) * (1.0 / (2.0 * a));
	} else {
		tmp = ((0.5 / a) * t_0) / (2.0 * ((a * (c / b)) - b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(2.0 * a)) <= -0.021)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) - b) * Float64(1.0 / Float64(2.0 * a)));
	else
		tmp = Float64(Float64(Float64(0.5 / a) * t_0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.021], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0210000000000000013

   1. Initial program 81.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 76.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 76.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 76.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 76.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 76.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 76.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 80.6%

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

   8. Simplified 80.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. div-inv 80.5%

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

      2. neg-mul-1 80.5%

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

      3. fma-define 80.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{a \cdot 2} \]

      4. pow1/3 76.2%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{a \cdot 2} \]

      5. pow-pow 81.1%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{a \cdot 2} \]

      6. metadata-eval 81.1%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{a \cdot 2} \]

      7. associate-*l* 81.1%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{1}{a \cdot 2} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}} \]
   11. Step-by-step derivation

       1. fma-undefine 81.1%

          \[\leadsto \color{blue}{\left(-1 \cdot b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \cdot \frac{1}{a \cdot 2} \]

       2. cancel-sign-sub-inv 81.1%

          \[\leadsto \left(-1 \cdot b + \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right) \cdot \frac{1}{a \cdot 2} \]

       3. unpow2 81.1%

          \[\leadsto \left(-1 \cdot b + \sqrt{\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2} \]

       4. metadata-eval 81.1%

          \[\leadsto \left(-1 \cdot b + \sqrt{b \cdot b + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2} \]

       5. *-commutative 81.1%

          \[\leadsto \left(-1 \cdot b + \sqrt{b \cdot b + -4 \cdot \color{blue}{\left(c \cdot a\right)}}\right) \cdot \frac{1}{a \cdot 2} \]

       6. fma-define 81.3%

          \[\leadsto \left(-1 \cdot b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right) \cdot \frac{1}{a \cdot 2} \]

   12. Applied egg-rr 81.3%

       \[\leadsto \color{blue}{\left(-1 \cdot b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \cdot \frac{1}{a \cdot 2} \]

   if -0.0210000000000000013 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 45.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 44.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 45.8%

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

   8. Simplified 45.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 45.6%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 45.6%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 46.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 44.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. associate-*l* 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. pow1/3 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow-pow 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. metadata-eval 47.5%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. associate-*l* 47.5%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 47.5%

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

       1. *-un-lft-identity 47.5%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

       2. associate-/l/ 47.5%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

       3. neg-mul-1 47.5%

          \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       4. unpow-prod-down 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       5. metadata-eval 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       6. *-un-lft-identity 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       7. cancel-sign-sub-inv 47.5%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       8. unpow2 47.5%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       9. fma-define 47.0%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       10. metadata-eval 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       11. *-commutative 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       12. *-commutative 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

   12. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 47.0%

          \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

       2. times-frac 47.0%

          \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

       4. associate-*l/ 47.0%

          \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]

   15. Taylor expanded in c around 0 88.5%

       \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
   16. Step-by-step derivation

       1. distribute-lft-out-- 88.5%

          \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}} \]

       2. associate-/l* 88.5%

          \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)} \]

   17. Simplified 88.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.021:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{2 \cdot a} \leq -0.021:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot t\_0}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* 2.0 a)) -0.021)
     (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* 2.0 a))
     (/ (* (/ 0.5 a) t_0) (* 2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (2.0 * a)) <= -0.021) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (2.0 * a);
	} else {
		tmp = ((0.5 / a) * t_0) / (2.0 * ((a * (c / b)) - b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(2.0 * a)) <= -0.021)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(0.5 / a) * t_0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.021], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0210000000000000013

   1. Initial program 81.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   if -0.0210000000000000013 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 45.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 44.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 45.8%

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

   8. Simplified 45.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 45.6%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 45.6%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 46.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 44.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. associate-*l* 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. pow1/3 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow-pow 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. metadata-eval 47.5%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. associate-*l* 47.5%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 47.5%

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

       1. *-un-lft-identity 47.5%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

       2. associate-/l/ 47.5%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

       3. neg-mul-1 47.5%

          \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       4. unpow-prod-down 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       5. metadata-eval 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       6. *-un-lft-identity 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       7. cancel-sign-sub-inv 47.5%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       8. unpow2 47.5%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       9. fma-define 47.0%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       10. metadata-eval 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       11. *-commutative 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       12. *-commutative 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

   12. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 47.0%

          \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

       2. times-frac 47.0%

          \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

       4. associate-*l/ 47.0%

          \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]

   15. Taylor expanded in c around 0 88.5%

       \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
   16. Step-by-step derivation

       1. distribute-lft-out-- 88.5%

          \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}} \]

       2. associate-/l* 88.5%

          \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)} \]

   17. Simplified 88.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.021:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ t_1 := \frac{\sqrt{b \cdot b - t\_0} - b}{2 \cdot a}\\ \mathbf{if}\;t\_1 \leq -0.021:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot t\_0}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))) (t_1 (/ (- (sqrt (- (* b b) t_0)) b) (* 2.0 a))))
   (if (<= t_1 -0.021) t_1 (/ (* (/ 0.5 a) t_0) (* 2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double t_1 = (sqrt(((b * b) - t_0)) - b) / (2.0 * a);
	double tmp;
	if (t_1 <= -0.021) {
		tmp = t_1;
	} else {
		tmp = ((0.5 / a) * t_0) / (2.0 * ((a * (c / b)) - b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (a * 4.0d0)
    t_1 = (sqrt(((b * b) - t_0)) - b) / (2.0d0 * a)
    if (t_1 <= (-0.021d0)) then
        tmp = t_1
    else
        tmp = ((0.5d0 / a) * t_0) / (2.0d0 * ((a * (c / b)) - b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double t_1 = (Math.sqrt(((b * b) - t_0)) - b) / (2.0 * a);
	double tmp;
	if (t_1 <= -0.021) {
		tmp = t_1;
	} else {
		tmp = ((0.5 / a) * t_0) / (2.0 * ((a * (c / b)) - b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 4.0)
	t_1 = (math.sqrt(((b * b) - t_0)) - b) / (2.0 * a)
	tmp = 0
	if t_1 <= -0.021:
		tmp = t_1
	else:
		tmp = ((0.5 / a) * t_0) / (2.0 * ((a * (c / b)) - b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_1 <= -0.021)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(0.5 / a) * t_0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 4.0);
	t_1 = (sqrt(((b * b) - t_0)) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_1 <= -0.021)
		tmp = t_1;
	else
		tmp = ((0.5 / a) * t_0) / (2.0 * ((a * (c / b)) - b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.021], t$95$1, N[(N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0210000000000000013

   1. Initial program 81.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -0.0210000000000000013 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 45.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 44.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 44.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 45.8%

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

   8. Simplified 45.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 45.6%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 45.6%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 46.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 44.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. associate-*l* 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. pow1/3 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow-pow 47.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. metadata-eval 47.5%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. associate-*l* 47.5%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 47.5%

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

       1. *-un-lft-identity 47.5%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

       2. associate-/l/ 47.5%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

       3. neg-mul-1 47.5%

          \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       4. unpow-prod-down 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       5. metadata-eval 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       6. *-un-lft-identity 47.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       7. cancel-sign-sub-inv 47.5%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       8. unpow2 47.5%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       9. fma-define 47.0%

          \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       10. metadata-eval 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       11. *-commutative 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

       12. *-commutative 47.0%

           \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

   12. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 47.0%

          \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

       2. times-frac 47.0%

          \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

       3. *-commutative 47.0%

          \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

       4. associate-*l/ 47.0%

          \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]

   15. Taylor expanded in c around 0 88.5%

       \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
   16. Step-by-step derivation

       1. distribute-lft-out-- 88.5%

          \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}} \]

       2. associate-/l* 88.5%

          \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)} \]

   17. Simplified 88.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.021:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (/ 0.5 a) (* c (* a 4.0)))
  (- (- b) (sqrt (* a (+ (* c -4.0) (/ (pow b 2.0) a)))))))

C

double code(double a, double b, double c) {
	return ((0.5 / a) * (c * (a * 4.0))) / (-b - sqrt((a * ((c * -4.0) + (pow(b, 2.0) / a)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((0.5d0 / a) * (c * (a * 4.0d0))) / (-b - sqrt((a * ((c * (-4.0d0)) + ((b ** 2.0d0) / a)))))
end function

Java

public static double code(double a, double b, double c) {
	return ((0.5 / a) * (c * (a * 4.0))) / (-b - Math.sqrt((a * ((c * -4.0) + (Math.pow(b, 2.0) / a)))));
}

Python

def code(a, b, c):
	return ((0.5 / a) * (c * (a * 4.0))) / (-b - math.sqrt((a * ((c * -4.0) + (math.pow(b, 2.0) / a)))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.5 / a) * Float64(c * Float64(a * 4.0))) / Float64(Float64(-b) - sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64((b ^ 2.0) / a))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((0.5 / a) * (c * (a * 4.0))) / (-b - sqrt((a * ((c * -4.0) + ((b ^ 2.0) / a)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.5 / a), $MachinePrecision] * N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 56.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 54.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 56.6%

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

8. Simplified 56.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. flip-+ 56.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow2 56.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. add-sqr-sqrt 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. pow1/3 54.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. metadata-eval 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. associate-*l* 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. pow1/3 58.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   9. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   10. metadata-eval 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   11. associate-*l* 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

10. Applied egg-rr 58.3%

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

    1. *-un-lft-identity 58.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

    2. associate-/l/ 58.3%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

    3. neg-mul-1 58.3%

       \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    4. unpow-prod-down 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    5. metadata-eval 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    6. *-un-lft-identity 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    7. cancel-sign-sub-inv 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    8. unpow2 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    9. fma-define 57.8%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    10. metadata-eval 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    11. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    12. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

12. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

    2. times-frac 57.8%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

    3. *-commutative 57.8%

       \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

    4. associate-*l/ 57.8%

       \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

14. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]

15. Taylor expanded in a around inf 99.3%

    \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\color{blue}{a \cdot \left(-4 \cdot c + \frac{{b}^{2}}{a}\right)}}\right) - b} \]

16. Final simplification 99.3%

    \[\leadsto \frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)}} \]
17. Add Preprocessing

Alternative 6: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (/ 0.5 a) (* c (* a 4.0))) (- (- b) (sqrt (fma -4.0 (* c a) (* b b))))))

C

double code(double a, double b, double c) {
	return ((0.5 / a) * (c * (a * 4.0))) / (-b - sqrt(fma(-4.0, (c * a), (b * b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.5 / a) * Float64(c * Float64(a * 4.0))) / Float64(Float64(-b) - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.5 / a), $MachinePrecision] * N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 56.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 54.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 56.6%

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

8. Simplified 56.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. flip-+ 56.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow2 56.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. add-sqr-sqrt 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. pow1/3 54.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. metadata-eval 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. associate-*l* 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. pow1/3 58.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   9. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   10. metadata-eval 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   11. associate-*l* 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

10. Applied egg-rr 58.3%

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

    1. *-un-lft-identity 58.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

    2. associate-/l/ 58.3%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

    3. neg-mul-1 58.3%

       \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    4. unpow-prod-down 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    5. metadata-eval 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    6. *-un-lft-identity 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    7. cancel-sign-sub-inv 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    8. unpow2 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    9. fma-define 57.8%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    10. metadata-eval 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    11. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    12. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

12. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

    2. times-frac 57.8%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

    3. *-commutative 57.8%

       \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

    4. associate-*l/ 57.8%

       \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

14. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]
15. Step-by-step derivation

    1. unpow2 99.3%

       \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, \color{blue}{b \cdot b}\right)}\right) - b} \]

16. Applied egg-rr 99.3%

    \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, \color{blue}{b \cdot b}\right)}\right) - b} \]

17. Final simplification 99.3%

    \[\leadsto \frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]
18. Add Preprocessing

Alternative 7: 81.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (/ 0.5 a) (* c (* a 4.0))) (* 2.0 (- (* a (/ c b)) b))))

C

double code(double a, double b, double c) {
	return ((0.5 / a) * (c * (a * 4.0))) / (2.0 * ((a * (c / b)) - b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((0.5d0 / a) * (c * (a * 4.0d0))) / (2.0d0 * ((a * (c / b)) - b))
end function

Java

public static double code(double a, double b, double c) {
	return ((0.5 / a) * (c * (a * 4.0))) / (2.0 * ((a * (c / b)) - b));
}

Python

def code(a, b, c):
	return ((0.5 / a) * (c * (a * 4.0))) / (2.0 * ((a * (c / b)) - b))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.5 / a) * Float64(c * Float64(a * 4.0))) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((0.5 / a) * (c * (a * 4.0))) / (2.0 * ((a * (c / b)) - b));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.5 / a), $MachinePrecision] * N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 56.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 54.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 56.6%

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

8. Simplified 56.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. flip-+ 56.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow2 56.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. add-sqr-sqrt 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. pow1/3 54.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. metadata-eval 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. associate-*l* 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. pow1/3 58.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   9. pow-pow 58.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   10. metadata-eval 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   11. associate-*l* 58.3%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

10. Applied egg-rr 58.3%

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

    1. *-un-lft-identity 58.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

    2. associate-/l/ 58.3%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

    3. neg-mul-1 58.3%

       \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    4. unpow-prod-down 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    5. metadata-eval 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    6. *-un-lft-identity 58.3%

       \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    7. cancel-sign-sub-inv 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    8. unpow2 58.3%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    9. fma-define 57.8%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    10. metadata-eval 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    11. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    12. *-commutative 57.8%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

12. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

    2. times-frac 57.8%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

    3. *-commutative 57.8%

       \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

    4. associate-*l/ 57.8%

       \[\leadsto \color{blue}{\frac{\left({b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]

14. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right) - b}} \]

15. Taylor expanded in c around 0 80.3%

    \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
16. Step-by-step derivation

    1. distribute-lft-out-- 80.3%

       \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}} \]

    2. associate-/l* 80.3%

       \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)} \]

17. Simplified 80.3%

    \[\leadsto \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}} \]

18. Final simplification 80.3%

    \[\leadsto \frac{\frac{0.5}{a} \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)} \]
19. Add Preprocessing

Alternative 8: 80.9% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 79.6%

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

   1. distribute-lft-out 79.6%

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

   2. associate-/l* 79.6%

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

7. Simplified 79.6%

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

8. Taylor expanded in b around inf 79.7%

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

   1. neg-mul-1 79.7%

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

   2. +-commutative 79.7%

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

   3. unsub-neg 79.7%

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

   4. mul-1-neg 79.7%

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

   5. associate-/l* 79.7%

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

   6. distribute-rgt-neg-in 79.7%

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

   7. unpow2 79.7%

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

   8. unpow2 79.7%

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

   9. times-frac 79.7%

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

   10. unpow2 79.7%

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

10. Simplified 79.7%

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

    1. unpow2 79.7%

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

12. Applied egg-rr 79.7%

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

13. Final simplification 79.7%

    \[\leadsto \frac{\left(\frac{c}{b} \cdot \frac{c}{b}\right) \cdot \left(-a\right) - c}{b} \]
14. Add Preprocessing

Alternative 9: 63.7% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 63.1%

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

   1. associate-*r/ 63.1%

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

   2. mul-1-neg 63.1%

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

7. Simplified 63.1%

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

8. Final simplification 63.1%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 56.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 54.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 54.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 56.6%

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

8. Simplified 56.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. div-inv 56.5%

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

   2. neg-mul-1 56.5%

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

   3. fma-define 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{a \cdot 2} \]

   4. pow1/3 54.0%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{a \cdot 2} \]

   5. pow-pow 57.0%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{a \cdot 2} \]

   6. metadata-eval 57.0%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{a \cdot 2} \]

   7. associate-*l* 57.0%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{1}{a \cdot 2} \]

10. Applied egg-rr 57.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}} \]

11. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
12. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

13. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))